On the asymptotic geometry of abelian-by-cyclic groups

Benson Farb^{∗} and Lee Mosher^{†}

September 19, 1999

1 Introduction

Gromov’s Polynomial Growth Theorem [Gro81] states that the property of having polynomial growth characterizes virtually nilpotent groups among all finitely generated groups.

Gromov’s theorem inspired the more general problem (see, e.g. [GdlH91]) of understanding to what extent the asymptotic geometry of a finitelygenerated solvable group determines its algebraic structure—in short, are solvable groups quasi-isometrically rigid? In general they aren’t: very recently A. Dioubina [Dio99] has found a solvable group which is quasi-isometric to a group which is not virtually solvable; these groups are finitely generated but not finitely presentable. In the opposite direction, first steps in identifying quasi-isometrically rigid solvable groups which are not virtually nilpotent were taken for a special class of examples, the solvable Baumslag-Solitar groups, in [FM98] and [FM99b].

The goal of the present paper is to show that a much broader class of solvable groups, the class of finitely-presented, nonpolycyclic, abelian-by-cyclic groups, is characterized among all finitely-generated groups by its quasi-isometry type. We also give a complete quasi-isometry classification of the groups in this class; such a classification for nilpotent groups remains a major open question. Motivated by these results, we offer a conjectural picture of quasi-isometric classification and rigidity for polycyclic abelianby-cyclic groups in §10.1.

∗Supported in part by NSF grant DMS 9704640, by IHES, and by the Alfred P. Sloan Foundation.

†Supported in part by NSF grant DMS 9504946 and by IHES.

The proofs of these results lead one naturally from a geometry of groups problem to the theory of dynamical systems via the asymptotic behavior of certain flows and their associated foliations.

1.1 Abelian-by-cyclic groups

A group Γ is abelian-by-cyclic if there is an exact sequence

1 → A → Γ → Z → 1

where A is an abelian group and Z is an infinite cyclic group. If Γ is finitely generated, then A is a finitely generated module over the group ring Z[Z], although A may not be finitely generated as a group.

By a result of Bieri and Strebel [BS78], the class of finitely presented, torsion-free, abelian-by-cyclic groups may be described in another way. Consider an n × n matrix M with integral entries and det M ≠ 0. Let Γ_{M} be the ascending HNN extension of Z^{n} given by the monomorphism ϕ_{M} with matrix M. Then Γ_{M} has a finite presentation

⟨t, a_{1}, …, a_{n} | [a_{i}, a_{j}] = 1, ta_{i}t^{−1} = ϕ_{M}(a_{i}), i, j = 1, …, n⟩

where ϕ_{M}(a_{i}) is the word a_{1}^{m_{1}} ⋯ a_{n}^{m_{n}} and the vector (m_{1}, …, m_{n}) is the i^{th} column of the matrix M. Such groups Γ_{M} are precisely the class of finitely presented, torsion-free, abelian-by-cyclic groups (see [BS78] for a proof involving a precursor of the Bieri-Neumann-Strebel invariant, or [FM99b] for a proof using trees). The group Γ_{M} is polycyclic if and only if |det M| = 1; this is easy to see directly, and also follows from [BS80].

1.2 Statement of results

The first main theorem in this paper gives a classification of all finitelypresented, nonpolycyclic, abelian-by-cyclic groups up to quasi-isometry. It is easy to see that any such group has a torsion-free subgroup of finite index, so is commensurable (hence quasi-isometric) to some Γ_{M}. The classification of these groups is actually quite delicate—the standard quasi-isometry invariants (ends, growth, isoperimetric inequalities, etc.) do not distinguish any of these groups from each other, except that the size of the matrix M can be detected by large scale cohomological invariants of Γ_{M}.

Given M ∈ GL(n, R), the absolute Jordan form of M is the matrix obtained from the Jordan form for M over C by replacing each diagonal entry with its absolute value, and rearranging the Jordan blocks in some canonical order.

Theorem 1.1 (Classification theorem). Let M_{1} and M_{2} be integral matrices with |det M_{i}| > 1 for i = 1, 2. Then Γ_{M_{1}} is quasi-isometric to Γ_{M_{2}} if and only if there are positive integers r_{1}, r_{2} such that M_{1}^{r_{1}} and M_{2}^{r_{2}} have the same absolute Jordan form.

Remark. Theorem 1.1 generalizes the main result of [FM98], which is the case when M_{1}, M_{2} are positive 1 × 1 matrices; in that case the result of [FM98] says even more, namely that Γ_{M_{1}} and Γ_{M_{2}} are quasi-isometric if and only if they are commensurable. When n ≥ 2, however, it’s not hard to find n × n matrices M_{1}, M_{2} such that Γ_{M_{1}}, Γ_{M_{2}} are quasi-isometric but not commensurable. Polycyclic examples are given in [BG96], and the same ideas are used to produce nonpolycyclic examples in [FM99c].

The following theorem shows that the algebraic property of being a finitely-presented, nonpolycyclic, abelian-by-cyclic group is in fact a largescale geometric property.

Theorem 1.2 (Quasi-isometric rigidity). Let Γ = Γ_{M} be a finitely presented abelian-by-cyclic group, determined by an n × n integer matrix M with |det M| > 1. Let G be any finitely generated group quasi-isometric to Γ. Then there is a finite normal subgroup N ⊂ G such that G/N is abstractly commensurable to Γ_{N}, for some n × n integer matrix N with |det N| > 1.

Remark. Theorem 1.2 generalizes the main result of [FM99b], which covers the case when M is a positive 1 × 1 matrix. The latter result was given a new proof in [MSW], and in §9 we follow the methods of [MSW] in proving Theorem 1.2.

Remark. The “finitely presented” hypothesis in Theorem 1.2 cannot be weakened to “finitely generated”. Dioubina shows [Dio99] that the wreath product Z wr Z, an abelian-by-cyclic group of the form Z[Z]-by-Z, is quasiisometric to the wreath product (Z ⊕ F) wr Z whenever F is a finite group. But (Z ⊕ F) wr Z has no nontrivial finite normal subgroups, and when F is nonabelian it is not abstractly commensurable to an abelian-by-cyclic group.

One of the key technical results used to prove Theorem 1.1 is the following theorem, which we believe is of independent interest. It describes a rigidity phenomenon for 1-parameter subgroups of GL(n, R) which generalizes work of Benardete [Ben88] (see also [Wit90]).

A 1-parameter subgroup M^{t} of GL(n, R) determines a 1-parameter family of quadratic forms Q_{M}(t) = (M^{−t})^{T} (M^{−t}) on R^{n}, where the superscript ^{T} denotes transpose. Each Q_{M}(t) determines a norm ∥ · ∥_{M,t} and a distance function d_{M,t} on R^{n}.

Theorem 5.11 (One-parameter subgroup rigidity) Let M^{t}, N^{t} be 1parameter subgroups of GL(n, R), such that M = M^{1} and N = N^{1} have no eigenvalues on the unit circle. If there exists a bijection f: R^{n} → R^{n} and constants K ≥ 1, C ≥ 0 such that for each t ∈ R and p, q ∈ R^{n}

−C + \frac{1}{K} · d_{M,t}(p, q) ≤ d_{N,t}(f(p), f(q)) ≤ K · d_{M,t}(p, q) + C

then M and N have the same absolute Jordan form.

The proof of Theorem 5.11 is given in §6, and shows that in fact f is a homeomorphism with a reasonably high degree of regularity; see Proposition 6.3.

Remark. The case of Theorem 5.11 when f is the identity map follows from a theorem of D. Benardete [Ben88]. See also D. Witte [Wit90]. Benardete’s theorem determines precisely when two one-parameter subgroups of GL(n, R) diverge, and it applies as well to matrices with eigenvalues on the unit circle.

1.3 Homogeneous spaces

Using coarse topological and geometrical methods, we reduce the study of quasi-isometries of Γ_{M} to that of a certain Lie group G_{M}.

After squaring M if necessary, we can assume that det M > 0 and that M lies on a 1-parameter subgroup M^{t} of GL(n, R). The group Γ_{M} is a cocompact subgroup of the solvable Lie group G_{M} = R^{n} ⋊_{M} R, where R acts on R^{n} by the 1-parameter subgroup M^{t}. The group Γ_{M} is discrete in G_{M} if and only if det M = 1. See section 4 for details.

The groups G_{M}, with their left invariant metrics, give a rich and familiar collection of examples, including: all real hyperbolic spaces, when M is a constant times the identity; many negatively curved homogeneous spaces, when M has all eigenvalues > 1 in absolute value; and 3-dimensional solv geometry, when M is a 2 × 2 hyperbolic matrix of determinant 1. The negatively curved examples associated to a real diagonal matrix with all eigenvalues > 1 were studied by Pansu [Pan89a] (and later Gromov [Gro93]), who analyzed their quasi-isometric geometry using the idea of “conformal dimension”.

We should mention also the result of Heintze [Hei74] that the class of connected, negatively curved homogeneous spaces consists precisely of those spaces of the form N ⋊ R where N is a nilpotent Lie group, and the action of R on the Lie algebra has all eigenvalues strictly outside the unit circle.

1.4 Outline of proofs

After preliminary sections, §3 on Linear Algebra, and §4 on The Solvable Lie Group G_{M}, the proof of Theorem 1.1 can be divided into 3 main parts: §5,6 on the Dynamics of G_{M}; §7 on Quasi-Isometries of Γ_{M} via Coarse Topology; and §8 on Finding the Integers, where the pieces of the proof are put together. The proof of Theorem 1.2 is contained in §9 on Quasi-Isometric Rigidity. Finally we pose some conjectures and problems in §10.

§5,6: Dynamics of G_{M}

In these two sections we classify the Lie groups G_{M} up to horizontal-respecting quasi-isometry, that is up to quasi-isometries ϕ: G_{M} → G_{N} which take each set of the form R^{m} × {t} to a set of the form R^{n} × {h(t)} for some function h called the induced time change.

Theorem 5.2′ (Horizontal respecting quasi-isometries: special case) Let M, N lie on 1-parameter subgroups M^{t}, N^{t} of GL(n, R), and suppose that det M, det N > 1. If there exists a horizontal respecting quasi-isometry ϕ: G_{M} → G_{N}, then there exist real numbers r, s > 0 so that M^{r} and N^{s} have the same absolute Jordan form.

Remark. In the special case where M, N are diagonalizable with all eigenvalues > 1, this can be extracted from work of Pansu [Pan89a] without the assumption that ϕ is horizontal respecting. This special case was later reconsidered by Gromov (see [Gro93] Section 7.C), as an application of his “infδim” invariant. Our statement and proof of Theorem 5.2 is inspired in part by the ideas of exponential growth rates built into the infδim invariant (see also comments after Proposition 5.8).

In §5 we give a slightly more general version of this statement, Theorem 5.2.

The proof of Theorem 5.2 uses a certain dynamical system on G_{M}, the “vertical flow” which flows upward at unit speed along flow lines of the form (point)×R ⊂ R^{m} ⋊_{M} R. When M has no eigenvalues on the unit circle this is a hyperbolic or Anosov flow, and in general it is a partially hyperbolic flow. We prove Theorem 5.2 in several steps, using stronger and stronger dynamical properties of flows in G_{M}.

Step 1 (Foliations Rigidity, Proposition 5.4) Using the Shadowing Lemma from hyperbolic dynamics we show that ϕ coarsely respects three dynamically defined foliations of G_{M} and G_{N}: the weak stable, weak unstable, and center foliations. This, together with a result of Bridson-Gersten that depends in turn on work of Pansu (see Corollary 5.6), allows reduction to the case where M, N have no eigenvalues on the unit circle.

Step 2 (Time rigidity, Proposition 5.8) We show that the induced time change map of ϕ is actually an affine map between the time parameters of G_{M} and G_{N}. After taking a real power of N and composing with a vertical translation, we can assume that ϕ preserves the time parameter, that is h(t) = t.

Step 3 (One-parameter subgroup rigidity, Theorem 5.11) From Step 2, ϕ induces a quasi-isometry between corresponding level sets of the time parameter on G_{M}, G_{N}, which reduces the proof to Theorem 5.11, One-Parameter Subgroup Rigidity. The latter theorem is proved in §6, by studying rigidity properties of certain flags of foliations of R^{n} associated to the absolute Jordan form of M ∈ GL(n, R).

§7: Quasi-Isometries of Γ_{M} via Coarse Topology

Given an integer matrix M ∈ GL(n, R) with det M > 1, we study the geometry of Γ_{M} by constructing a contractible metric cell complex X_{M} on which Γ_{M} acts freely, properly discontinuously and cocompactly by isometries, so that Γ_{M} is quasi-isometric to X_{M}. Topologically, X_{M} is a product of R^{m} with the homogeneous directed tree T_{M} with one edge entering and d edges leaving each vertex. Here d = det M. Metrically, for every coherently oriented line ℓ in T_{M}, the metric on X_{M} is such that R^{m} × ℓ is isometric to G_{M}.

The main result of this section, Proposition 7.1, says that a quasiisometry f: X_{M} → X_{N} induces a quasi-isometry ϕ: G_{M} → G_{N} which respects horizontal foliations. This is proved using coarse geometric and topological methods. This is precisely where the condition det M, det N > 1 is essential for our proof, since it gives that the trees T_{M}, T_{N} have nontrivial branching, and this branching allows us to show that f “remembers” the branch points (see Step 2 of §7.2).

While this proof is in the spirit of [FM98], further complications arise in this more general case (see §7.2). Also, for other applications (e.g. [FM99a], [MSW]), we shall derive Proposition 7.1 from a still more general result, Theorem 7.7, which applies to many graphs of groups whose vertex and edge groups are fundamental groups of aspherical manifolds of fixed dimension.

§8: Finding the integers

Given integer matrices M, N ∈ GL(n, R) with |det M| , |det N| > 1 such that Γ_{M} and Γ_{N} are quasi-isometric, a simple argument allows us to reduce to the case of positive determinant, and then the results of §5–7 combine to show that there are positive real numbers r, s so that M^{r} and N^{s} have the same absolute Jordan form. We need to show that integral r, s exist. This is done by showing that a quasi-isometry X_{M} → X_{N} induces a bilipschitz homeomorphism between certain self-similar Cantor sets attached to X_{M} and X_{N}. Applying a theorem of Cooper on bilipschitz types of these Cantor sets allows us to conclude that (det M)^{p} = (det N)^{q} for some integers p, q ≥ 1, from which the desired conclusion follows.

§9: Quasi-Isometric Rigidity

To prove Theorem 1.2, we use the coarse topology results from §7 to show that a group quasi-isometric to some Γ_{M} admits a quasi-action on a tree of n-dimensional Euclidean spaces. We then use the results of [MSW] to convert this quasi-action into a true action on a tree, whose edge and vertex stabilizers are finitely generated groups quasi-isometric to Z^{n}. The proof is completed by invoking well-known quasi-isometry invariants, combined with a brief study of injective endomorphisms of virtually abelian groups.

§10: Questions

We make some conjectures concerning possible extensions of this work to the polycyclic case. Also, we state some problems on the quasi-isometry group of Γ_{M}.

Acknowledgements We thank Kevin Whyte and Amie Wilkinson for all their help. We are also grateful to the IHES, where much of this work was done.

2 Preliminaries

This brief section reviews some basic material; see for example [GdlH91].

Given K ≥ 1, C ≥ 0, a (K, C) quasi-isometry between metric spaces is a map f: X → Y such that:

1. For all x, x′ ∈ X we have

\frac{1}{K} · d_{X}(x, x′) − C ≤ d_{Y} (f(x), f(x′)) ≤ K · d_{X}(x, x′) + C

2. For all y ∈ Y we have d_{Y} (y, f(X)) ≤ C.

If f satisfies (1) but not necessarily (2) then it is called a (K, C) quasiisometric embedding. If f satisfies only the right hand inequality of (1) then f is (K, C) coarsely lipschitz, and if in addition C = 0 then f is K-lipschitz.

A coarse inverse of a quasi-isometry f: X → Y is a quasi-isometry g: Y → X such that, for some constant C′ > 0, we have d(g ∘ f(x), x) < C′ and d(f ∘g(y), y) < C′ for all x ∈ X and y ∈ Y. Every (K, C) quasi-isometry f: X → Y has a K, C′ coarse inverse g: Y → X, where C′ depends only on K, C: for each y ∈ Y define g(y) to be any point x ∈ X such that d(f(x), y) ≤ C.

A fundamental fact observed by Efremovich, by Milnor [Mil68], and by Švarc, which we use repeatedly without mention, states that if a group G acts properly discontinuously and cocompactly by isometries on a proper geodesic metric space X, then G is finitely generated, and X is quasi-isometric to G equipped with the word metric.

Given a metric space X and A, B ⊂ X, we denote the Hausdorff distance by

d_{H}(A, B) = inf{r ∈ [0, ∞] | A ⊂ N_{r}(B) and B ⊂ N_{r}(A)}

The following lemma says that an ambient quasi-isometry induces a quasi-isometry between subspaces of a certain type. A map σ: S → X between geodesic metric spaces is uniformly proper if there is function ρ: [0, ∞) → [0, ∞) with lim_{t→∞} ρ(t) = +∞, and constants K ≥ 1, C ≥ 0, such that for all x, y ∈ S we have:

ρ(d_{S}(x, y)) ≤ d_{X}(σ(x), σ(y)) ≤ K · d_{S}(x, y) + C

The function ρ and the constants K, C are called uniformity data for σ.

Lemma 2.1. Given geodesic metric spaces X, Y, S, T, a quasi-isometry f: X → Y, and uniformly proper maps σ: S → X and τ: T → Y, suppose that d_{H}(fσ(S), τ(T)) < ∞. Then S, T are quasi-isometric. To be explicit, any function g: S → T such that d_{Y} (fσ(x), τg(x)) is uniformly bounded is a quasi-isometry; the quasi-isometry constants for g depend only on those for f, the uniformity data for σ and τ, and the bound for d_{Y} (fσ(x), gτ(x)).

Proof. Pick K ≥ 1, C ≥ 0 and ρ: [0, ∞) → [0, ∞) such that f is a (K, C) quasi-isometry, d_{Y} (fσ(x), gτ(x)) ≤ C, and ρ, K, C are uniformity data for σ, τ.

Consider x, y ∈ S such that d_{S}(x, y) ≤ 1. We have d_{Y} (fσ(x), fσ(y)) ≤ K^{2} + KC + C, and so d_{Y} (τg(x), τg(y)) ≤ K^{2} + KC + 3C, from which it follows that ρ(d_{T} (g(x), g(y))) ≤ K^{2} + KC + 3C. Since lim_{t→∞} ρ(t) = ∞ we obtain a bound d_{T} (g(x), g(y)) ≤ A depending only on K, C, ρ. The usual “rubber band” argument, using geodesics in S divided into subsegments of length 1 with a terminal subsegment of length ≤ 1, suffices to prove that g is (K′, C′) coarsely lipschitz, with K′, C′ depending only on K, C, ρ.

For any ξ ∈ T there is a point \overline{g}(ξ) ∈ S such that d_{Y} (fσ\overline{g}(ξ), τ(ξ)) ≤ C. For any ξ, η ∈ T with d(ξ, η) ≤ 1 we have

d_{Y} (fσḡ(ξ), fσḡ(η)) ≤ d_{Y} (fσḡ(ξ), τ(ξ)) + d_{Y} (τ(ξ), τ(η)) + d_{Y} (fσḡ(η), τ(η))
≤ K + 3C

and so ρ(d_{S}(ḡ(ξ), ḡ(η))) ≤ d_{X}(σḡ(ξ), σḡ(η)) ≤ K^{2} + 4KC. As above we obtain an upper bound for d_{S}(ḡ(ξ), ḡ(η)) and the rubber band argument shows that \overline{g} is coarsely lipschitz.

For any x ∈ S, setting ξ = g(x) ∈ T, we have

d_{Y} (fσ(x), fσ\overline{g}(ξ)) ≤ d_{Y} (fσ(x), τg(x)) + d_{Y} (τ(ξ), fσ\overline{g}(ξ))
≤ 2C

It follows that d_{X}(σ(x), σ\overline{g}(ξ)) ≤ 3KC and so ρ(d_{S}(x, \overline{g}g(x))) = ρ(d_{S}(x, \overline{g}(ξ))) ≤ 3KC, yielding an upper bound for d_{S}(x, \overline{g}g(x)). Similarly, d_{Y} (ξ, g\overline{g}(ξ)) is bounded for all ξ ∈ T.

Knowing that g: S → T and \overline{g}: T → S are coarse lipschitz maps which are coarse inverses of each other, it easily follows that g is a quasi-isometry, with quasi-isometry constants depending only on the coarse lipschitz constants for g and \overline{g}, and on the coarse inverse constants for g, \overline{g}. ♢

3 Linear Algebra

In this section we collect some basic results about canonical forms of matrices, and growth of vectors under the action of a matrix.

Let M(n, F) denote all n×n matrices over a field F, and let GL(n, F) be the group of invertible matrices. Let GL_{0}(n, R) be the identity component of GL(n, R), consisting of all matrices of positive determinant.

3.1 Jordan Forms

A matrix J ∈ M(k, C) is a Jordan block it it has the form J = J(k, λ) = λ · Id + N where λ ∈ C and N_{ij} = δ(i + 1, j), so N is the k × k matrix with 1’s on the superdiagonal and 0’s elsewhere.

A matrix M ∈ M(n, C) is in Jordan form if it is in block diagonal form

M = ( \begin{matrix}J_{1} & 0 & … & 0 \\ 0 & J_{2} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋮ & J_{I} \end{matrix} )

where each J_{i} is a Jordan block. Every matrix in M(n, C) is conjugate, via an invertible complex matrix, to a matrix in Jordan form, unique up to permutation of the Jordan blocks. When all eigenvalues are real, say J_{i} has eigenvalue ℓ_{i}, we resolve the nonuniqueness by requiring ℓ_{1} ≥ ℓ_{2} ≥ ⋯ ≥ ℓ_{I}, and for each i = 1, …, I − 1, if ℓ_{i} = ℓ_{i+1} then rk(J_{i}) ≥ rk(J_{i+1}).

A matrix J ∈ M(k, R) is a real Jordan block if it has one of the following two forms. The first form is an ordinary Jordan block J(k, ℓ) where ℓ ∈ R. The second form, which requires k to be even, has a 2×2 block decomposition of the form

J = J(k, a, b) = ( \begin{matrix}Q(a, b) & Id & … & 0 & 0 \\ 0 & Q(a, b) & … & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & … & Q(a, b) & Id \\ 0 & 0 & … & 0 & Q(a, b) \end{matrix} )

where Id is the identity, 0 is the zero matrix, Q(a, b) = ( \begin{matrix}a & −b \\ b & a \end{matrix} ), and b ≠ 0.

A matrix M ∈ M(n, R) is in real Jordan form if it is in block diagonal form as above where each block J_{i} is a real Jordan block. Every matrix in M(n, R) is conjugate, via an invertible real matrix, to a matrix in real Jordan form, unique up to permutation of blocks.

The absolute Jordan form of M ∈ M(n, R) is the matrix obtained from the Jordan form of M by replacing each diagonal entry λ by ℓ = |λ|, and permuting the blocks to resolve the nonuniqueness. If M is invertible then the absolute Jordan form of M can be written in block diagonal form

( \begin{matrix}J_{M}^{+} & 0 & 0 \\ 0 & J_{M}^{0} & 0 \\ 0 & 0 & J_{M}^{−} \end{matrix} )

where the diagonal entries of J_{M}^{+} are > 1, of J_{M}^{0} are = 1, and of J_{M}^{−} are < 1. We call J_{M}^{+} the expanding part of the absolute Jordan form, J_{M}^{0} the

unipotent part, and J_{M}^{−} the contracting part, and the block matrix ( \begin{matrix}J_{M}^{+} & 0 \\ 0 & J_{M}^{−} \end{matrix} ) is called the nonunipotent part. Of course, one or more of these parts might be empty.

Note that the Jordan form of the real matrix J(k, a, b) is

( \begin{matrix}J(k/2, a + bi) & 0 \\ 0 & J(k/2, a − bi) \end{matrix} )

and so the absolute Jordan form of J(k, a, b) is

( \begin{matrix}J(k/2, \sqrt{a^{2} + b^{2}}) & 0 \\ 0 & J(k/2, \sqrt{a^{2} + b^{2}}) \end{matrix} )

Given M ∈ M(n, R), this process may be applied block-by-block to the real Jordan form of M, and the blocks then permuted, to obtain the absolute Jordan form of M.

Let GL_{×}(n, R) denote the set of all matrices in GL(n, R) lying on a 1-parameter subgroup of GL(n, R), so GL_{×}(n, R) ⊂ GL_{0}(n, R). It is wellknown and easy to see, given a matrix M ∈ GL(n, R), that M ∈ GL_{×}(n, R) if and only if the negative eigenvalue Jordan blocks of M may be paired up so that the two blocks occuring in each pair are identical to each other, and this occurs if and only if M has a square root in GL(n, R). Thus, if M does not already lie on a 1-parameter subgroup then M^{2} does. We are therefore free to replace a matrix by its square in order to land on a 1-parameter subgroup.

Given a 1-parameter subgroup ρ(t) of GL(n, R), if M = ρ(1) then we will often abuse notation and write ρ(t) = M^{t}, despite the fact that M may not lie on a unique 1-parameter subgroup.

Given A ∈ M(n, R) in Jordan form—no J(k, a, b) blocks—we say that ρ(t) = e^{At} is a 1-parameter Jordan subgroup. Notice that the matrices e^{At} are not themselves in Jordan form. For example when A = J(n, ℓ) = ℓ·Id+N is a single n × n Jordan block then e^{At} is obtained by multiplying the scalar e^{ℓt} times the matrix

e^{N·t} = ∑_{i=0}^{n} \frac{1}{i!}N^{i} · t^{i} = ( \begin{matrix}1 & t & \frac{t^{2}}{2!} & ⋯ & \frac{t^{n−1}}{(n−1)!} & \frac{t^{n}}{n!} \\ & 1 & t & ⋯ & \frac{t^{n−2}}{(n−2)!} & \frac{t^{n−1}}{(n−1)!} \\ & & 1 & ⋯ & \frac{t^{n−3}}{(n−3)!} & \frac{t^{n−2}}{(n−2)!} \\ & & & ... & ⋮ & ⋮ \\ & & & & 1 & t \\ & & & & & 1 \end{matrix} )  (3.1)

Nevertheless, for any Jordan form matrix J = ℓ · Id + N with ℓ ∈ R, the Jordan form of e^{J} is e^{ℓ} · Id + N.

Given a general 1-parameter subgroup e^{μt} in GL(n, R), choose A so that A^{−1}μA is in real Jordan form, and so A^{−1}μA = δ+ν +η where δ is diagonal, ν is superdiagonal, and η is skew-symmetric. We then have

e^{μt} = (Ae^{(δ+ν)t}A^{−1})(Ae^{ηt}A^{−1})

Since η is skew symmetric it follows that e^{ηt} is in the orthogonal group O(n, R). We have therefore proved (see [Wit90] for this particular formulation):

Proposition 3.1 (1-parameter Real Jordan Form). Let M^{t} be a 1-parameter subgroup of GL(n, R). There exists a 1-parameter Jordan subgroup e^{Jt}, a matrix A ∈ GL(n, R), and a bounded 1-parameter subgroup P^{t} conjugate into the orthogonal group O(n, R), such that e^{J} is the absolute Jordan form of M, and letting \overline{M}^{t} = A^{−1}e^{Jt}A we have

M^{t} = \overline{M}^{t}P^{t} = P^{t}\overline{M}^{t}

Remark. In [Wit90] the subgroup \overline{M}^{t} is called the nonelliptic part of M^{t}, and P^{t} is called the elliptic part. These two 1-parameter subgroups, which commute with each other, are uniquely determined by M^{t}.

3.2 Growth of vectors under a linear transformation

Consider a 1-parameter subgroup M^{t} of GL(n, R), with real Jordan form M^{t} = (A^{−1}e^{Jt}A)P^{t} = \overline{M}^{t}P^{t}. Let

0 < λ_{1} < ⋯ < λ_{L}

be the eigenvalues of \overline{M}. Let V_{l} = ker((λ_{l} · Id − \overline{M})^{m}) be the root space of the eigenvalue λ_{l}, where m is the multiplicity of λ_{l}. Let n_{l} be the index of nilpotency of \overline{M} | V_{l}, the smallest integer such that V_{l} = ker((λ_{l} ·Id−\overline{M})^{n_{l}}). For i = 0, …, n_{l} − 1 let V_{l,i} = ker((λ_{l} · Id − \overline{M})^{i+1}), so V_{l,0} is the eigenspace of λ_{l} and V_{l,n_{l}−1} = V_{l}. We thus have the Jordan decomposition of \overline{M}, which consists of the direct sum of root spaces

R^{n} = V_{1} ⊕ ⋯ ⊕ V_{L}

together with the Jordan filtrations

V_{l,0} ⊂ ⋯ ⊂ V_{l,n_{l}−1} = V_{l}, l = 1, …, L

This decomposition is uniquely determined by \overline{M}, and hence by M.

Proposition 3.2 (Growth of vectors). With the above notation, there exist constants A, B > 0 with the following properties. Given l = 1, …, L with λ_{l} ≥ 1:

Exponential Lower Bound If v ∈ V_{l} and t ≥ 0 then

∥M^{t}v∥ ≥ Aλ_{l}^{t}∥v∥
In fact the same lower bound holds if v ∈ V_{l} ⊕ V_{l+1} ⊕ ⋯ ⊕ V_{L}.

Exponential·Polynomial Upper Bound Given i = 0, …, n_{l} − 1, if v ∈

V_{l,i} and t ≥ 1 then

∥M^{t}v∥ ≤ Bλ_{l}^{t}t^{i}∥v∥
In fact the same upper bound holds if v ∈ (V_{1} ⊕ ⋯ ⊕ V_{l−1}) ⊕ V_{l,i}.

Exponential·Polynomial Lower Bound Given i = 0, …, n_{l} − 1, if v ∈ V_{l,i} \ V_{l,i−1} then there exists C_{v} > 0 such that if t ≥ 1 then

∥M^{t}v∥ ≥ C_{v}λ_{l}^{t}t^{i}

Given l = 1, …, L with λ_{l} ≤ 0, similar statements are true with negative values of t.

Proof. We start with the case when M^{t} = e^{Jt} is a 1-parameter Jordan subgroup, and the proposition follows by examining each Jordan block (3.1).

The second case we consider is when M^{t} has all positive real eigenvalues. By Proposition 3.1 we have M^{t} = A^{−1}e^{Jt}A, and Proposition 3.2 follows immediately from the first case applied to e^{Jt}, together with the fact that A takes the Jordan decomposition of M^{t} to the Jordan decomposition of e^{Jt}.

In the general case, applying Proposition 3.1 we have M^{t} = (A^{−1}e^{Jt}A)P^{t} = \overline{M}^{t}P^{t}. We can the apply the second case to \overline{M}^{t} = A^{−1}e^{Jt}A. Since P^{t} commutes with \overline{M}^{t} it follows that P^{t} preserves the Jordan decomposition of \overline{M}^{t}. Proposition 3.2 then follows from the boundedness of P^{t}. ♢

4 The Solvable Lie Group G_{M}

Recall that GL_{×}(n, R) denotes those matrices in GL(n, R) which lie on a 1-parameter subgroup of GL(n, R). Also, each matrix in GL_{×}(n, R) has positive determinant.

Given a matrix M ∈ GL_{×}(n, R) lying on a 1-parameter subgroup M^{t} of GL(n, R), we associate a solvable Lie group denoted G_{M}. This is the semidirect product G_{M} = R^{n} ⋊_{M} R with multiplication defined by

(x, t) · (y, s) = (x + M^{t}y, t + s)

for all (x, t), (y, s) ∈ R^{n} × R. We will often identify G_{M} = R^{n} ⋊_{M} R with the underlying set R^{n} × R.

Remark. Although the Lie group G_{M} depends on more than just the matrix M = M^{1} itself—it depends on the entire 1-parameter subgroup M^{t}— we suppress this dependence in our notation G_{M} = R^{n} ⋊_{M} R. This is justified by the fact that the quasi-isometry type of G_{M} depends only on M, not on the 1-parameter subgroup containing M (see the remark after Proposition 4.1). Henceforth, when we say something like “Given M ∈ GL_{×}(n, R) … ”, we will either implicitly or explicitly choose a 1-parameter subgroup M^{t} < GL(n, R) with M^{1} = M, which in turn determines G_{M}.

If M has integer entries then there is a homomorphism Γ_{M} → G_{M} taking the commuting generators a_{1}, …, a_{n} to the standard basis of the integer lattice Z^{n} × 0 ⊂ R^{n} × 0 ⊂ R^{n} × R, and taking the stable letter t to the generator (0, 1) ∈ R^{n} × R. The relator ta_{i}t^{−1} = ϕ_{M}(a_{i}) is checked by noting that

(0, 1) · (x, 0) · (0, −1) = (Mx, 0), for all x ∈ R^{n}

Cocompactness of the image of this homomorphism is obvious. To see that Γ_{M} embeds in G_{M} one checks that in the abelian-by-cyclic extension 1 → A → Γ_{M} → Z → 1, the group A is identified with the nested union Z^{n} ∪ M^{−1}(Z^{n}) ∪ M^{−2}(Z^{n}) ∪ ⋯, in R^{n}. This also shows that discreteness of Γ_{M} in G_{M} is equivalent to det M = 1, which is equivalent to Z^{n} = M(Z^{n}).

For the next several sections we will investigate the geometry of the solvable Lie group G_{M}. In this section we begin by showing that G_{M} and G_{N} are quasi-isometric if M, N have powers with the same absolute Jordan form. Later in §7 we will see that when M has integer entries, much of the geometry of Γ_{M} is reflected in the geometry of G_{M}.

We endow G_{M} with the left invariant metric determined by taking the standard Euclidean metric at the identity of G_{M} ≈ R^{n} × R = R^{n+1}. At a point (x, t) ∈ R^{n} × R ≈ G_{M}, the tangent space is identified with R^{n} × R, and the Riemannian metric is given by the symmetric matrix

( \begin{matrix}Q_{M}(t) & 0 \\ 0 & 1 \end{matrix} )

where Q_{M}(t) = (M^{−t})^{T} M^{−t}. For each t ∈ R, the identification R^{n} ≈ R^{n} × t ⊂ G_{M} induces in R^{n} the metric determined by the quadratic form Q_{M}(t). This metric has distance formula

d_{M,t}(x, y) = ∥M^{−t}(x − y)∥

Remarks

1. When M is a 1 × 1 matrix with entry a > 1, the group G_{M} is isomorphic to Aff(R), the group of affine transformations of R, and as a Riemannian manifold G_{M} is isometric to a scaled copy of the hyperbolic plane with constant sectional curvature depending on a.

2. The eigenvalues of M are greater than 1 in absolute value if and only if all sectional curvatures of G_{M} are negative (see [Hei74]).

Proposition 4.1 (How the metric on G_{M} depends on choices). Given 1-parameter subgroups M^{t}, N^{t} in GL(n, R), suppose there exist real numbers r, s > 0 such that M^{r} and N^{s} have the same absolute Jordan form. Then the metric spaces G_{M} and G_{N} are quasi-isometric. To be explicit there exists A ∈ GL(n, R) and K ≥ 1 such that for each t ∈ R, the map v ↦ A(v) is a K-bilipschitz homeomorphism from the metric d_{M,t} to the metric d_{N, \frac{s}{r} ·t}; it follows that the map from G_{M} = R^{n} ⋊_{M} R to G_{N} = R^{n} ⋊_{N} R given by

(x, t) ↦ (Ax, \frac{s}{r} · t)

is a bilipschitz homeomorphism from G_{M} to G_{N}, with bilipschitz constant sup\{K, \frac{s}{r}, \frac{r}{s}\}.

Remark. The absolute Jordan form of M^{r} is uniquely determined by M and r: it is the r^{th} power of the absolute Jordan form of M. It follows in particular that the quasi-isometry type of G_{M} depends only on the matrix M = M^{1}, not on the choice of 1-parameter subgroup M^{t}.

Proof of Proposition 4.1. We proceed in cases.

Case 1: Assume that N^{t} = e^{Jt} is the unique 1-parameter Jordan subgroup such that N = e^{J} is conjugate to the absolute Jordan form of M. Applying Proposition 3.1 we have

M^{t} = (A^{−1}N^{t}A)P^{t}

where A ∈ GL(n, R) and the 1-parameter subgroup P^{t} is bounded.

Choose t ∈ R and v ∈ R^{n}. We must show that the two numbers ∥M^{−t}v∥ = ∥P^{−t}(A^{−1}N^{−t}A)v∥ and ∥N^{−t}Av∥ have ratio bounded away from 0 and ∞, with bound independent of t, v. Setting u = N^{−t}Av, it suffices to show that ∥P^{−t}A^{−1}u∥ and ∥u∥ have bounded ratio. But this is clearly true, with a bound of

(sup_{t} ∥P^{t}∥) · Max\{∥A∥, \frac{1}{∥A∥}\}

since the 1-parameter subgoup P^{t} is bounded.

Case 2: Assume that there exists a > 0 such that M^{t} = N^{at} for all t. Then the metrics d_{M,t} and d_{N,at} are identical.

General case: Applying Case 2 we may assume that det M = det N. Applying Case 1 twice we may go from G_{M} to G_{e^{J}} to G_{N}, where e^{J} is conjugate to the absolute Jordan form of M and of N. ♢

5 Dynamics of G_{M}, Part I:

Horizontal Respecting Quasi-isometries

In this section we begin studying the asymptotic geometry of the solvable Lie groups G_{M} associated to 1-parameter subgroups M^{t} of GL(n, R). As we saw in Section 4, the quasi-isometry type of G_{M} depends only on M, not on the choice of 1–parameter subgroup M^{t} passing through M; see the remark after Proposition 4.1. We therefore continue to suppress the choice of 1-parameter subgroup in our notation. Further, we do not restrict the determinant to be > 1: the results of this section hold even when det M = 1.

5.1 Definitions and statement of main theorem

Let X, Y be metric spaces. Let F be a decomposition of X, that is, a collection of disjoint subsets of X whose union is X. Let G be a decomposition of Y. Motivated by a foliation of a manifold, the elements of these decompositions are called leaves and the decomposition itself is called the leaf space. A quasi-isometry ϕ: X → Y is said to coarsely respect the decompositions F, G if there exists a number A ≥ 0 and a map of leaf spaces h: F → G such that for each leaf L ∈ F we have

d_{H}(ϕ(L), h(L)) ≤ A

For example, consider the space G_{M}. The coordinate function G_{M} ≈ R^{n} × R → R given by (x, t) ↦ t is called the time function of G_{M}. The level sets P_{t} ≈ R^{n} × t form the horizontal foliation of G_{M}, whose leaves are called horizontal leaves of G_{M}, and whose leaf space is R. Notice that d_{H}(P_{s}, P_{t}) = |s − t|, and so the time function induces an isometry between the horizontal leaf space equipped with the Hausdorff metric and R.

Consider another matrix N ∈ GL_{×}(n, R), and denote the horizontal leaves of G_{N} by P′_{t}.

Definition (Horizontal respecting). A quasi-isometry ϕ: G_{M} → G_{N} is said to be horizontal respecting if it coarsely respects the horizontal foliations of G_{M}, G_{N}. That is, there exists a function h: R → R and A ≥ 0 such that d_{H}(ϕ(P_{t}), P′_{h(t)}) ≤ A for all t ∈ R.

The function h: R → R is called an induced time change for ϕ, with Hausdorff constant A.

If h, h′ are two induced time changes for ϕ then sup_{t} |h(t) − h′(t)| ≤ A+A′ < ∞, where A, A′ are Hausdorff constants for h, h′ respectively. Also, if h: R → R is an induced time change for ϕ with Hausdorff constant A, if A′ ≥ 0, and if h′: R → R is any function satisfying sup_{t∈R} |h(t) − h′(t)| ≤ A′, then h′ is also an induced time change for ϕ, with Hausdorff constant A + A′.

Lemma 5.1. For each K, C, A there exists C′ such that if ϕ: G_{M} → G_{N} is a horizontal respecting (K, C) quasi-isometry, and h: R → R is an induced time change for ϕ with Hausdorff constant A, then h is a (K, C′) quasiisometry of R.

Proof. We have |h(t) − h(s)| ≤ d_{H}(P_{h(t)}, P_{h(s)}) + 2A ≤ K |t − s| + C + 2A. The reverse inequality is similar, and so h is a quasi-isometric embedding. Since ϕ is coarsely onto, an easy argument shows h is coarsely onto. ♢

A (K, C′) quasi-isometry h: R → R induces a bijection of the two-point set Ends(R) = {−∞, +∞}: given η_{1}, η_{2} ∈ Ends(R), we have h(η_{1}) = η_{2} if and only if h takes every sequence that diverges to η_{1} to a sequence that diverges to η_{2}. The following two properties of h are equivalent:

1. h induces the identity on Ends(R).

2. h is coarsely increasing, that is there exists L > 0 such that if t > s+L then h(t) > h(s).

That (2) implies (1) is obvious. The other direction is true with any L > 2C′K, for if there existed t ≥ s+L with h(t) ≤ h(s), then since h induces the identity on Ends(R) there would exist t′ > t such that |h(s) − h(t′)| ≤ C′, but also |h(s) − h(t′)| ≥ |s − t′| /K − C′ ≥ L/K − C′ > C′, a contradiction.

If h: R → R is an induced time change of a horizontal respecting quasiisometry ϕ: G_{M} → G_{N}, and if h satisfies the equivalent properties (1) and (2), then we say that ϕ coarsely respects the transverse orientation of the horizontal foliations.

Terminology (time vs. height): In some contexts the vertical parameter which we have been calling “time” will also be called height, as sometimes this terminology is more suggestive, for example in discussing horizontal foliations.

Here is the main result, whose proof will occupy the remainder of this section and the next section.

Theorem 5.2 (Horizontal respecting quasi-isometries). Let ϕ: G_{M} → G_{N} be a quasi-isometry which coarsely respects the transversely oriented horizontal foliations of G_{M} and G_{N}. Then there exist real numbers r, s > 0 so that M^{r} and N^{s} have the same absolute Jordan form.

Our proof of Theorem 5.2 proceeds in steps, following the outline given in the introduction.

5.2 Step 1a: Hyperbolic dynamics and the Shadowing Lemma

The Lie group G_{M} has a natural flow which fits into the theory of partially hyperbolic dynamical systems. From the dynamics we find that the flow has several invariant foliations, the “weak stable, weak unstable, and center” foliations. In §§5.2,5.3, by using the Shadowing Lemma ([HPS77], Lemma 7.A.2, page 133), we prove that a horizontal respecting quasi-isometry G_{M} → G_{N} also respects the dynamically defined foliations of G_{M}, G_{N}.

From this result we obtain the first piece of our rigidity theorem, by showing that expanding, contracting, and unipotent parts of the absolute Jordan forms of M and N have the same ranks respectively, and that the unipotent parts are identical.

5.2.1 Dynamically defined foliations

Consider a 1-parameter subgroup M^{t} ∈ GL(n, R), with real Jordan form M^{t} = \overline{M}^{t}P^{t}. Consider the Jordan decomposition of \overline{M}, and group the root spaces according to whether the corresponding eigenvalue is < 1, = 1, or > 1 (alternatively, a logarithm which is < 0, = 0, or > 0), to obtain a decomposition R^{n} = V^{−} ⊕ V^{0} ⊕ V^{+}.

Remark. It might happen that one or two of the factors V^{−}, V^{0}, V^{+} is trivial, that is, 0-dimensional, for instance when all eigenvalues of M lie outside the unit circle.

Now consider the Lie group G_{M} = R^{n}⋊_{M}R determined by a 1-parameter subgroup M^{t}. Define the vertical flow Φ on G_{M} to be

Φ_{t}(x, s) = (x, s + t)

The tangent bundle T G_{M} has a Φ-invariant splitting

T G_{M} = E^{s} ⊕ E^{c} ⊕ E^{u}

defined as follows. The tangent space at each point x ∈ G_{M} is identified with R^{n} ⊕ R, and we take

E_{x}^{s} = V^{+} ⊕ 0, E_{x}^{c} = V^{0} ⊕ R, E_{x}^{u} = V^{−} ⊕ 0

It is evident from the construction that each of the distributions E^{s} ⊕ E^{c}, E^{u}⊕E^{c}, E^{c} is integrable, tangent to foliations denoted W^{s}, W^{u}, W^{c}. We call these foliations the (weak) stable, unstable, and center foliations, respectively. The stable and unstable foliations are transverse, and the intersection of any stable leaf with any unstable leaf is a center leaf.

Applying the Exponential Lower Bound from Proposition 3.2, there exist constants A > 0, λ > 1 such that:

• If v ∈ E^{u}, then for t ≥ 0 we have ∥DΦ_{t}v∥ ≥ Aλ^{t}∥v∥, and for t ≤ 0 we have ∥DΦ_{t}v∥ ≤ \frac{1}{A}λ^{t}∥v∥.

• If v ∈ E^{s}, then for t ≤ 0 we have ∥DΦ_{t}v∥ ≥ Aλ^{−t}∥v∥, and for t ≥ 0 we have ∥DΦ_{t}v∥ ≤ \frac{1}{A}λ^{−t}∥v∥.

Also, applying the Exponential·Polynomial Upper Bound from Proposition 3.2, there exists B > 0 and an integer n ≥ 1 such that:

• If v ∈ E^{c}, then for |t| ≥ 1 we have ∥DΦ_{t}v∥ ≤ B |t|^{n} ∥v∥.

When we want to emphasize the dependence of the V ’s and E’s on the 1-parameter subgroup M^{t}, we will append a subscript, e.g. V_{M}^{+}, E_{M}^{s}, etc.

5.2.2 Shadowing Lemma

Consider a flow Φ on a metric space X. We write x · t as an abbreviation for Φ_{t}(x). Given ϵ, T > 0, an ϵ, T pseudo-orbit of Φ consists of a sequence of flow segments (x_{i} · [0, t_{i}]), where the index set i runs over an interval in Z, such that d_{X}(x_{i} · t_{i}, x_{i+1}) < ϵ and t_{i} > T for all i.

Lemma 5.3 (Shadowing Lemma). Consider a 1-parameter subgroup M^{t} of GL(n, R), and let Φ be the vertical flow on G_{M}. For every ϵ, T > 0 there exists δ, ϵ′, T′ > 0 such that every ϵ, T pseudo-orbit of Φ is δ-shadowed by an ϵ′, T′ pseudo-orbit of Φ which is contained in some center leaf. That is, if (x_{i} · [0, t_{i}]) is an ϵ, T pseudo-orbit, then there is an ϵ′, T pseudo-orbit (y_{i} · [0, t_{i}]) contained in some center leaf so that d(x_{i} · t, y_{i} · t) < δ for all i and all t ∈ [0, t_{i}].

Proof. By construction, the foliations W^{s} and W^{u} are coordinate foliations in R^{n+1}; this shows that the flow Φ has a “global product structure” in the language of hyperbolic dynamical systems. The lemma now follows the proof of the Shadowing Lemma in [HPS77], Lemma 7.A.2, page 133. A direct proof is also easy to work out, and is left to the reader. ♢

5.3 Step 1b: Foliations rigidity

The Shadowing Lemma implies further rigidity properties of horizontal respecting quasi-isometries:

Proposition 5.4 (Foliations rigidity). Suppose ϕ: G_{M} → G_{N} is a quasiisometry which coarsely respects the horizontal foliations and their transverse orientations. Then ϕ also coarsely respects the weak unstable foliations W_{M}^{u}, W_{N}^{u}, the weak stable foliations W_{M}^{s}, W_{N}^{s}, and the center foliations W_{M}^{c}, W_{N}^{c}. In particular

• dim(V_{M}^{−}) = dim(V_{N}^{−})

• dim(V_{M}^{+}) = dim(V_{N}^{+})

• dim(V_{M}^{0}) = dim(V_{N}^{0})

Remarks

1. In the case where neither M nor N has any eigenvalue on the unit circle, the center foliations of both G_{M} and G_{N} are simply the foliations by vertical flow lines, and Proposition 5.4 says that ϕ respects these foliations. But in the general case, it is not true that ϕ always respects the foliations by vertical flow lines. For a simple counterexample, consider the 1×1 matrix M = N = (1), which gives Γ_{M} = Γ_{N} = Z^{2}. There exist horizontal respecting quasiisometries of R^{2} = R × R which do not respect the vertical foliation.

2. If all eigenvalues of M and N are outside the unit circle, then both G_{M} and G_{N} are negatively curved, and the proposition follows from a standard fact: a quasigeodesic in a negatively curved space X is Hausdorff close to a geodesic (this was the approach taken in [FM98] in the case of a 1×1 matrix M, where G_{M} is isometric to a scaled copy of the hyperbolic plane). This “fact” is unavailable when X = G_{M} is not negatively curved, forcing us to study horizontal respecting quasi-isometries via the Shadowing Lemma.

Before proving Proposition 5.4, we use it to obtain some pieces of our classification theorem. Since rk(J_{M}^{−}) = dim(V_{M}^{−}) etc., we immediately have:

Corollary 5.5. If there is a quasi-isometry from G_{M} to G_{N} which coarsely respects the transversely oriented horizontal foliations, then rk(J_{M}^{−}) = rk(J_{N}^{−}), rk(J_{M}^{0}) = rk(J_{N}^{0}), and rk(J_{M}^{+}) = rk(J_{N}^{+}). ♢

We also have:

Corollary 5.6. The unipotent blocks of the absolute Jordan forms of M and N are identical.

Proof. Let L be some center leaf of G_{M}, of dimension k. From Proposition 5.4 it follows that ϕ(L) is Hausdorff close to some center leaf L′ of G_{N}, also of dimension k. By composition with nearest point projection (which moves points a uniformly bounded amount) we get an induced map L → L′. By Lemma 2.1 this map is a quasi-isometry. By Proposition 4.1, L and L′ are quasi-isometric to the nilpotent Lie groups R^{k−1} ⋊_{J_{M}^{0}} R and R^{k−1} ⋊_{J_{N}^{0}} R, respectively. As Bridson and Gersten have shown [BG96], Pansu’s invariant [Pan89b] may be used to prove that J_{M}^{0} = J_{N}^{0}. ♢

Proof of Proposition 5.4. We begin with:

Claim 5.7. For each vertical flow line γ = Φ_{R}(x) in G_{M}, there exists a center leaf τ_{γ} in G_{N} such that ϕ(γ) is contained in the α-neighborhood of τ_{γ}, where the constant α > 0 does not depend on γ.

Before proving the claim, we apply it to prove the proposition as follows. Consider any two vertical flow lines γ_{1}, γ_{2} in G_{M}. By the claim we have

that ϕ(γ_{1}) and ϕ(γ_{2}) lie, respectively, in bounded neighborhoods of center leaves σ_{1} and σ_{2} of G_{N}. Since h(t) → ±∞ as t → ±∞, for each choice of sign + or − the following two statements are equivalent, and the second statement implies the third:

1. The distance between the points γ_{1}∩P_{t} and γ_{2}∩P_{t} in P_{t} stays bounded as t → ±∞.

2. The distance between the points ϕ(γ_{1})∩P_{h(t)} and ϕ(γ_{2})∩P_{h(t)} in P_{h(t)} stays bounded as t → ±∞.

3. The Hausdorff distance between the sets σ_{1} ∩ P_{h(t)} and σ_{2} ∩ P_{h(t)} in P_{h(t)} stays bounded as t → ±∞.

Using + signs, the first statement is equivalent to saying that γ_{1}, γ_{2} are contained in the same unstable leaf of G_{M}, and the third statement is equivalent to saying that σ_{1}, σ_{2} are contained in the same unstable leaf of G_{N}. It follows that ϕ takes every unstable leaf of G_{M} into a bounded neighborhood of an unstable leaf of G_{N}. Applying the same argument to a coarse inverse \overline{ϕ} of ϕ gives the opposite inclusion. Since d(\overline{ϕ}∘ϕ, Id) < ∞ it follows that the image under ϕ of any unstable leaf of G_{M} lies a bounded Hausdorff distance from an unstable leaf of G_{N}, that is, ϕ coarsely preserves the unstable foliations. A similar argument using − signs shows that ϕ coarsely preserves stable foliations. By taking intersections of stable and unstable leaves it follows that ϕ coarsely preserves center foliations.

The final statements about dimensions follow from the fact that dimension is a quasi-isometry invariant, for leaves of the foliations in question; see [Ger93] or [BW97].

It remains to prove the claim. Applying Lemma 5.1, we have an induced time change h: R → R which is a (K, C′)-quasi-isometry with Hausdorff constant A, where C′ depends only on K, C, A. Furthermore by Lemma 5.1 and the comments following it, the map h is coarsely increasing: there exists L = L(K, C, A) > 0 such that if t ≥ s + L then h(t) > h(s).

We can furthermore increase L, depending only on K, C′, A, so that:

if t′ ≥ t + L, x ∈ P_{t′}, y ∈ P_{t}, ϕ(x) ∈ P_{s′} and ϕ(y) ∈ P_{s}, then s′ ≥ s + 1. (5.1)

In fact taking L > (C′ + 2A + 1)K will do, for then we have

h(t′) ≥ h(t) + (t′ − t)/K − C′ ≥ h(t′) + L/K − C′
≥ h(t) + 2A + 1

and, since P_{s′} is A-Hausdorff close to P_{h(t′)} and P_{s} is A-Hausdorff close to P_{h(t)}, it follows that s′ ≥ s + 1.

To prove the claim, we first show that ϕ(γ) is Hausdorff close to some pseudo-orbit in G_{N}, and then we apply the Shadowing Lemma to show that the pseudo-orbit lies in a bounded neighborhood of some center leaf.

To be more precise, fix a point x_{0} ∈ γ and consider the sequence x_{i} = Φ_{i·L}(x_{0}) for i ∈ Z. Let y_{i} = ϕ(x_{i}), and let s_{i} be such that y_{i} ∈ P_{s_{i}}. From (5.1) it follows that s_{i+1} ≥ s_{i} + 1. Let t_{i} = s_{i+1} − s_{i} ≥ 1.

We claim that there exists ϵ > 0, depending ultimately only on K, C, so that (y_{i} · [0, t_{i}]) is an ϵ, 1 pseudo-orbit; in other words, d(y_{i} · t_{i}, y_{i+1}) is bounded. To see why, first note that

d(y_{i} · t_{i}, y_{i}) = t_{i} = s_{i+1} − s_{i}
≤ 2A + h(L · (i + 1)) − h(L · i)
≤ 2A + KL + C′

and then

d(y_{i}, y_{i+1}) ≤ K · d(x_{i}, x_{i+1}) + C ≤ KL + C

so we may take ϵ = 2A + 2KL + C + C′.

Applying the Shadowing Lemma, there exists β, ϵ′, T′ such that (y_{i} · [0, t_{i}]) is β-Hausdorff close to an ϵ′, T′ pseudo-orbit (y′_{i}, [0, t_{i}]) contained in some center leaf of G_{N}. On the other hand, since every point of γ is within distance L of some x_{i}, it follows that ϕ(γ) is uniformly Hausdorff close to (y_{i}·[0, t_{i}]), and so it is also uniformly close to the pseudo-orbit (y′_{i}·[0, t_{i}]). ♢

5.4 Step 2: Time rigidity

The main result of this subsection says that a horizontal respecting quasiisometry has an induced time change function which is be affine.

Proposition 5.8 (Time rigidity). Consider the Lie groups G_{M}, G_{N} where M, N ∈ GL_{×}(n, R) each have an eigenvalue of absolute value greater than 1. Then there exists m ∈ R_{+} with the following properties. For all K ≥ 1, C, A ≥ 0 there exists A′ ≥ 0 such that if ϕ: G_{M} → G_{N} is a (K, C) quasiisometry which coarsely respects horizontal foliations and their transverse orientations, with an induced time change of Hausdorff constant A, then there exists b ∈ R such that h(t) = mt + b is an induced time change with Hausdorff constant A′. In fact, m can be computed as follows: Let α (resp. β) be the least eigenvalue greater than 1 of the absolute Jordan form of M (resp. N); the numbers α, β exist by the assumption on eigenvalues. Then m = log α/ log β.

Remarks

1. In the case of self-quasi-isometries of Aff(R) = G_{(e^{1})} = H^{2} which coarsely respect the horizontal foliation, this result is part of Proposition 5.3 of [FM98], where the conclusion is that the induced time change is a translation of R.

2. One of the delicate points in Gromov’s development of the infδim invariant is the rescaling problem discussed at the beginning of Section 7.C_{1} of [Gro93]: rates of exponential growth change when the parameter is rescaled. Time Rigidity allows us to avoid the rescaling problem altogether, by showing that the time parameter is “natural” with respect to quasi-isometries.

Proof. This proof will define a sequence of constants which will depend on K, C, A and on the matrices M and N. We will indicate the dependence on K, C, A by writing, for example, C_{1} = C_{1}(K, C, A), but we will suppress the dependence on M, N. Although each constant in the sequence will depend on previous constants in the sequence, by induction it will ultimately depend only on K, C, A, M, N.

Claim 5.9. For each fixed time t_{0}, and for each t ≤ t_{0}, we have

h(t) ≥ m(t − t_{0}) + h(t_{0}) − C_{1}

for some C_{1} = C_{1}(K, C, A) ≥ 0.

Accepting this claim for the moment, we prove the proposition. The idea is simply that the conclusion of the claim, applied to both h and its coarse inverse h̄, with t_{0} → +∞, implies the proposition.

Let s be a time parameter for G_{N}. Let ϕ̄: G_{N} → G_{M} be a coarse inverse for ϕ, also a quasi-isometry which coarsely respects the horizontal foliations and their transverse orientations, and with an induced time change h̄(s). The constants for ϕ̄ and h̄ depend only on K, C, A. The claim therefore applies as well to h̄ and we obtain, for each fixed time s_{0} and each s ≤ s_{0},

h̄(s) ≥ \frac{1}{m}(s − s_{0}) + h̄(s_{0}) − C_{2}

for some C_{2} = C_{2}(K, C, A) ≥ 0.

It is clear h̄ is a coarse inverse for h, that is:

|h̄(h(t)) − t| ≤ C_{3}, |h(h̄(s)) − s| ≤ C_{3}

for some C_{3} = C_{3}(K, C, A) ≥ 0.

Also, by Lemma 5.1 and the comments after it, the map h is coarsely increasing: there exists L = L(K, C, A) ≥ 0 such that if t′ > t + L then h(t′) > h(t).

We reverse the inequality in the claim as follows. Fix t_{0}. Let s_{0} = t_{0}. Consider for the moment some t ≤ t_{0} − L. Letting s = h(t) it follows that s ≤ s_{0} and so we have

h̄(h(t)) ≥ \frac{1}{m}(h(t) − h(t_{0})) + h̄(h(t_{0})) − C_{2}

But t + C_{3} ≥ h̄(h(t)) and h̄(h(t_{0})) ≥ t_{0} − C_{3} and so we obtain

t ≥ \frac{1}{m}(h(t) − h(t_{0})) + t_{0} − (2C_{3} + C_{2})
h(t) ≤ m(t − t_{0}) + h(t_{0}) + m(2C_{3} + C_{2})

This has been derived only for t ≤ t_{0} −L, but for t_{0} −L ≤ t ≤ t_{0} we obtain a similar inequality with another constant in place of m(2C_{3} +C_{2}). Therefore, for all t ≤ t_{0} we obtain

m(t − t_{0}) + h(t_{0}) − C_{4} ≤ h(t) ≤ m(t − t_{0}) + h(t_{0}) + C_{4}

for some C_{4} = C_{4}(K, C, A). Note that this is true for all t_{0}, with C_{4} independent of t_{0}.

In particular, taking t_{0} = 0, for all t ≤ 0 we obtain

mt + h(0) − C_{4} ≤ h(t) ≤ mt + h(0) + C_{4}

Now take any t_{1} ≥ 0, and since 0 ≤ t_{1} we obtain

m(0 − t_{1}) + h(t_{1}) − C_{4} ≤ h(0) ≤ m(0 − t_{1}) + h(t_{1}) + C_{4}

and so

mt_{1} + h(0) − C_{4} ≤ h(t_{1}) ≤ mt_{1} + h(0) + C_{4}

Taking b = h(0), this proves that mt + b is an induced time change for ϕ, with Hausdorff constant A′ = C_{4} + A.

Now we turn to the proof of Claim 5.9.

Let M^{t} = \overline{M}^{t}Q^{t}, N^{t} = \overline{N}^{t}Q′^{t} be the real Jordan forms. Let U (resp. U′) be the root space with eigenvalue 1 for \overline{M} (resp. \overline{N}). Let W (resp. W′) be the direct sum of root spaces with eigenvalue ≥ 1 for \overline{M} (resp. \overline{N}). Recall that α is the smallest eigenvalue > 1 for \overline{M}, and β is the smallest eigenvalue > 1 for \overline{N}. Let V be the direct sum of U and the eigenspace with eigenvalue α for \overline{M}. We have U ⊂ V ⊂ W; let F(U), F(V), F(W) be the corresponding foliations of G_{M} ≈ R^{n} × R, whose leaves are parallel to U × R, V × R, W × R respectively. We also have U′ ⊂ W′; let F(U′), F(W′) be the corresponding foliations of G_{N}.

Here is the idea for proving Claim 5.9. Each leaf of F(V) is foliated by leaves of F(U). Because V is the direct sum of U with the α eigenspace of \overline{M}, it follows that as t → −∞ distinct leaves of F(U) in F(V) diverge from each other exactly as α^{−t}, measured in the time t horizontal plane of G_{M}. This is a consequence of the Exponential Lower Bound and the Exponential·Polynomial Upper Bound in Proposition 3.2; notice that it is critical here that V not be the direct sum of U with the α root space, for then Exponential·Polynomial Upper Bound would be at best α^{−t} times some polynomial, which would mess up the following calculations. Mapping over via the quasi-isometry ϕ: G_{M} → G_{N}, distinct leaves of F(U) in a single leaf of F(V) must (coarsely) map to distinct leaves of F(U′) in a single leaf of F(W′), which as s → −∞ diverge from each other at least as fast as β^{−s}, by the Exponential Lower Bound. The time change map t ↦ h(t) = s therefore cannot grow slower than s = \frac{log α}{log β} · t, as t → −∞.

To make this precise, pick a leaf L_{V} of F(V), contained in some leaf L_{W} of F(W). We use the symbol γ to denote a general leaf of F(U), which we will typically take to be a subset of L_{V}. By Proposition 5.4, there exists a leaf L_{W′} of F(W′) such that

d_{H}(f(L_{W}), L_{W′}) ≤ C_{5} = C_{5}(K, C, A)

and for each leaf γ of F(U) there exists a leaf γ′ of F(U′) such that

d_{H}(f(γ), γ′) ≤ C_{5}

Moreover, if γ ⊂ L_{V} then γ′ ⊂ L_{W′}, because L_{V} ⊂ L_{W} and so γ′ stays in a bounded neighborhood of L_{W′}, but any leaf of F(U′) which is not a subset of L_{W′} has points which are arbitrarily far from L_{W′}.

Let P_{t} be the horizontal subset of G_{M} at height t ∈ R, and let d_{t} denote Hausdorff distance in P_{t} between closed subsets of P_{t}. Let P′_{s} be the horizontal subset of G_{N} at height s ∈ R, and let d′_{s} denote Hausdorff distance in P′_{s}.

Since the Hausdorff distance in G_{N} between ϕ(P_{t}) and P′_{h(t)} is at most A, the vertical projection from ϕ(P_{t}) to P′_{h(t)} induces a quasi-isometry between P_{t} and P′_{h(t)}; the multiplicative constant of this quasi-isometry is K, and its additive constant depends only on K, C, A. It follows that there exists a “coarseness constant” C_{6} = C_{6}(K, C, A) so that for any t, and for any x, y ∈ P_{t} with d_{t}(x, y) ≥ C_{6}, if x′, y′ ∈ P′_{h(t)} are the vertical projections of ϕ(x), ϕ(y) then

\frac{1}{2K} d_{t}(x, y) ≤ d′_{h(t)}(x′, y′) ≤ 2Kd_{t}(x, y)  (5.2)

To prove Claim 5.9, fix a time t_{0} and let s_{0} = h(t_{0}). Let γ_{1}, γ_{2} be two leaves of F(U) contained in L_{V}, and let γ′_{i} be the unique leaf of F(U′) within bounded Hausdorff distance of ϕ(γ_{i}); this bound depends only on K, C, A, as shown in Proposition 5.4.

In G_{M}, apply the Exponential Lower Bound and the Exponential·Polynomial Upper Bound of Proposition 3.2, and so for all t ≤ t_{0} we have

A · α^{−t+t_{0}}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}}) ≤ d_{t}(γ_{1} ∩ P_{t}, γ_{2} ∩ P_{t})
≤ B · α^{−t+t_{0}}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}})

where A, B depend only on G_{M} (note that t = t_{0} gives A ≤ 1 ≤ B).

We want the distance between γ_{1} and γ_{2} in P_{t} to be greater than the coarseness constant C_{6}, for each t ≤ t_{0}, in order that property (5.2) may be applied. We therefore impose a condition on γ_{1} and γ_{2}, namely that

d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}}) ≥ \frac{C_{6}}{A}

which implies, for all t ≤ t_{0}, that

d_{t}(γ_{1} ∩ P_{t}, γ_{2} ∩ P_{t}) ≥ C_{6}

and so

\frac{1}{2K} · d_{t}(γ_{1} ∩ P_{t}, γ_{2} ∩ P_{t}) ≤ d′_{h(t)}(γ′_{1} ∩ P′_{h(t)}, γ′_{2} ∩ P′_{h(t)})
≤ 2K · d_{t}(γ_{1} ∩ P_{t}, γ_{2} ∩ P_{t})

which implies

\frac{A}{2K} α^{−t+t_{0}}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}}) ≤ d′_{h(t)}(γ′_{1} ∩ P′_{h(t)}, γ′_{2} ∩ P′_{h(t)})

≤ 2BKα^{−t+t_{0}}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}})

Next, applying the Exponential Lower Bound of Proposition 3.2 in G_{N}, for each s ≤ s_{0} we have

d′_{s}(γ′_{1} ∩ P′_{s}, γ′_{2} ∩ P′_{s}) ≥ A · β^{−s+s_{0}}d′_{s_{0}}(γ′_{1} ∩ P′_{s_{0}}, γ′_{2} ∩ P′_{s_{0}})

Taking s = h(t), and using the fact that s_{0} = h(t_{0}), this implies

β^{−h(t)+h(t_{0})}d′_{h(t_{0})}(γ′_{1}∩P′_{h(t_{0})}, γ′_{2}∩P′_{h(t_{0})}) ≤ \frac{2BK}{A} ·α^{−t+t_{0}} ·d_{t_{0}}(γ_{1}∩P_{t_{0}}, γ_{2}∩P_{t_{0}})

Therefore,

β^{−h(t)+h(t_{0})}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}}) ≤ \frac{4BK^{2}}{A} α^{−t+t_{0}}d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}})

Now divide both sides by d_{t_{0}}(γ_{1} ∩ P_{t_{0}}, γ_{2} ∩ P_{t_{0}}), and take logarithms, obtaining

(−h(t) + h(t_{0})) log(β) ≤ log (\frac{4BK^{2}}{A} ) + (−t + t_{0}) log(α)

and so

h(t) ≥ \frac{log(α)}{log(β)}(t − t_{0}) + h(t_{0}) − \frac{log (\frac{4BK^{2}}{A} )}{log(β)}

proving Claim 5.9 and therefore completing the proof of Proposition 5.8. ♢

5.5 Interlude: The induced boundary map

The upper boundary ∂^{u}G_{M} is defined to be the leaf space of the weak stable foliation, identified with V^{−}. The lower boundary ∂_{ℓ}G_{M} is the leaf space of the weak unstable foliation, identified with V^{+}. The internal boundary ∂_{int}G_{M} is defined as

∂_{int}G_{M} = ∂_{ℓ}G_{M} × ∂^{u}G_{M} = V^{+} × V^{−} ≈ R^{n}/V^{0}

which is identified with the leaf space of the center foliation.

As a consequence of Proposition 5.4, a quasi-isometry ϕ: G_{M} → G_{L} which respects the transversely oriented horizontal foliations induces a bijection

∂_{int}ϕ: ∂_{int}G_{M} → ∂_{int}G_{L}

which preserves the factors, that is,

∂_{int}ϕ = ∂_{l}ϕ × ∂^{u}ϕ: ∂_{ℓ}G_{M} × ∂^{u}G_{M} → ∂_{ℓ}G_{L} × ∂^{u}G_{L}

Recall the 1-parameter family of metrics d_{M,t} on R^{n} given by the quadratic form Q_{M,t} = (M^{−t})^{T} M^{−t}. The internal boundary ∂_{int}G_{M} is identified with R^{n}/V^{0} and with V^{+} × V^{−}, and we consider two 1-parameter families of metrics.

First, regarding points of R^{n}/V^{0} as affine subspaces parallel to V^{0}, there is a 1-parameter family of Hausdorff metrics induced from d_{M,t} which we denote dh_{M,t}. Second, restrict the action of M^{t} to the subspace V^{+} × V^{−} to get a 1-parameter subgroup of GL(V^{+} × V^{−}), and by choosing a basis for V^{+} × V^{−} we obtain a 1-parameter subgroup M̂^{t} of GL(k, R), where k is the dimension of V^{+} ×V^{−}. We obtain a 1-parameter family of metrics d_{M̂,t}. There is a canonical identification V^{+} × V^{−} ≈ R^{n}/V^{0}, and with respect to this identification the metrics d_{M̂,t} and dh_{M,t} are bilipschitz equivalent, with a uniform bilipschitz constant independent of t.

Note that the absolute Jordan form of M̂ is identical with the nonunipotent part of the absolute Jordan form of M, and similarly for N.

Lemma 5.10. Given two 1-parameter subgroups M^{t}, N^{t} of GL(n, R), for all K ≥ 1, C, A ≥ 0, there exist K′ ≥ 1, C′ ≥ 0 with the following properties. If ϕ: G_{M} → G_{N} is a K, C quasi-isometry which coarsely respects the transversely oriented horizontal foliations, with Hausdorff constant A, then for every t ∈ R the induced map ∂_{int}ϕ: ∂_{int}G_{M} → ∂_{int}G_{N} is a K′, C′ quasi-isometry from the metric d_{M̂,t} to the metric d_{N̂,h(t)}.

Proof. With what we know, the proof is mostly a matter of chasing through definitions.

The quasi-isometry ϕ is a bounded distance from a quasi-isometry ψ: G_{M} → G_{N} which takes the horizontal leaf P_{t} to the horizontal leaf P′_{h(t)}, and which simultaneously takes center leaves of G_{M} to center leaves of G_{N}. Now restrict the center foliations of G_{M}, G_{N} to P_{t}, P′_{h(t)}, and denote the respective leaf spaces as Q_{t}, Q′_{h(t)}.

In order to apply Lemma 2.1, consider each horizontal leaf P_{t} of G_{M} as a geodesic metric space, with respect to the Riemannian metric induced by restriction from G_{M}. The inclusion map P_{t} ↪ G_{M} is evidently (1, 0) coarsely lipschitz, and it is uniformly proper, with a uniformity function s(r) = a^{r} where a > 1 is larger than the maximum of the absolute values of all eigenvalues of M and their multiplicative inverses. Note in particular that the coarse Lipschitz constants and the uniformity functions of the maps P_{t} ↪ G_{M} depend only on K, C, A and on the matrix M, but not on t. Similar remarks apply to the inclusion map P′_{h(t)} ↪ G_{N}. Applying Lemma 2.1, restricting ψ to P_{t} results in a map ψ_{t}: P_{t} ↦ P′_{h(t)} which is a quasi-isometry. There is in turn an induced map θ_{t}: Q_{t} ↦ Q′_{h(t)} which is a quasi-isometry with respect to the associated Hausdorff metric. The quasi-isometry constants of the maps ψ_{t} and θ_{t} depend only on K, C, A.

Now consider the coordinate identifications G_{M} ≈ R^{n} × R, G_{N} ≈ R^{n} × R. By construction of the left invariant metrics, for each t the space P_{t} is identified with R^{n} × t ≈ R^{n} with the metric d_{M,t}, and the space P′_{h(t)} is identified with R^{n} with metric d_{N,h(t)}, and so the maps ψ_{t}: R^{n} → R^{n} are uniform quasi-isometries from d_{M,t} to d_{N,h(t)} for all t. Also, Q_{t} is identified with R^{n}/V_{M}^{0} with the associated Hausdorff metric dh_{M,t}, and Q′_{h(t)} is identified with R^{n}/V_{N}^{0} with the associated Hausdorff metric dh_{N,h(t)}, and so the maps θ_{t}: R^{n}/V_{M}^{0} → R^{n}/V_{N}^{0} are uniform quasi-isometries from dh_{M,t} to dh_{N,h(t)} for all t. This implies that θ_{t}: V_{M}^{+} × V_{M}^{−} → V_{N}^{+} × V_{N}^{−} is a quasiisometry from d_{M̂,t} to d_{N̂,t} for all t. But for all t the map θ_{t} is identical to ∂_{int}ϕ: ∂_{int}G_{M} → ∂_{int}G_{N}, proving the lemma. ♢

5.6 Step 3: Reduction of Theorem 5.2 to Theorem 5.11

Assume the hypotheses of Theorem 5.2, namely that we have 1-parameter subgroups M^{t}, N^{t}, and a quasi-isometry ϕ: G_{M} → G_{N} which coarsely respects the transversely oriented horizontal foliations. Applying Proposition 5.8, there is an induced time change of the form h(t) = mt + b with m > 0. Applying Proposition 4.1, there is a horizontal respecting quasi-isometry G_{N} → G_{N^{m}} with an induced time change of the form s ↦ s/m. By composition we obtain a horizontal respecting quasi-isometry G_{M} → G_{N^{m}} with an induced time change of the form t ↦ t+b′. Changing the coordinates in G_{M} by a translation of the time coordinate t, we have a horizontal respecting quasi-isometry G_{M} → G_{N^{m}} for which the identity map t ↦ t is an induced time change. Applying Lemma 5.10, we obtain a map ∂_{int}ϕ: R^{n} → R^{n} which, for each t, is a (K′, C′)-quasi-isometry from d_{M̂,t} to d_{N̂^{m},t}.

Now apply the following theorem (with N in place of N^{m}), which will be proved in the next section:

Theorem 5.11 (One-parameter subgroup rigidity). Let M^{t}, N^{t} be 1-parameter subgroups of GL(n, R), such that M = M^{1} and N = N^{1} have no eigenvalues on the unit circle. If there exists a bijection f: R^{n} → R^{n} and constants K ≥ 1, C ≥ 0 such that for each t ∈ R and p, q ∈ R^{n} we have

−C + \frac{1}{K} d_{M,t}(p, q) ≤ d_{N,t}(f(p), f(q)) ≤ Kd_{M,t}(p, q) + C

then M and N have the same absolute Jordan form.

Returning to the previous discussion, this theorem allows us to conclude that M̂ and N̂^{m} have the same absolute Jordan form, and so the nonunipotent parts of the absolute Jordan forms of M, N^{m} are identical. We have already proved in Corollary 5.6 that the unipotent parts are identical, and so M and N^{m} have the same absolute Jordan forms, finishing the proof of Theorem 5.2. ♢

6 Dynamics of G_{M}, Part II:

1-parameter subgroup rigidity

In this section we give a proof of Theorem 5.11.

Let M^{t}, N^{t} be 1-parameter subgroups of GL(n, R) with no eigenvalues on the unit circle. Let M^{t} = \overline{M}^{t}P^{t}, N^{t} = \overline{N}^{t}Q^{t} be the real Jordan forms, so \overline{M} and \overline{N} have all positive eigenvalues, none equal to 1. Let f: R^{n} → R^{n} be a bijection which satisfies

−C + \frac{1}{K} d_{M,t}(p, q) ≤ d_{N,t}(f(p), f(q)) ≤ Kd_{M,t}(p, q) + C  (6.1)

for all t ∈ R, p, q ∈ R^{n}.

The bijection f: R^{n} → R^{n} must in fact be a homeomorphism. To see why, for each p ∈ R^{n}, R > 0, T > 0 let

F_{p,R}(T) = {q ∈ R^{n} | d_{M,t}(p, q) < R for all t ∈ (−T, T)}

In other words, F_{p,R}(T) is the intersection of open balls of radius R about p in each of the metrics d_{M,t}, for t ∈ (−T, T). Since the eigenvalues of \overline{M} are all positive real numbers, none equal to 1, it follows from Proposition 3.2 that for each p ∈ R^{n} and each R > 0 the collection of sets F_{p,R}(T) as T ranges in (0, ∞) is a neighborhood basis for p, in the standard topology on R^{n}. We define a similar neighborhood basis using matrix N, denoted G_{p,R}(T). Since f(F_{p,R}(T)) ⊆ G_{f(p),KR+C}(T) for each p ∈ R^{n}, R > 0, T > 0, it follows that f is continuous. The same argument applies to f^{−1}, and so f is a homeomorphism.

The idea of the proof of Theorem 5.11 is to show that f respects certain “flags of foliations” which are closely related to the Jordan decompositions of R^{n} with respect to M^{t} and N^{t}. We begin by setting up the notation needed to define and study these foliations.

Definition (Flags of foliations). If V is a vector subspace of R^{n}, define a foliation F(V) of R^{n} whose leaves are the affine subspaces of R^{n} parallel to V. Given a flag of subspaces V_{1} ⊂ ⋯ ⊂ V_{r}, it follows that if 1 ≤ i < j ≤ r then each leaf of F(V_{i}) is contained in some leaf of F(V_{j}); we denote this relation by saying that F(V_{1}) ≺ ⋯ ≺ F(V_{r}) is a flag of foliations of R^{n}.

Recall the root space decompositions of R^{n} with respect to \overline{M} and \overline{N}. We denote the eigenvalues of \overline{M} and \overline{N} by

0 < μ_{m}^{−} < ⋯ < μ_{1}^{−} < 1 < μ_{1}^{+} < ⋯ < μ_{r}^{+}

and

0 < ν_{n}^{−} < ⋯ < ν_{1}^{−} < 1 < ν_{1}^{+} < ⋯ < ν_{s}^{+}

respectively. The corresponding root space decompositions are denoted

V_{m}^{−} ⊕ ⋯ ⊕ V_{1}^{−} ⊕ V_{1}^{+} ⊕ ⋯ ⊕ V_{r}^{+}

and

W_{n}^{−} ⊕ ⋯ ⊕ W_{1}^{−} ⊕ W_{1}^{+} ⊕ ⋯ ⊕ W_{s}^{+}.

As in section 4 we set

V^{−} = V_{m}^{−} ⊕ ⋯ ⊕ V_{1}^{−},  V^{+} = V_{1}^{+} ⊕ ⋯ ⊕ V_{r}^{+}
W^{−} = W_{n}^{−} ⊕ ⋯ ⊕ W_{1}^{−},  W^{+} = W_{1}^{+} ⊕ ⋯ ⊕ W_{s}^{+}

Define the root space flags

U_{i}^{−} = V_{i}^{−} ⊕ ⋯ ⊕ V_{1}^{−}, i = 1, …, m
U_{j}^{+} = V_{1}^{+} ⊕ ⋯ ⊕ V_{j}^{+}, j = 1, …, r
Y_{i}^{−} = W_{i}^{−} ⊕ ⋯ ⊕ W_{1}^{−}, i = 1, …, n
Y_{j}^{+} = W_{1}^{+} ⊕ ⋯ ⊕ W_{j}^{+}, j = 1, …, s

and by convention we take U_{0}^{−}, U_{0}^{+}, Y_{0}^{−}, Y_{0}^{+} each to be the trivial subspace. Associated to the root space flags we have root space foliation flags

F(U_{1}^{−}) ≺ ⋯ ≺ F(U_{m}^{−}) = F(V^{−})
F(U_{1}^{+}) ≺ ⋯ ≺ F(U_{r}^{+}) = F(V^{+})
F(Y_{1}^{−}) ≺ ⋯ ≺ F(Y_{n}^{−}) = F(W^{−})
F(Y_{1}^{+}) ≺ ⋯ ≺ F(Y_{s}^{+}) = F(W^{+})

Step 1: f respects contracting and expanding foliations. First we show that f(F(V^{−})) = F(W^{−}) and f(F(V^{+})) = F(W^{+}).

Given p, q ∈ R^{n} we have the following chain of equivalences:

1. p, q are in the same leaf of F(V^{+}).

2. d_{M,t}(p, q) = ∥M^{−t}(p − q)∥ → 0 as t → +∞.

3. d_{M,t}(p, q) is bounded for t ∈ [0, +∞).

4. d_{N,t}(f(p), f(q)) is bounded for t ∈ [0, +∞).

5. d_{N,t}(f(p), f(q)) = ∥N^{−t}(f(p) − f(q))∥ → 0 as t → +∞.

6. f(p), f(q) are in the same leaf of F(W^{+}). The equivalence of (1–3) follows from Proposition 3.2, and similarly for (4–6). The equivalence of (3) and (4) follows from (6.1). This shows f(F(V^{+})) = F(W^{+}). A similar argument with t ∈ (−∞, 0] shows f(F(V^{−})) = F(W^{−}). Step 2: f respects root space foliation flags. Next we show: Claim 6.1. f: R^{n} → R^{n} respects the root space foliation flags, and corresponding root spaces have the same eigenvalues. More precisely we have:

1. r = s.

2. μ_{j}^{+} = ν_{j}^{+} for j = 1, …, r.

3. f(F(U_{j}^{+})) = F(U_{j}^{+}) for j = 1, …, r.

4. m = n.

5. μ_{i}^{−} = ν_{i}^{−} for i = 1, …, m.

6. f(F(U_{i}^{−})) = F(Y_{i}^{−}) for i = 1, …, m. It follows that M, N have the same eigenvalues with the same multiplicities.

We give the proof of 1,2,3; the proof of 4,5,6 is similar.

We know by Step 1 that f(F(V^{+})) = F(W^{+}). Consider points p, q in the same leaf of F(V^{+}), so f(p), f(q) are in the same leaf of F(W^{+}). From Proposition 3.2 it follows that as t → −∞ both of the quantities d_{M,t}(p, q) and d_{N,t}(f(p), f(q)) approach +∞. It follows that for sufficiently large t, in the inequality (6.1) we can absorb the additive constant C, yielding

\frac{1}{K + 1}d_{M,t}(p, q) ≤ d_{N,t}(f(p), f(q)) ≤ (K + 1)d_{M,t}(p, q)  (6.2)

Define displacement vectors v = p − q, w = f(p) − f(q). Taking natural logarithms, dividing by t, and taking limsup, we have

lim sup_{t→−∞} \frac{log(d_{M,t}(p, q))}{t} = lim sup_{t→−∞} \frac{log(d_{N,t}(f(p), f(q)))}{t}
lim sup_{t→+∞} \frac{log ∥M^{t}v∥}{t} = lim sup_{t→+∞} \frac{log ∥N^{t}w∥}{t}  (6.3)

To evaluate these limits, let I(p, q) = I(v) be the unique integer such that

v ∈ U_{I(v)}^{+} − U_{I(v)−1}^{+}

or, equivalently, the unique integer such that p, q are in the same leaf of F(U_{I(p,q)}^{+}) but not in the same leaf of F(U_{I(p,q)−1}^{+}) (recall the convention that U_{0}^{+} = 0, and so I(p, q) = 0 if and only if p = q). Define J(f(p), f(q)) = J(w) similarly by

w ∈ Y_{J(w)}^{+} − Y_{J(w)−1}^{+}

Applying Proposition 3.2 we have

lim sup_{t→+∞} \frac{log ∥M^{t}v∥}{t} = μ_{I(v)}^{+}
lim sup_{t→+∞} \frac{log ∥N^{t}w∥}{t} = ν_{J(w)}^{+}

and so by (6.3) we have

μ_{I(p,q)}^{+} = μ_{I(v)}^{+} = ν_{J(w)}^{+} = ν_{J(f(p),f(q))}^{+}

Since f is a bijection from each leaf of F(V^{+}) to some leaf of F(W^{+}), items (1) and (2) of Claim 6.1 now follow, and it also follows that

I(p, q) = J(f(p), f(q))

for all p, q contained in the same leaf of F(V^{+}).

We now prove item (3) of Claim 6.1 by induction on j. If p, q are in the same leaf of F(U_{1}^{+}) then I(p, q) = 1 and so J(p, q) = 1 which implies that f(p), f(q) are in the same leaf of F(Y_{1}^{+}). A similar argument with f^{−1} proves that f(F(U_{1}^{+})) = F(Y_{1}^{+}), proving the base step of the induction. Now assume that f(F(U_{j}^{+})) = F(Y_{j}^{+}), and suppose p, q are in the same leaf of F(U_{j+1}^{+}). There are two cases to consider. If p, q lie in the same leaf of F(U_{j}^{+}) then by the induction hypothesis f(p), f(q) lie in the same leaf of F(Y_{j}^{+}), in particular they lie in the same leaf of F(Y_{j+1}^{+}). If p, q do not lie in the same leaf of F(U_{j}^{+}) then I(p, q) = j + 1 and so J(f(p), f(q)) = j + 1 and so f(p), f(q) lie on the same leaf of F(Y_{j+1}^{+}). A similar argument with f^{−1} shows that f(F(U_{j+1}^{+})) = Y_{j+1}^{+}, completing the induction.

As mentioned earlier, (4–6) are proved similarly, completing the proof of Claim 6.1.

Step 3: f respects Jordan foliation flags.

From Step 2, for each fixed j = 1, …, r the matrices M, N have μ_{j}^{+} root spaces V_{j}^{+}, W_{j}^{+} respectively. As part of their root space flags we have

U_{j}^{+} = U_{j−1}^{+} ⊕ V_{j}^{+}
Y_{j}^{+} = Y_{j−1}^{+} ⊕ W_{j}^{+}

Let c_{j} be the index of nilpotency of μ_{j} · I − M, and let d_{j} be the index of nilpotency of μ_{j} · I − N. Then we have Jordan filtrations

V_{j,0}^{+} ⊂ ⋯ ⊂ V_{j,c_{j}}^{+} = V_{j}^{+}
W_{j,0}^{+} ⊂ ⋯ ⊂ W_{j,d_{j}}^{+} = W_{j}^{+}

and we set U_{j,k}^{+} = U_{j−1}^{+} ⊕V_{j,k}^{+} and Y_{j,k}^{+} = Y_{j−1}^{+} ⊕W_{j,k}^{+}, yielding subspace flags

U_{j−1}^{+} ⊂ U_{j,0}^{+} ⊂ ⋯ ⊂ U_{j,c_{j}−1}^{+} = U_{j}^{+}
Y_{j−1}^{+} ⊂ Y_{j,0}^{+} ⊂ ⋯ ⊂ Y_{j,d_{j}−1}^{+} = Y_{j}^{+}

Corresponding to these subspace flags are foliation flags,

F(U_{j−1}^{+}) ≺ F(U_{j,0}^{+}) ≺ ⋯ ≺ F(U_{j,c_{j}−1}^{+}) = F(U_{j}^{+})
F(Y_{j−1}^{+}) ≺ F(Y_{j,0}^{+}) ≺ ⋯ ≺ F(Y_{j,d_{j}−1}^{+}) = F(Y_{j}^{+})

called the expanding Jordan foliation flags associated to the corresponding root space foliations F(U_{j}^{+}), F(Y_{j}^{+}) respectively. The contracting Jordan foliation flags associated to each root space foliation F(U_{i}^{−}), F(Y_{i}^{−}) are similarly defined.

Claim 6.2. f: R^{n} → R^{n} respects the Jordan foliation flags associated to corresponding root space foliations. More precisely, for each j = 1, …, r we have:

1. c_{j} = d_{j}.

2. f(F(U_{j,k}^{+})) = F(Y_{j,k}^{+}) for k = 0, …, c_{j} − 1.

and similarly for the contracting Jordan foliation flags.

From this claim, for each j = 1, …, r it immediately follows that \overline{M}, \overline{N} have the same Jordan blocks with eigenvalue μ_{j}^{+}, and so the expanding parts of the Jordan forms for \overline{M}, \overline{N} are identical; similarly for the contracting parts. Since M, N have no eigenvalues on the unit circle, it now follows that M, N have the same absolute Jordan forms, completing the proof of Theorem 5.11.

Proof of Claim 6.2. Consider p, q ∈ R^{n} in the same leaf of F(U_{j}^{+}) but not in the same leaf of F(U_{j−1}^{+}), and so f(p), f(q) are in the same leaf of F(Y_{j}^{+}) but not in the same leaf of F(Y_{j−1}^{+}). Define displacement vectors v = p − q, w = f(p) − f(q), and so v ∈ U_{j}^{+} − U_{j−1}^{+} and w ∈ Y_{j}^{+} − Y_{j−1}^{+}. We know that

lim sup_{t→+∞} \frac{log ∥M^{t}v∥}{t} = \frac{log ∥N^{t}w∥}{t} = μ_{j}^{+}

We also know that (6.2) is true for t sufficiently close to −∞, and so for t sufficiently close to +∞ we have

\frac{1}{K + 1}∥M^{t}v∥ ≤ ∥N^{t}w∥ ≤ (K + 1)∥M^{t}v∥  (6.4)

By induction on k = 0, 1, …, we shall prove that v ∈ U_{j,k}^{+} if and only if w ∈ Y_{j,k}^{+}, or equivalently that f(F(U_{j,k}^{+})) = F(Y_{j,k}^{+}).

For the basis step k = 0, divide the inequality (6.4) by μ^{t} to obtain, for all t sufficiently close to +∞:

\frac{1}{K + 1} \frac{∥M^{t}v∥}{μ^{t}} ≤ \frac{∥N^{t}w∥}{μ^{t}} ≤ (K + 1)\frac{∥M^{t}v∥}{μ^{t}}  (6.5)

By the Exponential Lower Bound and the Exponential·Polynomial Upper

Bound of Proposition 3.2, the quantity \frac{∥M^{t}v∥}{μ^{t}} is bounded for t ≥ 0 if and

only if v ∈ U_{j,0}^{+}; and the quantity \frac{∥N^{t}w∥}{μ^{t}} is bounded on t ≥ 0 if and only

if w ∈ Y_{j,0}^{+}. However by (6.5) the boundedness of these two quantities on t ≥ 0 are equivalent.

For the induction step, assume that f(F(U_{j,k−1}^{+})) = F(Y_{j,k−1}^{+}), that is, v ∈ U_{j,k−1}^{+} if and only if w ∈ Y_{j,k−1}^{+}. We must prove that v ∈ U_{j,k}^{+} − U_{j,k−1}^{+} if and only if w ∈ Y_{j,k}^{+} − Y_{j,k−1}^{+}. From (6.4), for t sufficiently close to +∞ we have

\frac{1}{K + 1} \frac{∥M^{t}v∥}{μ^{t}t^{k}} ≤ \frac{∥N^{t}w∥}{μ^{t}t^{k}} ≤ (K + 1)\frac{∥M^{t}v∥}{μ^{t}t^{k}}  (6.6)

and

\frac{1}{K + 1} \frac{∥M^{t}v∥}{μ^{t}t^{k−1}} ≤ \frac{∥N^{t}w∥}{μ^{t}t^{k−1}} ≤ (K + 1)\frac{∥M^{t}v∥}{μ^{t}t^{k−1}}  (6.7)

By the Exponential·Polynomial Upper and Lower Bounds of Proposition 3.2, the following two statements are equivalent:

(1) v ∈ U_{j,k}^{+} − U_{j,k−1}^{+}

(2) For t ≥ 0, the quantity \frac{∥M^{t}v∥}{μ^{t}t^{k}} is bounded, but the quantity \frac{∥M^{t}v∥}{μ^{t}t^{k−1}}

is not bounded.

Similarly, the following two statements are equivalent:

(3) w ∈ Y_{j,k}^{+} − Y_{j,k−1}^{+}.

(4) For t ≥ 0, the quantity \frac{∥N^{t}w∥}{μ^{t}t^{k}} is bounded, but the quantity \frac{∥N^{t}w∥}{μ^{t}t^{k−1}}

is not bounded.

But by inequalities (6.6) and (6.7), statements (2) and (4) are equivalent, and so statements (1) and (3) are equivalent, completing the inductive proof of item 2 of Claim 6.2 for all k ≥ 0.

The foliation flag F(U_{j,0}^{+}) ≺ ⋯ ≺ F(U_{j,k}^{+}) ≺ ⋯ must terminate at F(U_{j}^{+}) for the same value of k for which the flag F(Y_{j,0}^{+}) ≺ ⋯ ≺ F(Y_{j,k}^{+}) ≺ ⋯ terminates at F(Y_{j}^{+}), proving that c_{j} = d_{j}, and completing the proof of Claim 6.2. ♢

Our proof of Theorem 5.11 actually provides for some regularity of f. We record the statement here, although it is not used at all in this paper.

Proposition 6.3 (Regularity). With the assumptions as in Theorem 5.11, f is a homeomorphism which respects the contracting and expanding root space foliation flags of \overline{M}, \overline{N}, and for each corresponding pair of root space foliations f also respects the associated Jordan foliation flags. ♢

Remark. Even stronger regularity properties should hold. For instance, f should satisfy lipschitz conditions in directions parallel to a root space, by arguments similar to the results of [FM98]. Understanding what happens transverse to root spaces will require new ideas.

7 Quasi-isometries of Γ_{M} via Coarse Topology

Recall the notation for abelian-by-cyclic Lie groups: given M ∈ GL_{×}(m, R), a 1-parameter subgroup M^{t} ⊂ GL(m, R) with M^{1} = M determines a Lie group denoted G_{M} = R^{m} ⋊_{M} R.

This entire section will be devoted to a proof of the following.

Proposition 7.1 (Induced quasi-isometries). Consider integral matrices M ∈ GL_{×}(m, R), N ∈ GL_{×}(n, R) and suppose that det M, det N > 1. If there exists a quasi-isometry f: Γ_{M} → Γ_{N} then m = n and there exists a quasi-isometry ϕ: G_{M} → G_{N} which coarsely respects horizontal foliations and their transverse orientations. Furthermore, all associated constants for ϕ depend only on those for f.

7.1 A geometric model for Γ_{M}

Let M ∈ GL_{×}(m, R) be an integral matrix lying on a 1-parameter subgroup M^{t} of GL(m, R) with M^{1} = M and with associated Lie group G_{M}. We assume that det M > 1 and we denote d = det M.

We start by constructing a contractible, (m+1)-dimensional metric complex X_{M} on which Γ_{M} acts properly discontinuously and cocompactly by isometries, and so the group Γ_{M} will be quasi-isometric to the geodesic metric space X_{M}.

The description of Γ_{M} as an ascending HNN extension shows that Γ_{M} is the fundamental group of the mapping torus of an injective endomorphism of the m-dimensional torus. Let X_{M} be the universal cover of this mapping torus. Topologically, there is a fibration

\xymatrix{R^{n−1} \ar[r] & X_{M} \ar[d] \\  & T_{M}}

where T_{M} is the homogeneous directed tree with one edge coming into each vertex and d = det M edges going out of each vertex. Hence X_{M} is a topological product X_{M} ≈ R^{n−1} × T_{M}.

The action of Γ_{M} on X_{M} by deck tranformations induces an action of Γ_{M} on T_{M}. This action is equivalent to the usual action of the HNN extension Γ_{M} on its Bass-Serre tree T_{M}.

Before constructing a metric on X_{M}, let us describe the essential properties of such a metric. These are best described by giving the isometry types of natural subcomplexes of X_{M}.

Definition (Doubled horoballs). We define a doubled G_{M} horoball, denoted H_{M}, to be the metric space obtained by identifying two copies of {(x, t) ∈ G_{M} | t ≥ 0} along {(x, 0) ∈ G_{M}}, endowed with the path metric.

Definition (Hyperplanes in X_{M}). Let P_{ℓ} = π_{M}^{−1}(ℓ), where ℓ is a biinfinite line in the directed tree T_{M}. We call P_{ℓ} a hyperplane in X_{M}. There are two cases to consider:

• ℓ is coherently oriented in T_{M}. In this case P_{ℓ} is isometric to G_{M}, and we call P_{ℓ} a coherent hyperplane in X_{M}.

• ℓ is not coherently oriented in T_{M}, and thus switches orientation precisely once. In this case P_{ℓ} is isometric to H_{M}, and we call P_{ℓ} an incoherent hyperplane in X_{M}.

This definition nearly determines a metric on X_{M}. To specify a metric on X_{M}, one proceeds as follows. Fix a path metric on T_{M} so that each edge has length 1. Fix a base vertex on T_{M}. These choices determine a unique height function T_{M} → R taking the base vertex to the origin, and taking each edge to a segment of length 1 via an orientation preserving isometry. We have also defined a height function G_{M} → R. Note that the height function on G_{M} was previously called the “time function”; we will use both terms.

The complex X_{M} is the fiber product of the two height functions T_{M} → R, G_{M} → R, as shown in the following diagram:

\xymatrix{ & X_{M} \ar[dl]_{g_{M}} \ar[dd] \ar[dr]^{π_{M}} \\ G_{M} \ar[dr] &  & T_{M} \ar[dl] \\  & R}

There are induced projections g_{M}: X_{M} → G_{M} and π_{M}: X_{M} → T_{M}, and an induced height function X_{M} → R. There is a unique path metric on X_{M} so that each continuous cross section G_{M} → X_{M} of g_{M} is a path-isometric embedding; and hence each coherent hyperplane in X_{M} is an isometrically embedded copy of G_{M}.

Definition (horizontal leaf). A horizontal leaf L in X_{M} is a subset of the form L = π_{M}^{−1}(v) where v ∈ T_{M}.

Note that the collection of horizontal leaves on X_{M}, equipped with the Hausdorff metric, forms a metric space which is isometric to T_{M} via the projection map π_{M}: X_{M} → T_{M}.

Note that each hyperplane in X_{M} comes equipped with a foliation by horizontal leaves. For coherent hyperplanes P in X_{M}, which are isometric to G_{M}, the notion of horizontal leaf in P coincides with that of a horizontal leaf in G_{M}, given in §5.1.

7.2 Proof of Proposition 7.1

Let M, N be as in the statement of the proposition.

We begin by showing that M and N have the same size. Suppose that M ∈ GL(m, R) and N ∈ GL(n, R). In §7.1 we constructed finite classifying spaces for Γ_{M} and Γ_{N} of dimensions m+1, n+1 respectively, and by Lemma 5.2 of [FM99b] these numbers are the virtual cohomological dimensions of Γ_{M}, Γ_{N}. By a result of Block-Weinberger [BW97] and Gersten [Ger93], virtual cohomological dimension is a quasi-isometry invariant for groups with finite classifying spaces. It follows that m = n.

Now Γ_{M} acts properly discontinuously, freely, and cocompactly on X_{M}. This action is by isometries, because Γ_{M} acts on G_{M}, on T_{M}, and on R by isometries, and the fiber product diagram is equivariant with respect to these actions. It follows that Γ_{M} in any word-metric is quasi-isometric to X_{M}. Henceforth we will freely interchange Γ_{M} and X_{M} when discussing quasi-isometry type. The same discussion applies to Γ_{N} and X_{N}, and so the quasi-isometry f: Γ_{M} → Γ_{N} gives a quasi-isometry (perhaps with bigger constants) f: X_{M} → X_{N}.

Proposition 7.1 generalizes the case when M and N are 1 × 1 matrices, done in §4 and §5 of [FM98]. The proof here is more difficult, and the steps must be proved in different order. In Steps 1 and 2 we prove (in a more general context; see Theorem 7.7) that a quasi-isometry X_{M} → X_{N} coarsely respects hyperplanes and horizontal sets. However, we must still distinguish between coherent and incoherent hyperplanes. This is easy in the 1 × 1 case handled in [FM98], where G_{M} and G_{N} are (scaled versions of) H^{2}, and a doubled H^{2} horoball is evidently not quasi-isometric to H^{2}. In general we are unable to distinguish the quasi-isometry types of coherent and incoherent hyperplanes. To get around this, in Step 3, Proposition 7.11, we prove that there is no horizontal respecting quasi-isometry between a coherent and an incoherent hyperplane.

Step 1. Quasi-isometrically embedded hyperplanes are close to hyperplanes: Given integral matrices M, N ∈ GL_{×}(n, R), if P = G_{M} or H_{M}, then for all K ≥ 1, C ≥ 0 there exists A ≥ 0 such that if ϕ: P → X_{N} is a K, C-quasi-isometric embedding then there is a unique hyperplane Q ⊂ X_{N} with d_{H}(ϕ(P), Q) ≤ A.

This was proved for 1 × 1 matrices in [FM98]. Our proof of Step 1, while following the same outline as the 1 × 1 case, will actually apply in a much broader setting. The generalized versions of Steps 1 and 2, given in Theorems 7.3 and Theorem 7.7, are used for example in [FM99a] to study surface-by-free groups, and also in [MSW] to prove quasi-isometric rigidity theorems for various “homogeneous” graphs of groups (see the remark after Theorem 7.7).

The generalization of Step 1 given in Theorem 7.3 will require moving from the category of quasi-isometric embeddings into the category of uniformly proper embeddings. After a fair amount of work to establish the new setting, we then quote some theorems of coarse algebraic topology and follow the proof of [FM98].

Consider a finite graph Γ of finitely generated groups; each edge e is oriented, with initial and final vertices i(e), f(e). We say that Γ is geometrically homogeneous if each edge-to-vertex injection is a quasi-isometry with respect to the word metric, or equivalently, has finite index image. Ideally we would like to have a version of Step 1 for any geometrically homogeneous graph of groups in which each vertex and edge group is the fundamental group of a closed, aspherical n-manifold, or even more generally, an n-dimensional Poincaré duality group. This should come from a more careful reading of results in coarse algebraic topology such as [KK], but meanwhile we will use Theorems 7.5 and 7.6, which require us to impose additional assumptions on Γ.

Suppose that we have a category C of aspherical, closed, smooth manifolds such that C is closed under finite coverings and satisfies smooth rigidity, meaning that any homotopy equivalence between manifolds in C is homotopic to a diffeomorphism. Such categories include: the n-torus, n ≥ 1; hyperbolic surfaces; all other irreducible, nonpositively curved, locally symmetric spaces, by Mostow’s Rigidity Theorem [Mos73]; solvmanifolds, by earlier work of Mostow [Mos54]; nilmanifolds, by still earlier work of Malcev [Mal49]; and various generalizations due to Farrell and Jones [FJ89], [FJ97].

We shall assume that Γ is a geometrically homogeneous graph of groups where each vertex group Γ_{v} is the fundamental group of a manifold M_{v} in the category C. Construct a graph of aspherical manifolds M_{Γ}, with fundamental group π_{1}Γ, as follows. For each edge e, the two injections Γ_{e} → Γ_{i(e)}, Γ_{e} → Γ_{t(e)} determine two finite covering spaces of M_{v} each of whose fundamental group is identified with Γ_{e}, and so we obtain a diffeomorphism between the two covering spaces; identify these covering spaces and let M_{e} be the resulting smooth manifold. We have smooth, finite covering maps M_{e} → M_{i(e)}, M_{e} → M_{t(e)} inducing the corresponding edge-to-vertex group injections. Form M_{Γ} from the disjoint union

(⋃_{v} M_{v}) ∪ (⋃_{e} M_{e} × e)

by gluing M_{e} × i(e) to M_{i(e)} and M_{e} × f(e) to M_{f(e)} via the finite covering maps M_{e} → M_{i(e)} and M_{e} → M_{f(e)}. From the construction of M_{Γ} we obtain a map M_{Γ} → Γ, such that each fiber M_{x}, x ∈ Γ, is a manifold in the category C.

Let X_{Γ} be the universal cover of M_{Γ}. There is a Γ-equivariant fiber bundle X_{Γ} → T_{Γ} over the Bass-Serre tree T_{Γ} of Γ, whose fiber is a contractible n-manifold. Any geodesic metric on M_{Γ} lifts to a π_{1}Γ-equivariant geodesic metric on X_{Γ}. Smoothness allows us to impose additional geometric structure on X_{Γ} which we now describe.

A geodesic metric space is proper if closed balls are compact. A bounded geometry metric simplicial complex is a simplicial complex Σ equipped with a proper, geodesic metric such that for some constants 0 < C_{1} < C_{2} each positive dimensional simplex has diameter between C_{1} and C_{2}, and for some constant C > 0 the link of each simplex has ≤ C simplices. A subset S of Σ is rectifiable if for any p, q ∈ S there exists a path in S between p and q which is rectifiable in Σ, and which has the shortest Σ-length among all paths in S between p and q. The length of such a path defines a geodesic metric on S. A D-homotopy in Σ is a homotopy whose tracks all have diameter ≤ D. The space Σ is uniformly contractible if there exists a function δ: [0, ∞) → [0, ∞), such that for every bounded subset S ⊂ Σ, the inclusion map S ↪ Σ is δ(diam(S))-homotopic to a constant map. More precisely we say that Σ is δ-uniformly contractible.

Let T be a bounded geometry, metric simplicial tree, let X be a proper, geodesic metric, and let π: X → T be a surjective map. Denote X_{A} = π^{−1}(A) for each A ⊂ T. The map π is called a metric fibration if:

1. X is a uniformly contractible, bounded geometry, metric simplicial complex.

2. For each subtree T′ ∈ T, the subset X_{T′} is a subcomplex of X and is rectifiable in X.

3. For each t ∈ T the subspace X_{t} is uniformly contractible and is a bounded geometry, metric simplicial complex, with bounded geometry constants and uniform contractiblity data independent of t.

4. The map π: X → T is distance nonincreasing.

5. There is a homeomorphism Θ: X → F × T such that:

(a) For all t ∈ T, Θ(X_{t}) = F × t.

(b) For all x ∈ F, the map T → x × T ⟶^{Θ^{−1}} X is a locally isometric embedding.

(c) There exists K ≥ 1 such that for all edges e of T and t ∈ e, the retraction r: e → t induces a projection X_{e} ⟶^{Θ} F × e ⟶^{Id×r} F × t ⟶^{Θ^{−1}} X_{t} which is K-lipschitz.

Each fiber X_{t}, t ∈ T, is called a horizontal leaf in X. If L is a bi-infinite line in T then X_{L} is called a hyperplane in X. Items 4 and 5b combine to show that the map of item 5b is an isometric embedding; the image Θ^{−1}(x × T) is called a vertical leaf in X. For each subtree T′ ⊂ T, the closest point retraction r: T → T′ induces a map

X ⟶^{Θ} F × T ⟶^{Id×r} F × T′ ⟶^{Θ^{−1}} X_{T′}

called vertical projection of X to X_{T′}.

Remark. Suppose Γ is a graph of groups taken from a category C as above. Let M_{Γ} and X_{Γ} → T_{Γ} be as constructed above starting from Γ. Then elementary constructions produce a metric and a simplicial structure on M_{Γ} which lifts to a Γ-equivariant metric and simplicial structure on X such that X → T_{Γ} is a metric fibration. Item (1) follows by compactness of M_{Γ}.

Remark. The definition has some redundancy: item (1) is a formal consequence of item (3), as can be seen by elementary but mildly tedious arguments. But by the previous remark we may dispense with these arguments for the examples at hand.

The following lemma, applied to a bi-infinite lines in T, gives good geometric properties for hyperplanes:

Lemma 7.2. If π: X → T is a metric fibration then there exist functions δ′: [0, ∞) → [0, ∞) and ρ: [0, ∞) → [0, ∞) with lim_{t→∞} ρ(t) = ∞, such that for any subtree T′ ⊂ T we have:

(1) The embedding X_{T′} → X is ρ-uniformly proper.

(2) The geodesic metric space X_{T′} is δ′-uniformly contractible.

Proof. To prove (1), consider x, y ∈ X_{T′}, let D = d_{X}(x, y), and let γ: [0, D] → X be a geodesic connecting x and y. Let N_{D}(T′) be the D-neighborhood of T′ in T, so γ ⊂ X_{N_{D}(T′)}. Applying item 5 iteratively, projecting inward starting from the edges of N_{D}(T′) furthest from T′, it follows that vertical projection X_{N_{D}(T′)} → X_{T′} distorts any distance r by at worst K^{D}r, and so d_{X_{T′}} (x, y) ≤ K^{D}D.

To prove (2), suppose that A ⊂ X_{T′} and diam_{X_{T′}} (A) ≤ R and so A is R′-homotopic to a constant in X where R′ depends on R but not on A. This homotopy may then be mapped back to X_{T′} by vertical projection, distorting diameters of homotopy tracks by an amount bounded in terms of R′ as we saw above. The result is an R′′-homotopy of A to a constant in T′, with R′′ depending only on R and not on A. ♢

Here is our generalization of Step 1. It applies to any metric fibration of the form X_{Γ} → T_{Γ}, where Γ is a finite, geometrically homogeneous graph of fundamental groups of manifolds in any of the categories C described earlier.

Theorem 7.3. Let π: X → T be a metric fibration whose fibers are contractible n-manifolds for some n. Let P be a contractible (n + 1)-manifold which is a uniformly contractible, bounded geometry, metric simplicial complex. Then for any uniformly proper embedding ϕ: P → X, there exists a unique hyperplane Q ⊂ X such that ϕ(P) and Q have finite Hausdorff distance in X. The bound on Hausdorff distance depends only on the metric fibration data for π, the uniform contractibility data and bounded geometry data for P, and the uniform properness data for ϕ.

Proof. Uniqueness of Q follows obviously from the fact that distinct hyperplanes in X have infinite Hausdorff distance.

For existence of Q we follow closely the proof of Proposition 4.1 of [FM98], concentrating on details needed to explicate the difference between the “quasi-isometric” setting of [FM98] and the present “uniformly proper” setting.

Using the bounded geometry of P, uniform contractibility of X, and uniform properness of ϕ, we may replace ϕ by a continuous, uniformly proper map, moving values of ϕ a bounded distance. Henceforth we shall assume ϕ is continuous.

Pick a topologically proper embedding of T in an open disc D. For each component U of D − T, the frontier of U in D is a bi-infinite line L(U) in T. There is a homeomorphism of pairs (\overline{U}, L(U)) ≈ (L(U) × [0, ∞), L_{U} × 0).

Consider the topologically proper embedding X ⟶^{Θ} F × T ↪ F × D. Note that F × D is a contractible (n + 2)-manifold. For each component U of D − T we have a homeomorphism

F × \overline{U} ⟶^{≈} F × (L(U) × [0, ∞)) ⟶^{≈} (F × L(U)) × [0, ∞)

⟶_{Θ×Id}^{≈} X_{L(U)} × [0, ∞)

The frontier of this set in F ×D is F ×L(U) ≈ X_{L(U)}. Put a product metric and a product simplicial structure on X_{L(U)} × [0, ∞) and glue to F × L(U). Doing this for each U, we impose a proper geodesic metric on F × D for which the inclusion X ↪ F × D is an isometric embedding.

The simplicial structure on F ×D evidently has bounded geometry. Also, the metric space F ×D is uniformly contractible. To see this, let A ⊂ F ×D have diameter ≤ r. If A ∩ X ≠ ∅ then homotoping along products lines of X_{L(U)} × [0, ∞) for each U we obtain an r-homotopy of A into F × T ≈ X, and then we use uniform contractibility of X. Whereas if A ∩ X = ∅, then A ⊂ F × U ≈ X_{L(U)} × (0, ∞) for some component U of D − T; there is an r-homotopy of A into some X_{L(U)} × x, and the latter is uniform contractible by Lemma 7.2.

We now plug this setup into the coarse separation and packing methods of Farb–Schwartz [FS96] and Schwartz [Sch96]. We’ll use a generalization of the Coarse Separation Theorem with more easily applied hypotheses, due to Kapovich–Kleiner [KK]. We denote the r-ball about a subset A of a metric space M by B_{r}(A; M). In a metric space Z, a subset U ⊂ Z is deep in Z if for each r > 0 there exists x ∈ U such that B_{r}(x; Z) ⊂ U. A subset A ⊂ Z coarsely separates Z if for some D > 0 there are at least two components of Z − N_{D}(A; Z) which are deep in Z; the constant D is called a coarse separation constant for A. Note that if subsets A and B of Z have bounded Hausdorff distance from each other, then A coarsely separates Z if and only if B does.

Here is an elementary consequence of the definitions:

Lemma 7.4. Let f: X → Y be a quasi-isometry between geodesic metric spaces. If A ⊂ X coarsely separates X then f(A) coarsely separates Y, with separation constant depending only on the quasi-isometry constants of f and the separation constant for A. ♢

Here is the version of the Coarse Separation Theorem that we will use.

Theorem 7.5 ([KK]). Let P be a contractible (n + 1)-manifold, Z a contractible (n + 2)-manifold, and suppose that P, Z are uniformly contractible, bounded geometry, metric simplicial complexes. Let Φ: P → Z be a uniformly proper map. Then Φ(P) coarsely separates Z, with coarse separation constant D depending only on the uniform contractibility and bounded geometry data for P and Z and the uniform properness data for Φ. Moreover if Φ is continuous then we may take D = 0, that is, Z − Φ(P) has at least two components which are deep in Z. ♢

Remark. In fact there are exactly two components of Z − N_{D}(Φ(P); Z) which are deep in Z (see [KK]).

Following [FS96] we have a corollary:

Theorem 7.6 (Packing Theorem). Let Q, P be contractible (n+1)-manifolds, which are uniformly contractible, bounded geometry, metric simplicial complexes. Let ψ: Q → P be a uniformly proper map. Then there exists R > 0 such that N_{R}(ψ(Q); P) = P. The constant R depends only on the uniform contractibility data and bounded geometry data for Q, P and the uniform properness data for ψ.

Proof. If no such R exists then the image of the map Q ⟶^{ψ} P ↪ P × R does not coarsely separate P × R, violating Theorem 7.5. ♢

Continuing with the proof of Theorem 7.3, compose the continuous, uniformly proper map ϕ: P → X with the isometric embedding X → F × D to obtain a continuous, uniformly proper map Φ: P → F × D. By the Coarse Separation Theorem it follows that (F × D) − Φ(P) has at least two components which are deep in F × D.

Now take the argument of [FM98], Step 1, pages 426–427, and apply it verbatim, to produce a hyperplane Q ⊂ X such that Q ⊂ Φ(P). Next take the argument of Step 2, pages 427–428, and apply it verbatim, replacing “quasi-isometric embeddings” with “uniformly proper maps” and using the Packing Theorem above, to show the existence of R′ such that ϕ(P) ⊂ N_{R′}(Q; X), where R′ depends only on the metric fibration data for π, the uniform contractibility and bounded geometry data for P, and the uniform properness data for ϕ.

This finishes the proof of Theorem 7.3 and of Step 1. ♢

Step 2. A quasi-isometry takes hyperplanes and horizontal leaves in X_{M} to hyperplanes and horizontal leaves in X_{N}: Consider integral matrices M, N ∈ GL_{×}(n, R) with det M, det N > 1, and let f: X_{M} → X_{N} be a quasi-isometric embedding. Then there is a constant A ≥ 0, depending only on X_{M}, X_{N} and the quasi-isometry constants of f, such that:

1. For each hyperplane P ⊂ X_{M} there exists a unique hyperplane Q ⊂ X_{N} such that d_{H}(f(P), Q) ≤ A.

2. For each horizontal leaf L of X_{M} there exists a horizontal leaf L′ of X_{N} such that d_{H}(f(L), L′) ≤ A.

The proof of this step is the first place in our arguments where the assumption that det M, det N > 1 is crucial. Again we will investigate this step in the general setting of metric fibrations over trees.

Consider a metric fibration π: X → T. The tree T is bushy if there exists a constant β such that each point of T is within distance β of some vertex v such that T − v has at least 3 unbounded components. Note that if M is an integer matrix in GL_{×}(n, R), and if X_{M} → T_{M} is the associated metric fibration over the Bass-Serre tree T_{M} of the group Γ_{M}, then T_{M} is bushy if and only if det M > 1. In fact, for any graph of finitely generated groups, the Bass-Serre tree is either bounded, quasi-isometric to a line, or bushy, and the question of which alternative holds is easily decided by inspection of the graph of groups.

Here is our generalization of Step 2:

Theorem 7.7. Let π: X → T, π′: X′ → T′ be metric fibrations over βbushy trees T, T′, such that the fibers of π and π′ are contractible n-manifolds for some n. Let f: X → X′ be a quasi-isometry. Then there exists a constant A, depending only on the metric fibration data of π, π′, the quasiisometry data for f, and the constant β, such that:

(1) For each hyperplane P ⊂ X there exists a unique hyperplane Q ⊂ X′ such that d_{H}(f(P), Q) ≤ A.

(2) For each horizontal leaf L ⊂ X there is a horizontal leaf L′ ⊂ X′ such that d_{H}(f(L), L′) ≤ A.

Remark. This result is used in [MSW] to prove quasi-isometric ridigity for fundamental groups of geometrically homogeneous graphs of groups whose vertex groups are fundamental groups of manifolds in a category C as above, as long as that class of groups is itself quasi-isometrically rigid. For example, quasi-isometric rigidity is proved for graphs of Z’s, Z^{n}’s, surface groups, lattices in semisimple Lie groups, nilpotent groups, etc.

Proof. To prove (1), by Lemma 7.2 the inclusion map P ↪ X is uniformly proper and P is uniformly contractible, and clearly P is a contractible (n+1)- manifold. Composing with f we obtain a uniformly proper map P → X′. Now apply Theorem 7.3.

The idea of the proof of (2) is that bushiness of the tree allows one to gain quasi-isometric control over horizontal leaves by considering them as “coarse intersections” of hyperplanes.

Definition (Coarse intersection). A subset W of a metric space X is a coarse intersection of subsets U, V ⊂ X, denoted W = U ∩_{C} V, if there exists K_{0} such that for every K ≥ K_{0} there exists K′ ≥ 0 so that

d_{H}(Nbhd_{K}(U) ∩ Nbhd_{K}(V), W ) ≤ K′

Note that although such a set W may not exist, when it does exist then any two such sets are a bounded Hausdorff distance from each other.

The following fact is an elementary consequence of the definitions.

Figure 1: Possible coarse intersections of distinct hyperplanes in X, projected to T. In (a), P_{1} ∩_{C} P_{2} = P_{1} ∩ P_{2} is a half-plane. In (b–d), P_{1} ∩_{C} P_{2} is a horizontal leaf; P_{1} ∩ P_{2} can be empty (b), a horizontal leaf (c), or a finite strip of horizontal leaves (d).

Lemma 7.8. For any quasi-isometry f: X → Y of metric spaces, and U, V ⊂ X, if U ∩_{C} V exists then f(U ∩_{C} V) is a coarse intersection of f(U), f(V), with constants depending only on the quasi-isometry constants for f and the coarse intersection constants for U and V. ♢

Consider now a metric fibration π: X → T. A subset of X of the form X_{σ} = π^{−1}(σ), where σ is an infinite ray in T, will be called a half-plane in X. The next lemma is an easy observation—see Figure 1.

Lemma 7.9. Let π: X → T be a metric fibration over a tree T. Let P_{1} and P_{2} be distinct hyperplanes in X. Then P_{1} ∩_{C} P_{2} exists and is a bounded Hausdorff distance from either a half-plane or a horizontal leaf in X. Moreover, P_{1} ∩_{C} P_{2} is a bounded Hausdorff distance from a half-plane if and only if P_{1} ∩ P_{2} is a half-plane. ♢

We remark that P_{1} ∩_{C} P_{2} can be an arbitrarily large finite Hausdorff distance from a horizontal leaf; see Figure 1(b,d).

Lemma 7.10. Let π: X → T, π′: X′ → T′ be metric fibrations. Let f: X → X′ be a quasi-isometry. Suppose P_{1} and P_{2} are distinct hyperplanes in X which intersect in a half-plane. Then f(P_{1}) and f(P_{2}) are a uniformly bounded Hausdorff distance from distinct hyperplanes Q_{1}, Q_{2} in X′ which intersect in a half-plane in X′.

Proof. By Theorem 7.3, there exists a constant A so that f(P_{i}) is within Hausdorff distance A of a unique hyperplane Q_{i} in X′. Since P_{1}, P_{2} are distinct they have infinite Hausdorff distance, so Q_{1} and Q_{2} have infinite Hausdorff distance and hence Q_{1} ≠ Q_{2}.

By Lemma 7.9, it is enough to prove that Q_{1} ∩_{C} Q_{2} is not a bounded Hausdorff distance from a horizontal leaf in X′. If Q_{1} ∩_{C} Q_{2} is a bounded Hausdorff distance from a horizontal leaf, then since any horizontal leaf in Q_{1} coarsely separates Q_{1} it must be that Q_{1} ∩_{C} Q_{2} coarsely separates Q_{1}. But P_{1}∩_{C}P_{2} does not coarsely separate P_{1}. This contradicts Lemma 7.4. ♢

We now prove Theorem 7.7. Consider the quasi-isometry f: X → X′. Since T is bushy, any horizontal leaf L in X can be realized as a coarse intersection of three hyperplanes P_{1}, P_{2}, P_{3}, such that the pairwise intersections P_{1} ∩ P_{2}, P_{2} ∩ P_{3}, P_{3} ∩ P_{1} form three half-planes, any two of which have infinite Hausdorff distance. Moreover, d_{H}(L, P_{1} ∩ P_{2} ∩ P_{3}) ≤ β where β is a bushiness constant for T (see Figure 2).

Consider the unique hyperplane Q_{i} which lies a Hausdorff distance of at most A from f(P_{i}), i = 1, 2, 3. By Lemma 7.10, the pairwise intersections Q_{1} ∩ Q_{2}, Q_{2} ∩ Q_{3}, Q_{3} ∩ Q_{1} are all half-planes, any two of which have infinite Hausdorff distance. The following elementary fact about trees, applied to T′, now shows that Q_{1} ∩ Q_{2} ∩ Q_{3} is a horizontal leaf L′ in X′:

Fact about trees: Let ℓ_{1}, ℓ_{2}, ℓ_{3} be bi-infinite lines in a simplicial tree T′, such that the pairwise intersections ℓ_{1} ∩ℓ_{2}, ℓ_{2} ∩ℓ_{3}, ℓ_{3} ∩ℓ_{1} are all infinite rays in T′, any two of which have infinite Hausdorff distance. Then ℓ_{1} ∩ ℓ_{2} ∩ ℓ_{3} is a vertex of T′.

Since L ⊂ N_{β}(P_{i}) it follows that

f(L) ⊂ N_{Kβ+C}(f(P_{i})) ⊂ N_{Kβ+C+A}(Q_{i}), i = 1, 2, 3

But clearly we have ⋂_{i=1}^{3} N_{Kβ+C+A}(Q_{i}) = N_{Kβ+C+A}(L′).

To summarize, given a horizontal leaf L of X, we have found a horizontal leaf L′ of X′ such that L ⊂ N_{A′}(L′) where A′ = Kβ + C + A. A similar argument using a coarse inverse for f provides the desired bound for d_{H}(f(L), L′). This completes the proofs of Theorem 7.7 and of Step 2. ♢ Figure 2: Any point x ∈ T is a bounded distance β from a vertex v ∈ T that separates T into at least three unbounded components. The vertex v is the (coarse) intersection of three proper lines ℓ_{1}, ℓ_{2}, ℓ_{3}, such that the pairwise intersections ℓ_{1} ∩ ℓ_{2}, ℓ_{2} ∩ ℓ_{3}, ℓ_{3} ∩ ℓ_{1} are rays in T, any two of which have infinite Hausdorff distance. Moreover, d(x, ℓ_{1} ∩ ℓ_{2} ∩ ℓ_{3}) ≤ β.

Step 3. A quasi-isometry takes coherent hyperplanes in X_{M} to coherent hyperplanes in X_{N}.

Let M, N be as in the statement of Proposition 7.1, and fix a quasiisometry f: X_{M} → X_{N}.

Let P be any coherent hyperplane in X_{M}. By Step 2 it follows that f(P) is within a Hausdorff distance A from a unique hyperplane Q in X_{N}. By composing f | P with vertical projection X_{N} → Q we obtain a map ϕ: P → Q. The inclusion maps P ↪ X_{M} and Q ↪ X_{N} are coarsely lipschitz and uniformly proper; indeed they are isometric embeddings with respect to the induced path metrics on P, Q. By Lemma 2.1, ϕ is a quasiisometry, with quasi-isometry constants depending only on those for f. By Step 2, f coarsely respects the horizontal foliations of X_{M} and X_{N}; vertical projection X_{N} → Q takes horizontal leaves to horizontal leaves, and so ϕ coarsely respects the horizontal foliations of P and Q, with a coarseness constant depending only on the quasi-isometry constants of f.

Since P is a coherent hyperplane it is isometric to G_{M}. Since Q is a hyperplane it is isometric to either G_{N} or H_{N}, and we now show that the second possibility cannot occur.

Proposition 7.11. Given matrices M, N ∈ GL_{×}(n, R) with det M, det N > 1, there is no quasi-isometry ϕ: G_{M} → H_{N} which coarsely respects horizontal foliations.

Proof. The idea of the proof is to compare the growth types of the filling area functions for “quasivertical bigons” in G_{M} and in H_{N}. In G_{M} this growth type will be quadratic, while in H_{N} it will be exponential.

Let H = G_{M}, H_{M}, G_{N}, or H_{N}. There is a quotient map H → R whose point pre-images give the horizontal foliation of H, and such that the Hausdorff distance between two horizontal leaves equals the distance between the corresponding points in R. A path γ in H is said to be (K, C)-quasivertical if its projection to R is a (K, C)-quasigeodesic. Define a (K, C)-quasivertical bigon in H to be a pair of (K, C)-quasivertical paths γ, γ′ which begin and end at the same point.

If K, C are fixed, we define a filling area function A(L) for (K, C)- quasivertical bigons in H. Given a (K, C)-quasivertical bigon γ, γ′, its filling area is the infimal area of a Lipschitz map D^{2} → H whose boundary is a reparameterization of the closed curve γ^{−1} ∗ γ′; such a map D^{2} → H is called a filling disc for γ^{−1} ∗ γ′. For each L ≥ 0 define A(L) to be the supremal filling area over all (K, C)-quasivertical bigons γ, γ′ in H such that Length(γ) + Length(γ′) ≤ L.

Suppose there is a quasi-isometry ϕ: G_{M} → H_{N} which coarsely respects horizontal foliations. Let ϕ̄: H_{N} → G_{M} be a coarse inverse for ϕ, also coarsely respecting horizontal foliations. Clearly ϕ̄ takes any K, Cquasivertical bigon in H_{N} to a (K′, C′)-quasivertical bigon in G_{M}, distorting lengths by at worst an affine function; this affine function, and the constants K′, C′, depend only on K, C, the quasi-isometry constants for ϕ, and the Hausdorff constant for the induced height function. Fill the resulting bigon in G_{M} as efficiently as possible, and map back to H_{N} via ϕ, distorting area by at worst an affine function which again has the same dependencies. We thereby obtain a filling of the original bigon in H_{N}. If A_{1}(L) denotes the filling area function for (K′, C′)-quasivertical bigons in G_{M}, and if A_{2}(L) denotes the filling area function for (K, C)-quasivertical bigons in H_{N}, it follows that the growth type of A_{2}(L) is dominated by the growth type of A_{1}(L), that is,

A_{2}(L) ≤ α · A_{1}(βL + δ) + ζ

for some positive constants α, β, δ, ζ independent of L.

However, we shall now show that A_{1}(L) has a quadratic upper bound while A_{2}(L) has an exponential lower bound, contradicting the above inequality.

Consider a K′, C′-quasivertical bigon γ, γ′ in G_{M}. Applying the argument of Claim 5.7, there are center leaves τ, τ′ in G_{M} and quasivertical paths ρ ⊂ τ, ρ′ ⊂ τ′ which stay uniformly close to γ, γ′, respectively. The initial points of ρ, ρ′ are at a uniformly bounded distance, as are the terminal points, and it follows that ρ′ stays uniformly close to a quasivertical path ρ′′ ⊂ τ. Connecting initial and terminal endpoints with short paths η, η′ we thus obtain a closed curve ρ^{−1} ∗ η ∗ ρ′′ ∗ η′, contained in a center leaf of G_{M}, which stays uniformly close to γ^{−1} ∗ γ′. Since center leaves of G_{M} are isometric to Euclidean space, in which the filling function is quadratic, it follows that A_{1}(L) has a quadratic upper bound.

To show that A_{2}(L) has an exponential lower bound, we now construct quasivertical bigons in H_{N} which can be filled only by discs of exponential area. In the case where N is a 1 × 1 matrix such loops are given explicitly in [ECH^{+}92], Chapter 7.4; examples for general N are simple modifications of this example. To be explicit, choose an eigenvalue of N of absolute value α > 1; such an eigenvalue exists because det N > 1. Choose an affine subspace A ⊂ R^{n} parallel to the α-eigenspace of N. Consider the subspace A × R ⊂ R^{n} × R ≈ G_{N}.

For each fixed L ≥ 0, choose two vertical segments g, g′ in A × [0, ∞) whose upper endpoints are in A×L and whose lower endpoints are in A×0, and so that the distance in A × L between the upper endpoints, measured using the Riemannian metric on G_{N}, is equal to 1; it follows that the distance in A×0 between the lower endpoints, measured using the Riemannian metric on G_{N}, is within a constant multiple of α^{L}.

Now double this picture, in the doubled G_{N} horoball H_{N}, to get a closed loop in H_{N}, that is: in one horoball go up g, across 1 unit, and down g′, and then in the other horoball go up g′, across 1 unit, and down g; let ρ be the resulting closed curve in H_{N}. We have Length(ρ) = 4L + 2. To see that the filling area of ρ is exponential in L, note that any filling disc for ρ must contain a path in A×0 connecting the lower endpoints of g, g′, because A×0 separates the two halves of ρ in H_{N}. This path has length exponential in L; and a neighborhood of this path in the filling disc has area exponential in L. ♢

Step 4. A horizontal respecting quasi-isometry preserves transverse orientation

Let M, N, and f: Γ_{M} → Γ_{N} be as in the statement of Proposition 7.1. By Step 3 there is a quasi-isometry ϕ: G_{M} → G_{N}, and by Step 2, ϕ coarsely respects the horizontal foliations of G_{M} and G_{N}. Suppose that ϕ reverses the transverse orientation. There is a quasi-isometry G_{N} → G_{N^{−1}} which coarsely respects horizontal foliations, reversing transverse orientations. Precomposing with ϕ: G_{M} → G_{N} and applying Steps 1–3, we obtain a quasi-isometry G_{M} → G_{N^{−1}} which coarsely respects the transversely oriented horizontal foliations. Applying Theorem 5.2, it follows that M and N^{−1} have positive real powers with the same absolute Jordan form, and so these powers also have the same determinant. But each positive power of M has determinant > 1, whereas every positive power of N^{−1} has determinant < 1, a contradiction showing that ϕ must preserve the transverse orientation.

This completes the proof of Proposition 7.1. ♢

Remark. Note in the proof of Proposition 7.1 that different choices of coherent hyperplanes in X_{M} yield different quasi-isometries ϕ. In some cases ϕ is well-defined up to some constant A, that is, for any two choices of coherent hyperplane in X_{M}, the induced maps ϕ_{1}, ϕ_{2} : G_{M} → G_{N} satisfy sup_{x} d(ϕ_{1}(x), ϕ_{2}(x)) ≤ A. This is true, for example, in the “centerless” case where M, N have no eigenvalues on the unit circle. In the general case, the best that can be said is that the map induced by ϕ from the center leaf space of G_{M} to the center leaf space of G_{N} is well defined up to a constant, with respect to the Hausdorff metrics on the center leaf spaces.

8 Finding the Integers

In this section we prove Theorem 1.1. Let M, N be n × n integral matrices with |det M| , |det N| > 1. We must prove that Γ_{M} is quasi-isometric to Γ_{N} if and only if there exist positive integers a, b such that M^{a} and N^{b} have the same absolute Jordan form.

First we show that the groups Γ_{M^{a}} and Γ_{M} are quasi-isometric, for any positive integer a, by showing that Γ_{M^{a}} is a subgroup of finite index in Γ_{M}, specifically of index a. To see why, consider the presentations

Γ_{M} = ⟨Z^{n}, t | t^{−1}xt = M(x), x ∈ Z^{n}⟩
Γ_{M^{a}} = ⟨Z^{n}, s | s^{−1}xs = M^{a}(x), x ∈ Z^{n}⟩

Define a homomorphism Γ_{M} ↦ Z/aZ by Z^{n} ↦ 0, t ↦ 1. This homomorphism is onto, and its kernel is generated by Z^{n}, t^{a}. This kernel is isomorphic to Γ_{M^{a}} under the injection Γ_{M^{a}} ↪ Γ_{M} given by x ↦ x, s ↦ t^{a}.

Similarly, Γ_{N^{b}} is quasi-isometric to Γ_{N}, for any positive integer b.

By squaring M, N if necessary, we may therefore assume that det M, det N > 1, and that M and N lie on 1-parameter subgroups; we continue with this assumption up through the end of the proof in §8.2. Choose 1-parameter subgroups M^{t}, N^{t} of GL(n, R) with M = M^{1}, N = N^{1}, let G_{M}, G_{N} be the associated Lie groups constructed in §4, and let X_{M}, X_{N} be the associated geodesic metric spaces constructed in §7. The group Γ_{M} is quasi-isometric to X_{M}, and Γ_{N} is quasi-isometric to X_{N}.

8.1 The first half of the classification

Assuming that M^{a} and N^{b} have the same absolute Jordan form, where a, b are positive integers, we must prove that Γ_{M} and Γ_{N} are quasi-isometric. We have shown above that Γ_{M^{a}} and Γ_{M} are quasi-isometric, and that Γ_{N^{b}} and Γ_{N} are quasi-isometric. Replacing M by M^{a} and N by N^{b}, we may therefore assume that M, N have the same absolute Jordan form. We shall prove that Γ_{M}, Γ_{N} are quasi-isometric by constructing a bilipschitz homeomorphism between X_{M} and X_{N}.

Since the absolute Jordan forms of M, N are equal it follows that det M = det N; let d be the common value. Applying Proposition 4.1, there is a bilipschitz homeomorphism from G_{M} = R^{n} ⋊_{M} R to G_{N} = R^{n} ⋊_{M} R of the form (x, t) ↦ (Ax, t) for some A ∈ GL(n, R). In the fiber product description of X_{M}, X_{N}, the trees T_{M} and T_{N} may both be identified with the homogeneous, oriented tree T_{d} with one incoming and d outgoing edges at each vertex. The bilipschitz homeomorphism G_{M} → G_{N} and the identity homeomorphism T_{d} → T_{d} both respect the height functions, and so these two homeomorphisms combine to give the desired bilipschitz homeomorphism X_{M} → X_{N}.

8.2 Quasi-isometric implies integral powers have same absolute Jordan forms

Assuming Γ_{M}, Γ_{N} are quasi-isometric, there is a quasi-isometry f: X_{M} → X_{N}. Combining Proposition 7.1 and Theorem 5.2 gives r ∈ R_{+} such that M^{r} and N have the same absolute Jordan form. We must show that there exist a, b ∈ Z_{+} so that M^{a} and N^{b} have the same absolute Jordan form.

Since M^{r} and N have the same absolute Jordan form, listing the absolute values of the eigenvalues of M and N in increasing order we obtain

μ_{−a} ≤ ⋯ ≤ μ_{0} ≤ 1 < μ_{1} =: α_{M} ≤ ⋯ ≤ μ_{b}
ν_{−a} ≤ ⋯ ≤ ν_{0} ≤ 1 < ν_{1} =: α_{N} ≤ ⋯ ≤ ν_{b}

with μ_{i}^{r} = ν_{i}, −a ≤ i ≤ b. From this it follows that

\frac{log α_{N}}{log α_{M}} = r = \frac{log det N}{log det M}

Let Q_{M} denote the set of coherent hyperplanes in X_{M}, and let h_{M} denote the height function on M. We define a metric on Q_{M} as follows: given coherent hyperplanes P_{1}, P_{2}, let L denote the horizontal leaf L = ∂(P_{1} ∩P_{2}). Then we set

d_{Q_{M}} (P_{1}, P_{2}) = (det M)^{−h_{M} (L)}

It is easy to check that this defines a metric on Q_{M}, and since the tree T_{M} branches m = det M times as h_{M} increases by 1, the metric space (Q_{M}, d_{Q_{M}}) is isometric to the m-adic rational numbers in their usual metric of Hausdorff dimension 1. Similarly, attached to X_{N} is a metric space (Q_{N}, d_{Q_{N}}) isometric to the n-adic rational numbers, with n = det N.

From Step 3 in the proof of Proposition 7.1 (see §7.2), the quasi-isometry f : X_{M} → X_{N} takes each coherent hyperplane in X_{M} to within a uniform Hausdorff distance of a unique coherent hyperplane in X_{N}, hence induces a bijection ψ: Q_{M} → Q_{N}. For each ℓ ∈ Q_{M}, setting ℓ′ = ψ(ℓ), there is an induced horizontal-respecting quasi-isometry P_{ℓ} → P′_{ℓ′}, and by Time Rigidity (Proposition 5.8) this quasi-isometry has an induced time change of the form t ↦ mt + b where

m = \frac{log α_{M}}{log α_{N}} = 1/r

and where b depends ostensibly on ℓ. However, for another ℓ_{1}, P_{ℓ} and P_{ℓ_{1}} coincide below some value of t, and so t ↦ mt + b is an induced time change for both P_{ℓ} ↦ P′_{ℓ′} and P_{ℓ_{1}} ↦ P′_{ℓ′_{1}}, possibly with a larger coarseness constant (this argument is taken from Claim 6.3 on p. 436 of [FM98]). Therefore, there is a uniform induced time change t ↦ mt + b with b independent of ℓ, and with a uniform Hausdorff constant A.

We now claim that ψ is a bilipschitz homeomorphism. To this end, let P_{1}, P_{2} ∈ Q_{M} be given. Let L = ∂(P_{1} ∩ P_{2}) and let L′ = ∂(ψ(P_{1}) ∩ ψ(P_{2})). Then

h_{N}(L′) ≥ m · h_{M}(L) + b − A

Hence

\frac{d_{Q_{N}} (ψ(P_{1}), ψ(P_{2}))}{d_{Q_{M}} (P_{1}, P_{2})} = \frac{(det N)^{−h_{N} (L′)}}{(det M)^{−h_{M} (L)}}

≤ \frac{(det N)^{−mh_{M} (L)−b+A}}{(det M)^{−h_{M} (L)}}

= \frac{((det N)^{(log det M/ log det N))})^{−h_{M} (L)}(det N)^{−b+A}}{(det M)^{−h_{M} (L)}}

= (det N)^{−b+A} which is a constant not depending on P_{1} or P_{2}. Hence ψ is Lipschitz. The same argument applied to ψ^{−1} shows that ψ is bilipschitz.

Applying Cooper’s Theorem (appendix to [FM98], Corollary 10.11) on bilipschitz homeomorphisms of Cantor sets, we obtain that there exist integers a, b > 0 such that (det M)^{a} = (det N)^{b}. Since M^{r} and N have the same absolute Jordan form, we have

\frac{b}{a} = \frac{log det M}{log det N} = r

and so (M^{r})^{a} = M^{b} and N^{a} have the same absolute Jordan form.

9 Quasi-isometric rigidity

In this section we prove Theorem 1.2 in a series of steps. Recall the hypotheses: M is an integer matrix in GL(n, R) with |det M| > 1, and G is a finitely generated group quasi-isometric to Γ_{M}. By squaring M if necessary we may assume that M ∈ GL_{×}(n, R) and det M > 1, and therefore Γ_{M} is quasi-isometric to X_{M}. It follows that G is quasi-isometric to X_{M}.

Step 1. The action of G on itself by left multiplication can be conjugated by the quasi-isometry G → X_{M} to give a proper, cobounded quasi-action of G on X_{M} (see [FM99b], Proposition 2.1). Since det M > 1 we may apply Theorem 7.7, concluding that the quasi-action of G on X_{M} coarsely respects the fibers of the uniform metric fibration X_{M} → T_{M}.

Step 2. Now we use the following result of [MSW]. Suppose π: X → T is a uniform metric fibration over a bushy tree T. If G is a finitely presented group with a cobounded, proper quasi-action on X, and if the quasi-action coarsely respects the fibers, then G is the fundamental group of a graph of groups whose vertex and edge groups are quasi-isometric to a fiber X_{t} = π^{−1}(t).

By Step 1, this result applies to the quasi-action of G on X_{M}, because G is quasi-isometric to the finitely presented group Γ_{M} and so G is finitely presented. The fibers of the map X_{M} → T_{M} are isometric to R^{n}, and it follows that G is the fundamental group of a graph of groups with each vertex and edge group quasi-isometric to R^{n}.

Step 3. Any finitely-generated group quasi-isometric to R^{n} is virtually Z^{n} (see [Ger95]), and so G is the fundamental group of a graph of groups whose vertex and edge groups are virtually Z^{n}.

Step 4. Applying the argument in Section 5 of [FM99b] to G gives that either G contains a noncyclic free group or G is an ascending HNN extension of the form

G = A_{ϕ} = ⟨A, t | tat^{−1} = ϕ(a), ∀a ∈ A⟩

where A is virtually Z^{n} and ϕ: A → A is an injective endomorphism. Since Γ_{M} is amenable, and since G is quasi-isometric to Γ_{M}, then G is amenable, and so G cannot contain a noncyclic free group. The second possibility must therefore occur: G = A_{ϕ} as above.

Step 5. Now we turn to an analysis of injective endomorphisms of virtually abelian groups. Suppose A is a finitely generated, virtually abelian group. Any injective endomorphism of A has finite index image.

A subgroup B ⊂ A is characteristic for endomorphisms if, for any injective endomorphism ϕ: A → A, we have ϕ(B) ⊂ B.

Given a group A and g ∈ A, the centralizer of g in A is denoted C_{A}(g). The virtual center of A, denoted V (A), is the set of all g ∈ A such that [A : C_{A}(g)] < ∞. This is a subgroup, because if g, h ∈ V (A) then the subgroup C_{A}(gh), which contains C_{A}(g) ∩ C_{A}(h), has finite index.

Lemma 9.1 (Some characteristic subgroups). Let A be a finitely generated, virtually abelian group. Then the virtual center V (A), its center ZV (A), and its torsion subgroup T ZV (A), are all characteristic for endomorphisms of A. Moreover, V (A) and ZV (A) both have finite index in A, whereas T ZV (A) is finite.

Lemma 9.1 is proved below.

Step 6. Consider the HNN extension G = A_{ϕ} above. Let V (A), ZV (A), T V Z(A) be as in Lemma 9.1, so all these subgroups are taken into themselves by ϕ. Since T V Z(A) is finite we in fact have ϕ(T V Z(A)) = T V Z(A), and so K = T V Z(A) is a finite, normal subgroup of G.

Replacing G by G/K, we may assume that T V Z(A) is trivial, and it follows that ZV (A) is torsion-free abelian, and so is isomorphic to Z^{n}. Since ϕ(ZV (A)) ⊂ ZV (A), the action of ϕ on ZV (A) is given by some n×n matrix of integers N. Thus, G/K has a finite-index subgroup isomorphic to Γ_{N}, finishing the proof of Theorem 1.2.

Proof of Lemma 9.1. To see [A : V (A)] < ∞, if B is any finite-index abelian subgroup of A then obviously B ⊂ V (A).

Consider an endomorphism ϕ: A → A. We now show that ϕ(V (A)) ⊂ V (A). Consider g ∈ V (A), so [A : C_{A}(g)] < ∞. It follows that [ϕ(A) : C_{ϕ(A)}(ϕ(g))] < ∞, and so [A : C_{ϕ(A)}(ϕ(g))] < ∞. But C_{ϕ(A)}(ϕ(g)) ⊂ C_{A}(ϕ(g)), and so ϕ(g) ∈ V (A).

Next we claim that V (V (A)) = V (A). To see why, if g ∈ V (A) then [A : C_{G}(g)] < ∞, and so [V (A) : C_{G}(g) ∩ V (A)] < ∞. But C_{G}(g) ∩ V (A) ⊂ C_{V (A)}(g), and so [V (A) : C_{V (A)}(g)] < ∞, i.e. g ∈ V (V (A)).

Next we claim that [V (A) : ZV (A)] < ∞. In fact if V is any finitely generated group which is its own virtual center, then [V : ZV] < ∞ (the converse is also true, trivially). To see why, let g_{1}, …, g_{k} be a generating set for V. Since V (V) = V, each of the groups C_{V} (g_{1}), …, C_{V} (g_{k}) has finite index in V. It follows that their intersection has finite index in V; but their intersection is precisely ZV.

Now we claim that ZV (A) is characteristic for endomorphisms of V (A) (and so is also characteristic for endomorphisms of A). In fact, if V is any finitely generated group whose center ZV has finite index, then ZV is characteristic for any injective endomorphism ϕ: V → V whose image has finite index. To see why, we have Z(ϕ(V)) = ϕ(ZV), and so

[ϕ(V) : Z(ϕ(V))] = [ϕ(V) : ϕ(ZV)] = [V : ZV] < ∞

Clearly ϕ(V) ∩ ZV ⊂ Z(ϕ(V)), and so

[ϕ(V) : Z(ϕ(V))] ≤ [ϕ(V) : ϕ(V) ∩ ZV]

The quotient group V/ZV is finite, and the quotient homomorphism V → V/ZV, when restricted to the subgroup ϕ(V), has kernel ϕ(V) ∩ ZV. It follows that

[ϕ(V) : ϕ(V) ∩ ZV] ≤ |V/ZV | = [V : ZV] = [ϕ(V) : Z(ϕ(V))]

All of the above inequalities are therefore equalities, and so

ϕ(ZV) = Z(ϕ(V)) = ϕ(V) ∩ ZV

which implies ϕ(ZV) ⊂ ZV.

Finally, it is clear that for any finitely generated abelian group, the torsion subgroup is characteristic for injective endomorphisms.

10 Questions

10.1 Remarks on the polycyclic case

Given an integer matrix M ∈ GL(n, R), the group Γ_{M} is polycyclic if and only if |det M| = 1, and if M ∈ GL_{×}(n, R) this occurs if and only if Γ_{M} is a cocompact discrete subgroup of G_{M}. In this case it follows that Γ_{M} is quasi-isometric to G_{M}, and the notion of horizontal-respecting quasiisometry clearly transfers to Γ_{M}. The techniques of this paper do not provide a quasi-isometric classification in this case, however they do yield the following partial result:

Theorem 10.1. Given M, N ∈ SL(n, Z) lying on 1-parameter subgroups of GL(n, R), there is a horizontal respecting quasi-isometry Γ_{M} → Γ_{N} if and only if there is a horizontal respecting quasi-isometry G_{M} → G_{N}, and this occurs if and only if there are real numbers a, b ≠ 0 such that M^{a}, N^{b} have the same absolute Jordan form. ♢

This raises the question: Is every quasi-isometry Γ_{M} → Γ_{N} horizontal respecting? Equivalently, is every quasi-isometry G_{M} → G_{N} horizontal respecting? The answer is obviously no, for example when M, N are identity matrices and G_{M}, G_{N} are Euclidean spaces. However, we conjecture:

Conjecture 10.2. If M, N ∈ SL(n, Z) lie on 1-parameter subgroups of GL(n, R), and if M, N have no eigenvalues on the unit circle, then any quasi-isometry G_{M} → G_{N} is horizontal respecting.

Moreover, Theorem 10.1 and Conjecture 10.2 together would imply the following (see [FM99c]):

Conjecture 10.3. Suppose M ∈ SL(n, Z) has no eigenvalues on the unit circle. If G is any finitely generated group quasi-isometric to Γ_{M}, then there is a finite normal subgroup F of G so that G/F is abstractly commensurable to Γ_{N}, for some N ∈ SL(n, Z) with no eigenvalues on the unit circle.

10.2 The quasi-isometry group of Γ_{M}

Given a finitely generated group G, the set of quasi-isometries from G to itself, modulo the identification of quasi-isometries which differ by a bounded amount, forms a group called the quasi-isometry group of G, denoted QI(G). Given a 1 × 1 matrix M = (m) with m ≥ 2, the quasi-isometry group of the solvable Baumslag-Solitar group Γ_{M} ≈ BS(1, m) was computed in [FM98]:

QI(BS(1, m)) ≈ Bilip(R) × Bilip(Q_{m}) where Q_{m} is the metric space of m-adic rational numbers, and Bilip(X) denotes the group of bilipschitz self maps of a metric space X.

Problem 10.4. Compute the quasi-isometry group of Γ_{M} in general.

The strongest result we have on this problem so far is Proposition 6.3, but see the remarks after that proposition.

In [FM99b] the computation of QI(BS(1, m)) was applied to prove quasiisometric rigidity of BS(1, m), using techniques of Hinkkanen [Hin85] and Tukia [Tuk86]. While quasi-isometric rigidity of BS(1, m) now has a completely different proof [MSW], which we have here generalized to Γ_{M}, one might still pursue:

Problem 10.5. Give a proof of quasi-isometric rigidity of Γ_{M}, generalizing the results of [FM99b].

This should lead to a deeper understanding of the geometry of Γ_{M}. For example, Tukia [Tuk86] characterizes subgroups of the quasiconformal group of a sphere which are conjugate into the Mobiüs group. We have analogous results for lattices in three-dimensional solv-geometry [FM99c], and there should be generalizations to solvable Baumslag-Solitar groups and to Γ_{M}.

References

[Ben88] D. Benardete, Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups, Trans. AMS 306 (1988), 499–527.

[BG96] M. Bridson and S. Gersten, The optimal isoperimetric inequality for torus bundles over the circle, Quart. J. Math. Oxford Ser. (2) 47 (1996), 1–23.

[BS78] R. Bieri and R. Strebel, Almost finitely presented soluble groups, Comm. Math. Helv. 53 (1978), 258–278.

[BS80] R. Bieri and R. Strebel, Valuations and finitely presented metabelian groups, Proc. London Math. Soc. 41 (1980), no. 3, 439–464.

[BW97] J. Block and S. Weinberger, Large scale homology theories and geometry, AMS/IP Stud. in Adv. Math. 2 (1997), 522–569.

[Dio99] A. Dioubina, On some properties of groups not preserved by quasi-isometry, preprint, St. Petersburg State University, July 1999.

[ECH^{+}92] D.B.A. Epstein, J. Cannon, D.F. Holt, S. Levy, M.S. Paterson, and W.P. Thurston, Word processing and group theory, Jones and Bartlett, 1992.

[FJ89] F. T. Farrell and L. E. Jones, A topological analogue of Mostow’s rigidity theorem, Jour. AMS 2 (1989), no. 2, 257–370.

[FJ97] F. T. Farrell and L. E. Jones, Compact infrasolvmanifolds are smoothly rigid, Geometry from the Pacific Rim (Singapore, 1994), de Gruyter, Berlin, 1997, pp. 85–97.

[FM98] B. Farb and L. Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups, Invent. Math. 131 (1998), no. 2, 419– 451.

[FM99a] B. Farb and L. Mosher, The geometry of surface-by-free groups, in preparation, 1999.

[FM99b] B. Farb and L. Mosher, Quasi-isometric rigidity for the solvable Baumslag-Solitar groups, II, Invent. Math. 137 (1999), no. 3, 613–649.

[FM99c] B. Farb and L. Mosher, Some problems on the geometry of solvable groups, in preparation, 1999.

[FS96] B. Farb and R. Schwartz, The large-scale geometry of Hilbert modular groups, J. Diff. Geom. 44 (1996), no. 3, 435–478.

[GdlH91] E. Ghys and P. de la Harpe, Infinite groups as geometric objects (after Gromov), Ergodic theory, symbolic dynamics and hyperbolic spaces (T. Bedford, M. Keane, and C. Series, eds.), Oxford Univ. Press, 1991.

[Ger93] S. M. Gersten, Quasi-isometry invariance of cohomological dimension, Comptes Rendues Acad. Sci. Paris Série 1 Math. 316 (1993), 411–416.

[Ger95] S. M. Gersten, Isoperimetric functions of groups and exotic cohomology, Combinatorial and geometric group theory, London Mathematical Society Lecture Note Series, vol. 204, Cambridge University Press, 1995, pp. 87–104.

[Gro81] M. Gromov, Groups of polynomial growth and expanding maps, IHES Sci. Publ. Math. 53 (1981), 53–73.

[Gro93] M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory (G. Niblo and M. Roller, eds.), LMS Lecture Notes, vol. 182, Cambridge Univ. Press, 1993.

[Hei74] E. Heintze, On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23–24.

[Hin85] A. Hinkkanen, Uniformly quasisymmetric groups, Proc. London Math. Soc. 51 (1985), 318–338.

[HPS77] M. Hirsch, C. Pugh, and M. Shub, Invariant manifolds, Springer Lecture Notes, no. 583, Springer, 1977.

[KK] M. Kapovich and B. Kleiner, Coarse Alexander duality and duality groups, preprint.

[Mal49] A. Malcev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR Ser., Mat 13 (1949), 9–32, AMS Translation No. 39 (1951).

[Mil68] J. Milnor, A note on curvature and fundamental group, J. Diff. Geom. 2 (1968), 1–7.

[Mos54] G. D. Mostow, Factor spaces of solvable groups, Ann. of Math. 60 (1954), no. 1, 1–27.

[Mos73] G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies, no. 78, Princeton Univ. Press, 1973.

[MSW] L. Mosher, M. Sageev, and K. Whyte, Quasi-actions on trees, in preparation.

[Pan89a] P. Pansu, Dimension conforme et sphère à l’infini des variété à courbure négative, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 2, 177–212.

[Pan89b] P. Pansu, Metriques de Carnot-Caratheodory et quasiisometries des espaces symmetriques de rang un, Ann. of Math. 129 (1989), 1–60.

[Sch96] R. Schwartz, Quasi-isometric rigidity and Diophantine approximation, Acta Math. 177 (1996), 75–112.

[Tuk86] P. Tukia, On quasi-conformal groups, J. D’Anal. Math. 46 (1986), 318–346.

[Wit90] Dave Witte, Topological equivalence of foliations of homogeneous spaces, Trans. AMS 317 (1990), 143–166.

Benson Farb: Department of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

Lee Mosher: Department of Mathematics, Rutgers University, Newark Newark, NJ 07102 E-mail: mosher@andromeda.rutgers.edu