QUASI-ISOMETRIC RIGIDITY OF NONUNIFORM LATTICES IN HIGHER RANK SYMMETRIC SPACES

ALEX ESKIN

Abstract. We compute the quasi-isometry group of a nonuniform lattice in a semisimple Lie group with no rank one factors, and show that any two such lattices are quasi-isometric if and only if they are commensurable up to conjugation.

We recall the following:

Definition Let X and Y be metric spaces. A (κ, C) quasi-isometric embedding is a map ϕ : X ⟶ Y, such that

\frac{1}{κ}d_{X}(x_{1}, x_{2}) − C ≤ d_{Y} (ϕ(x_{1}), ϕ(x_{2})) ≤ κd_{X}(x_{1}, x_{2}) + C (1)

Thus ϕ is bi-Lipschitz on large scales.

If there exists a constant C′ such that every point of Y is within a distance C′ of a point of ϕ(X), then there is a quasi-isometric embedding ψ : Y → X which is a “coarse inverse of ϕ”, i.e. sup_{x∈X} d_{X}(x, ψ(ϕ(x))) < ∞, and sup_{y∈Y} d_{Y} (y, ϕ(ψ(y))) < ∞. In this case ϕ is called a quasiisometry and the spaces X and Y are said to be quasi-isometric.

A basic example to keep in mind is that the fundamental group π_{1}(M) (endowed with the word metric) of a compact Riemannian manifold M is quasi-isometric to the universal cover M̃ of M.

In this paper we prove the following theorem:

Theorem 0.1. Let Γ be an irreducible nonuniform lattice in a semisimple Lie group G with finite center and no rank 1 factors. Let ϕ be a quasi-isometry of Γ to itself, in the word metric. Then there exists an element g in the commensurator of Γ so that ϕ is a bounded distance away from left multiplication by g. In particular, QI(Γ) ≅ Comm(Γ).

Corollary 0.2. Suppose Γ is an irreducible nonuniform lattice in a semisimple Lie group G with finite center and no rank 1 factors, and suppose Λ is a finitely generated group quasi-isometric to Γ. Then there exists a short exact sequence

1 → F → Λ → Γ′ → 1

where F is a finite group, and Γ′ is a lattice commensurable to Γ up to conjugation. In particular, two nonuniform irreducible lattices in G are quasi-isometric if and only if they are commensurable up to conjugation.

We note that the rigidity result for non-uniform lattices is much stronger then the analogous result for uniform lattices (in the higher rank case the result for uniform lattices was first proved in [KL]; see also [EF] for another proof). This phenomenon was first noticed by Richard Schwartz who did much of the work on the subject, and conjectured Theorem 0.1 and Corollary 0.2 in full generality.

Date: September 15, 1996; revised October 23, 1997.

1991 Mathematics Subject Classification. Primary 22E40, 20F32.

Key words and phrases. Lie groups, discrete subgroups, geometric group theory.

supported in part by an N.S.F. Postdoctoral Fellowship.

In [S1] Theorem 0.1 and Corollary 0.2 were proved using geometric methods in the case where the rank of G is 1, so that the symmetric space has negative curvature (the case G = SL(2, ℝ) has to be excluded). In [FS] the same result was proved for the case of quadratic Hilbert modular groups, and it was extended to arbitrary Hilbert modular groups in [S2]. The passage to higher rank and non-positive curvature required development of new methods, in particular the ”coarse topology” and ”coarse separation theorem” of [FS] and [S2]. Even though these methods are quite general, and have proved to be useful in many other questions of coarse geometry (for example [FM]), they do not seem to be suitable for the case when the ℚ-rank of the lattice is bigger then 1, e.g. SL(3, ℤ). Our proof is in fact more geometric then topological, and follows a different scheme from the earlier results.

The basic ingredients in our proof are the ”quasi-flats with holes” result of [EF], the theorem of Lubotzky-Mozes-Raghunathan establishing the equivalence of the word metric with the induced metric, the Moore ergodicity theorem, and Tits’s theory of buildings. The idea of the proof is to prove Theorem 0.1 without extending the map ϕ to the entire symmetric space. Instead (working in the induced metric), we use the quasi-flats with holes result and the Moore ergodicity theorem to construct a map ϕ_{0} of the boundary defined almost everywhere. We then show that this map is bi-Hölder on its domain of definition. This allows us to extend ϕ_{0} to a homeomorphism of the boundaries. We then show that this extended map induces an order preserving bijection of the associated Tits buildings, which implies by Tits’ theorem that ϕ_{0} is induced by an isometry. Then we show that ϕ is within a bounded distance of that isometry. A final argument shows that the element g ∈ G inducing the isometry belongs in fact to the commensurator of Γ.

Most of the argument (through and including the proof of Proposition 6.1) does not use the assumption that G has no rank one factors but only the weaker assumption rank G ≥ 2. In the case when rank G ≥ 2 but G contains rank one factors, we write down an intermediate result, Proposition 10.1, which follows naturally from the argument in this paper. Proposition 10.1 can then be combined with results from [FS], [S1] and [S2] to prove the analogues of Theorem 0.1 and Corollary 0.2 for this case: see [Fa] for an outline of the argument.

Acknowledgements: Thanks to Bob Zimmer for his support and encouragement, to Benson Farb with whom I collaborated on parts of this project, to Richard Schwartz for several useful conversations and to Nimish Shah for supplying the proof of Lemma 8.1.

1. Preliminaries

Notational conventions. We denote by μ(·) the Haar measure on Γ\G, normalized so that μ(Γ\G) = 1. The Euclidean measure on ℝ^{n} is denoted by ν or | · |. We denote the measure of the unit ball in ℝ^{n} by ω_{n}. The complement of a set U is denoted U^{c}.

Let π denote the natural projection from G to the symmetric space G/K = X, and let p denote the natural projection from G to Γ\G. The natural projection from X to Γ\X is denoted by p̄. Let μ̄ denote the measure on Γ\X which is the push-forward of μ; then μ̄(Γ\X) = 1.

We denote the identity element of G by 1, and π(1) ∈ X by e.

1.1. Theorems used in the proof. The proof of Theorem 0.1 is based on the following results:

Theorem 1.1 (Lubotzky-Mozes-Raghunathan). Let Γ be an irreducible lattice in a semisimple Lie group G, with rank G ≥ 2. Then the word metric on Γ is equivalent to the metric induced from the embedding of Γ in the symmetric space G/K.

Proof. See [LMR].

Theorem 1.2 (Moore ergodicity). Let Γ be an irreducible lattice in a semisimple Lie group G with finite center, H ⊂ G a non-compact subgroup. Then the H action on Γ\G is ergodic.

Proof. See [Zim, Theorem 2.2.6].

Theorem 1.3 (Birkhoff ergodic Theorem for ℝ^{n} action). Let (Y, m) be a finite measure space, and suppose A ≅ ℝ^{n} acts on Y, preserving m. If the action is ergodic, then for any f ∈ L^{1}(Y) and almost all y ∈ Y,

lim_{r→∞} \frac{1}{ω_{n}r^{n}} ∫_{B_{r}} f(a · y) dν(a) = ∫_{Y} f dm

where B_{r} = B(0, r) is the ball of radius r centered at 0.

Maximal flats and Cartan subgroups. The symmetric space X is a quotient G/K, where G is the isometry group of X and K is a maximal compact subgroup. The Killing form ⟨·, ·⟩ is a nondegenerate quadratic form on the Lie algebra 𝔤 of G defined by ⟨X, Y ⟩ = trace(ad X ad Y). The Lie algebra 𝔤 can be decomposed as 𝔨 ⊕ 𝔭 where 𝔨 is the Lie algebra of K and 𝔭 is the orthogonal complement relative to the Killing form. We fix a maximal Abelian subalgebra 𝔞 in 𝔭, and denote exp 𝔞 by A. The group A is Abelian and is called a Cartan subgroup of G; 𝔞 is a Cartan subalgebra. The dimension of 𝔞 is the rank of X (or of G), which we denote by n. For any g ∈ G, π(gA) ⊂ X is a maximal flat subspace of X; also every maximal flat can be written as π(gA) for some g ∈ G.

As an immediate corollary of Theorem 1.2 and Theorem 1.3 one has the following:

Theorem 1.4 (Birkhoff ergodic theorem for Cartan flow). Let G, 𝔞 be as above. Then, for any f ∈ L^{1}(Γ\G) and for μ-a.e. x ∈ Γ\G,

lim_{r→∞} \frac{1}{|B(0, r)|} ∫_{B(0,r)⊂𝔞} f(x exp v) dν(v) = ∫_{Γ\G} f dμ

Swiss cheeses. Let U be a subset of ℝ^{n}. We say that U is “an infinite swiss cheese” if it contains a fraction of the volume, i.e. there exists δ < 1 so that for every x ∈ U,

lim inf_{r→∞} \frac{|U ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − δ (2)

We note if the complement of U (i.e. the “holes”) contains balls of arbitrarily large radius, the bound in (2) cannot be uniform in the choice of x ∈ U. Given ϵ > 0 and R > 0, let

U_{(ϵ,R)} = {x ∈ U : for all r > R, |U ∩ B(x, r)| ≥ (1 − ϵ)|B(x, r)|}

Hence, U_{(ϵ,R)} is a subset of U on which a uniform bound exists.

Theorem 1.5 (Quasi-flats with holes). Suppose ϕ is a (κ, C) quasi-isometric embedding from U ⊂ ℝ^{n} to X, where U has the induced Euclidean metric. Let ϵ > 0 and R > 0 be constants, with ϵ sufficiently small (depending only on κ), and U_{(ϵ,R)} non-empty. Then ϕ(U_{(ϵ,R)}) lies in the N neighborhood of the union of at most M flats in X; here M = M(κ) and N = N(κ, ϵ, R, C).

Proof. See [EF, Theorem 8.1].

1.2. Symmetric spaces and quasi-isometries. Most of this subsection is standard: we review the results mostly to fix notation. In what follows, we follow the notation of [EF] where possible. Throughout, by a flat we mean a maximal flat.

Weyl chambers, the Weyl group and the longest element: We denote the root system associated to (G, A) by Σ, and fix a positive system Σ^{+} and associated simple system Δ for Σ. Let 𝔞^{+} denote the subset {H ∈ 𝔞 : α(H) > 0 for all α ∈ Δ}; the set 𝔞^{+} is called the positive Weyl chamber. Let A^{+} = exp 𝔞^{+} ⊂ A. For any g ∈ G, the orbit gA^{+} is a Weyl chamber in G; the set π(gA^{+}) is a Weyl chamber in X.

Let M denote the centralizer of A in K. The Weyl group W is the quotient of the normalizer in K of A by M. This is a finite group which acts transitively on the Weyl chambers of A. Let w_{0} ∈ W denote the longest element; it takes 𝔞^{+} into −𝔞^{+}.

Subflats, hyperplanes, polyhedra and orbits of subgroups of A: For a subset σ of Σ, let

A_{σ} = {a ∈ A | α(log a) = 0, ∀α ∈ σ}

If σ ⊂ Δ, let

A_{σ}^{+} = {a ∈ A : α(log a) = 0, ∀α ∈ σ, and α(log a) > 0, ∀α ∈ Δ − σ.}

We call the orbits of A_{σ} “σ-subflats in G” (or just subflats) and the orbits of A_{σ}^{+} “σ-walls” (or just walls) in G. If σ = ∅ then the orbit of A_{∅}^{+} (denoted A^{+}) is a Weyl chamber in G; if σ = {α} we write A_{α} instead of A_{{α}} and call the orbit a hyperplane in G. By a subflat, wall, Weyl chamber, or hyperplane in X we mean the projection to X of the corresponding object in G via π.

It is easy to check that the intersection of two flats is either empty, or a point or a subflat. Conversely, for every subflat S ⊂ A passing through the identity 1 there exists k_{S} ∈ K such that k_{S}Ak_{S}^{−1} ∩ A = S.

By a polyhedron we mean a (possibly unbounded) convex subset of a flat bounded by hyperplanes. A flat is also considered a polyhedron. If U is a polyhedron in X, we denote by ∂U its boundary as a subset of ℝ^{n}, not as a subset of X (otherwise we will always have ∂U = U).

The Furstenberg boundary X̂ and the Tits building T (X): The Furstenberg boundary X̂ is defined as the equivalence classes of Weyl chambers in X, with two chambers considered equivalent if the Hausdorff distance between them is finite. Each Weyl chamber is equivalent to a Weyl chamber with origin at the point e. Since every Weyl chamber with origin at e is of the form π(kA^{+}) for some k ∈ K, this identifies X̂ with K/M.

If σ ⊂ Δ, we denote by P_{σ} the standard parabolic subgroup associated to σ. We have the Langlands decomposition P_{σ} = M_{σ}A_{σ}N_{σ}, where A_{σ} is as above, M_{σ} is semisimple and N_{σ} is unipotent. Write K_{σ} = M_{σ} ∩ K. P_{∅} is a Borel subgroup of G, denoted by B.

Any parabolic subgroup is of the form gP_{σ}g^{−1} for some σ ⊂ Σ and some g ∈ G. The Tits building T (X) is the set of parabolic subgroups of G, partially ordered by reverse inclusion. Hence the maximal dimension simplices of T (X) correspond to conjugates of B. Since B is its own normalizer, the set of conjugates of B is G/B ≅ K/M, where the last identification used the Iwasawa decomposition G = KB, and the fact that M = K ∩ B.

Thus every maximal dimension simplex of T (X) corresponds to a point in K/M. The same argument shows that every simplex D in T (X) corresponds to a point in K/K_{σ}, with dim(D) = n − |σ|.

This description of the T (X) is convenient for the argument in this paper. As in [EF] we denote the K-invariant distance on K/M by d_{K}.

Galleries, combinatorial distance and the singular subset. Each pair of n-simplices in the Tits building lies in an apartment. A gallery is a sequence of n-simplices, with successive simplices sharing a codimension 1 face. By the Tits combinatorial distance d_{c}(k_{1}, k_{2}) we mean the length of a minimal gallery in the Tits building connecting the n-simplices k_{1}M and k_{2}M. This implies that sup_{k_{1},k_{2}} d_{c}(k_{1}, k_{2}) < ∞. Let ℓ = max_{k_{1},k_{2}} d_{c}(k_{1}, k_{2}), and as in [EF], let S = {k ∈ K : d_{c}(1, k) < ℓ}. The singular subset S ⊂ K is closed, and S = K ∩⋃_{w∈W −{w_{0}}} BwB, where B is the Borel subgroup. This exhibits S as a union of cells, of varying dimensions.

Graded quasi-isometric embeddings. If X_{1} and X_{2} are metric spaces, and z_{0} ∈ X_{1}, we say that a map q : X_{1} → X_{2} is a (κ, ϵ, ρ) graded quasi-isometric embedding centered at z_{0} if for all x, y ∈ X_{1} with d(y, z_{0}) ≥ d(x, z_{0}),

\frac{1}{κ}d(x, y) − max(ρ, ϵd(x, z_{0})) ≤ d(q(x), q(y)) ≤ κd(x, y) + max(ρ, ϵd(x, z_{0})) (3)

Thus any (κ, ρ) quasi-isometric embedding is also a (κ, ϵ, ρ) graded quasi-isometric embedding. Graded quasi-isometric embeddings are more general in the sense that the “coarseness constant” is allowed to grow linearly with distance to the origin z_{0}.

Graded q.i. embeddings (c.f [EF, §3.1]) are a useful technical device both for the proof of Theorem 1.5 and for some of the arguments below. Two (κ, ϵ, ρ) graded q.i. embeddings q_{1} and q_{2} centered at the same point z_{0} are equivalent if for all x ∈ X_{1}, d(q_{1}(x), q_{2}(x)) ≤ max(ρ, ϵd(x, z_{0})).

Neighborhoods, Conical neighborhoods.

We denote the r-tubular neighborhood of a set U by Nbhd(U, r). The metric interior Int(U, r) is defined to be U \ Nbhd(∂U, r).

Let U be a subset of a metric space Y, and y_{0} ∈ Y a point. We let U[c; y_{0}] be a neighborhood of U whose size is proportional to the distance from y_{0}, namely:

U[c; y_{0}] = {y ∈ Y : ∃u ∈ U with d(y, u) < cd(y, y_{0})}

As in [EF], if U ⊂ ℝ^{n}, we write U[c] as shorthand for U[c; 0], and if U ⊂ X, we write U[c] as shorthand for U[c; e].

Hausdorff distance and ϵ-equivalence: If U, V are subsets of a metric space, we denote the Hausdorff distance between their closures by hd(U, V). If hd(U, V) < ∞, we say that U is equivalent to V and write U ∼ V.

Let x_{0} be a point and let A_{R}(x_{0}) denote the annulus {x : R < d(x, x_{0}) < 2R}. We say that two subsets U and V are ϵ-equivalent and write U ∼_{ϵ} V if for all sufficiently large R,

hd(U ∩ A_{R}(x_{0}), V ∩ A_{R}(x_{0})) ≤ ϵR. (4)

(clearly this is meaningful only if ϵ is small).

Because of the triangle inequality, the notion of ϵ-equivalence is (almost) independent of the point x_{0}. Indeed, if U ∼_{ϵ} V with origin x_{0}, and z_{0} is any other point, then U ∼_{δ} V with origin z_{0} for any δ > ϵ.

We will occasionally need a more quantitative version. We write U ∼_{ϵ,x_{0},ρ} V if for all R > ρ, (4) holds. It is easy to check that if U ∼_{ϵ,x,ρ} V then outside the ball B(x, ρ)

U ⊂ V [2ϵ; x] and V ⊂ U[2ϵ; x]. (5)

Conversely, if (5) holds outside the ball B(x, ρ), then U ∼_{4ϵ,x,ρ} V.

Multiplicative constants depending only κ and X: As in [EF], the argument produces a proliferation of constants depending only on κ and X. (Essentially there is an extra factor of κ each time (1) or (3) is used). Thus we use the ≫, ≪, ≺, ≻ and O(·) notations in the same way as defined in [EF, §3.1].

1.3. Structure of the proof. The proof of Theorem 0.1 is a succession of steps; in each step we gain more control over the map ϕ. Each step is proved in its own section; only the results stated in the introductory subsection of each section are used in subsequent sections.

We now begin the proof of Theorem 0.1. In view of Theorem 1.1, we may assume that ϕ is a (κ, C) quasi-isometry Γ → Γ in the induced metric.

Step 1 (Extending ϕ to the “neutered space”). For any compact subset C of Γ\X, ϕ can be extended to a (κ, C′) quasi-isometry of p̄^{−1}(C) to p̄^{−1}(C). Here C′ depends on κ, C and C.

Proof. This construction is now standard; see e.g. [S1], [FS], [S2]. Since for every point y of p^{−1}(C) is within a uniformly bounded distance of a point in Γ, one may just define ϕ(y) to be ϕ(γ) where γ is the point in Γ closest to y, (one chooses randomly if there are several possibilities for γ). The resulting map is easily verified to be a quasi-isometry.

For notational convenience, we define ϕ on π^{−1}(p̄^{−1}(C)) ⊂ G by making ϕ constant along the K fibers.

Choice of C and ϵ. We choose ϵ > 0 so that Theorem 1.5 holds, and also so that ϵ ≪ 1, i.e. for any multiplicative constant λ = λ(κ) which arises in §§2-§7 λϵ < 1/100. Note that ϵ depends only on κ.

We now choose C so that μ̄(C) > 1 − ϵ/4. The set C and the constant ϵ are fixed throughout the argument.

2. Quasi-flats with holes and the Moore ergodicity theorem

Definition 2.1 (Γ-invariant family of flats). A family of F of flats is Γ-invariant if for every γ ∈ Γ and every flat F ∈ F, γF ∈ F. A Γ-invariant family F of flats is said to have full measure if there exists a set E ⊂ Γ\G with μ(E) = 1 such that for any y ∈ p^{−1}(E), π(yA) ∈ F.

The approach in this paper is based on the following lemma, which follows trivially from the ergodic theorem Theorem 1.4:

Lemma 2.2. There exists constants R > 0, ρ > 0, a family F of flats in X and for each F ∈ F subsets Ω_{F} ⊂ Ω′_{F} ⊂ F with the following properties:

(a) F is a Γ-invariant family of full measure.

(b) For every F ∈ F, U_{F} = F ∩ p̄^{−1}(C) is not empty. In fact, for any x ∈ F,

lim inf_{r→∞} \frac{|U_{F} ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − \frac{ϵ}{4}

The quasi-isometry ϕ is defined on U_{F}.

(c) For every F ∈ F, Ω′_{F} ⊂ (U_{F})_{(ϵ,R)} is not empty. In fact, for any x ∈ F,

lim inf_{r→∞} \frac{|Ω′_{F} ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − \frac{ϵ}{2} (6)

where the notation (U_{F})_{(ϵ,R)} is as in Theorem 1.5.

(d) For every F ∈ F, Ω_{F} ⊂ (Ω′_{F})_{(ϵ,ρ)} is not empty. In fact for any x ∈ F,

lim inf_{r→∞} \frac{|Ω_{F} ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − \frac{ϵ}{2}

Remark. From (c) and Theorem 1.5, it follows that for all F ∈ F, ϕ(Ω′_{F}) is close to a finite union of flats. Thus, in view of (6) we have information about the image under ϕ of “most of the volume in almost all flats”.

We need (d) for a technical reason which will become apparent shortly.

Proof of Lemma 2.2. Let χ denote the characteristic function of the set π^{−1}(C) ⊂ Γ\G. Then, in our normalizations, ∫_{Γ\G} χ dμ ≥ 1 − ϵ/4. Then by Theorem 1.4, there exists a set E′ ⊂ Γ\G such that μ(E′) = 1, and for all x ∈ E′,

lim_{r→∞} \frac{1}{|B_{r}|} ∫_{B_{r}} χ(x exp v) dν(v) ≥ 1 − ϵ/4.

where B_{r} denotes the ball B(0, r) ⊂ 𝔞.

Suppose y ∈ p^{−1}(E′), and let F = π(yA). Note that for v ∈ 𝔞,

(7) π(y exp v) ∈ U_{F} = F ∩ p̄^{−1}(C) precisely when χ(y exp v) = 1.

Thus if we let F_{1} = {π(yA) : y ∈ p^{−1}(E′)}, (a)-(b) of Lemma 2.2 hold for F_{1} instead of F. To get (c) we may argue as follows: For any j > 0, let

(8) E′_{j} = \{x ∈ E′ : for all r > j, \frac{1}{|B_{r}|} ∫_{B_{r}} χ(x exp v) dν(v) ≥ 1 − ϵ/2\}

Then E′_{j} is an increasing family of sets and ⋃_{j=1}^{∞} E′_{j} = E′. Thus, there exists an integer R such that μ(E′_{R}) ≥ 1 − ϵ/4. Let E_{R} = E′_{R} ∩ π^{−1}(C); then μ(E_{R}) ≥ 1 − ϵ/2.

Note that if y ∈ p^{−1}(E′) and F = π(yA), then in view of (7) and (8), π(yA∩p^{−1}(E_{R})) ⊂ (U_{F})_{(ϵ,R)}. Thus we let Ω′_{F} = π(yA ∩ p^{−1}(E_{R})).

Let χ_{R} denote the characteristic function of E_{R}. Then by the ergodic theorem Theorem 1.4, there exists a set E′′ ⊂ E′ with μ(E′′) = 1 such that for x ∈ E′′,

lim_{r→∞} \frac{1}{|B_{r}|} ∫_{B_{r}} χ_{R}(x exp v) dν(v) = μ(E_{R}) ≥ 1 − ϵ/2 (9)

Thus we can let F_{2} = {π(yA) : y ∈ p^{−1}(E′′)}. Since μ(E′′) = 1, part (a) holds for F_{2}. Since F_{2} ⊂ F_{1}, part (b) also holds. Finally part (c) holds for F_{2} in view of (9).

To get (d) we repeat the argument for (c). Let

(10) E′′_{j} = \{x ∈ E′′ : for all r > j, \frac{1}{|B_{r}|} ∫_{B_{r}} χ_{R}(x exp v) dν(v) ≥ 1 − ϵ\}

Then E′′_{j} is an increasing family, with ⋃_{j=1}^{∞} E′′_{j} = E′′. Thus there exists an integer ρ such that μ(E′′_{ρ}) ≥ 1 − ϵ/4. Let Ω = E′′_{ρ} ∩ π^{−1}(C), then μ(Ω) ≥ 1 − ϵ/2. For F = π(yA) ∈ F_{2}, we set Ω_{F} = π(yA ∩ Ω). Then in view of (10), Ω_{F} ⊂ (Ω′_{F})_{(ϵ,ρ)}. Let χ_{Ω} denote the characteristic function of Ω. Thus π(y exp v) ∈ Ω_{F} precisely when p(y exp v) ∈ Ω.

By Theorem 1.4, there exists a subset E′′′ ⊂ E′′ with μ(E′′′) = 1 such that for any x ∈ E′′′,

(11)  lim_{r→∞} \frac{1}{|B_{r}|} ∫_{B_{r}} χ_{Ω}(x exp v) dν(v) = μ(Ω) ≥ 1 − ϵ/2

Thus we can let F = {π(yA) : y ∈ p^{−1}(E′′′)}. Part (a) holds since μ(E′′′) = 1, parts (b)-(c) hold since F ⊂ F_{2}, and part (d) holds because of (11).

Combining Lemma 2.2 with Theorem 1.5 we get the following:

Step 2 (ϕ maps almost all flats to finite unions of flats). There exist constants M = M(κ), ρ = ρ(κ, ϵ, C) and N = N(κ, ϵ, C, ρ), a family of flats F, and for each F ∈ F nonempty subsets Ω_{F} ⊂ Ω′_{F} ⊂ F, such that the following holds:

(a) F is a Γ-invariant family of full measure.

(b) For every F ∈ F, there exist at most M flats F′_{1}, …, F′_{M} such that ϕ(Ω′_{F}) lies within a distance N of ⋃_{j=1}^{M} F′_{j}.

(c) For every F ∈ F, every x ∈ Ω_{F}, and every r > ρ,

\frac{|B(x, r) ∩ Ω′_{F} |}{|B(x, r)|} ≥ 1 − ϵ (12)

Hence there exists a map ϕ̃ : F → Nbhd(⋃_{j=1}^{M} F′_{j}, N), which agrees with ϕ on Ω′_{F}, and is a (κ, 2κϵ, ρ) graded quasi-isometric embedding centered at x for any x ∈ Ω_{F}.

(d) For every F ∈ F, and any x ∈ F,

lim inf_{r→∞} \frac{|Ω_{F} ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − ϵ (13)

(e) There exists a subset Ω ⊂ Γ\G with μ(Ω) ≥ 1 − ϵ/2 such that for any x ∈ p^{−1}(Ω), F = π(xA) ∈ F and π(x) ∈ Ω_{F}.

The map ϕ̃ of part (c) is just the map obtained by ϕ̃(u) = ϕ(v) where v a point of Ω′_{F} nearest u (c.f. Step 1). In view of (12), ϕ̃ is indeed a (κ, 2κϵ, ρ) graded q.i. embedding.

The set Ω of part (e) is the same as in the proof of Lemma 2.2. Part (e) will be used in §7 to “return from the boundary”.

3. Mapping flats to flats

Step 3 (ϕ maps most flats to single flats). There exists a full-measure Γ-invariant family of flats F^{∗} ⊂ F such that for every F ∈ F^{∗} the following hold:

(i) Parts (b) and (c) of Step 2 hold with M = 1; i.e. there exists a flat F′ such that ϕ(Ω′_{F}) ⊂ Nbhd(F′, N). (the sets Ω_{F}, Ω′_{F} and the constant N are the same as in Step 2).

(ii) There exists a constant λ = λ(κ) such that for every Weyl chamber ℭ ⊂ F there exists a chamber ℭ′ ⊂ F′ such that ϕ(ℭ ∩ Ω′_{F}) ∼_{λϵ} ℭ′.

Remark. The proof of this step is somewhat involved.

Transverse Flats. For each α ∈ Σ, we pick k_{α} ∈ K such that:

k_{α}Ak_{α}^{−1} ∩ A = A_{α} = {a ∈ A : α(a) = 0}.

We fix the k_{α} for the remainder of the paper. Now if F = π(zA) is a flat, and x = π(za) ∈ F, then we define

trans_{α}(x; z) = π(zak_{α}A)

Then F_{α} = trans_{α}(x; z) is a flat, which intersects F in the hyperplane L_{α} = π(zaA_{α}) passing through x. Let

Trans_{F} (x) = {trans_{α}(x; z) : α ∈ Σ, and F = π(zA)}

We call any flat F_{0} ∈ Trans_{F} (x) a transverse flat to F at x. The name is motivated by the following lemma:

Lemma 3.1 (Transverse flats). Suppose F is a flat, x ∈ F, and F_{0} ∈ Trans_{F} (x). Then

(a) There exists a hyperplane L ⊂ F with x ∈ L and F ∩ F_{0} = L.

(b) There exists constants λ_{0}, λ_{1} depending only on X (and the choice of the k_{α}) such that for r > λ_{0},

hd(F ∩ Nbhd(F_{0}, r), L) ≤ λ_{1}r

This lemma expresses in a quantitative way the fact that the flats F and F_{0} come close only near L. Part (a) follows from the definition of k_{α}. Part (b) is essentially the fact that geodesics in F_{1} and F_{2} emanating from a point of L and orthogonal to L diverge; cf. the discussion at the beginning of §5.

The sets Ω_{z}^{∗} and Ω_{F}^{∗}: Suppose F = π(zA) ∈ F. Let

(14) Ω_{z}^{∗} = {x ∈ Ω_{F} : trans_{α}(x; z) ∈ F and x ∈ Ω_{trans_{α}(x;z)} for all α ∈ Σ}

Hence if x ∈ Ω_{z}^{∗}, then for every hyperplane L passing through x there exists a flat F_{L} ∈ Trans_{F} (x) such that x ∈ Ω_{F_{L}}.

As we defined it, the set Ω_{z}^{∗} depends on z, and not only on F = π(zA). Even though it is not essential for the argument, to simplify some notation we set

Ω_{F}^{∗} = ⋃_{{z with π(zA)=F }} Ω_{z}^{∗}

Thus Ω_{F}^{∗} ⊂ Ω_{F} ⊂ Ω′_{F} ⊂ F.

Lemma 3.2 (the family F^{∗}). There exists a full-measure Γ-invariant family F^{∗} ⊂ F such that for any F ∈ F^{∗} and any x ∈ F,

lim inf_{r→∞} \frac{|Ω_{F}^{∗} ∩ B(x, r)|}{|B(x, r)|} ≥ 1 − |Σ|ϵ

Proof. Let

Ω^{∗} = ⋂_{α∈Σ} Ωk_{α}^{−1}

where Ω is as in the proof of Lemma 2.2. Then μ(Ω^{∗}) ≥ 1 − |Σ|ϵ. Suppose z ∈ G. Note that if p(za) ∈ Ω^{∗} for some a ∈ A, then p(zak_{α}) ∈ Ω for all α ∈ Σ, hence x = π(za) = π(zak_{α}) ∈ Ω_{π(zak_{α}A)} = Ω_{trans_{α}(x;z)} for all α ∈ Σ. Hence:

x = π(za) ∈ Ω_{z}^{∗} if p(za) ∈ Ω^{∗} (15)

Let χ_{Ω^{∗}} be the characteristic function of Ω^{∗}. By the ergodic theorem Theorem 1.4, there exists a set E^{∗} ⊂ E′′′ ⊂ Γ\G with μ(E^{∗}) = 1 (where E′′′ is as in the proof of Lemma 2.2) such that for all x ∈ E^{∗},

(16)  lim_{r→∞} \frac{1}{|B_{r}|} ∫_{B_{r}} χ_{Ω^{∗}}(x exp v) dν(v) = μ(Ω^{∗}) ≥ 1 − |Σ|ϵ

Let F^{∗} = {π(zA) : z ∈ p^{−1}(E^{∗})}. Then for F ∈ F^{∗} Lemma 3.2 holds in view of (15) and (16).

Step 3 will follow from the following:

Proposition 3.3. Let F^{∗} be the family defined in Lemma 3.2. Then for every flat F ∈ F^{∗}, (i) and (ii) of Step 3 hold.

The proof of Proposition 3.3, which involves no ergodic theory, occupies the rest of this section. Along the way, we will prove the following result, which will be used in §5:

Proposition 3.4. There exist constants λ = λ(κ) and ρ′ = ρ′(κ, C, ϵ, X) such that the following holds: Suppose F ∈ F^{∗}, and x ∈ Ω_{F}^{∗}. Then:

(a) For every hyperplane L ⊂ F passing through x there exists a hyperplane L′ such that ϕ̃(L) ∼_{λϵ,ϕ̃(x),ρ′} L′.

(b) For every chamber ℭ ⊂ F based at x there exists a chamber ℭ′ such that ϕ̃(ℭ) ∼_{λϵ,ϕ̃(x),ρ′} ℭ′.

3.1. Intersections of quasi-flats. Our aim in this subsection is to prove the following:

Lemma 3.5 (Hyperplanes map to finite unions of hyperplanes). Suppose F ∈ F^{∗}, x_{0} ∈ F and R is sufficiently large (depending on F). Let A_{R} = A(R, 2R; ϕ(x_{0})) ⊂ X, Then there exists M′ > 0 (depending only on κ, X) such that for any hyperplane L in F, there exist at most M′ hyperplanes L′_{1} ⋯ L′_{M′} such that

ϕ̃(L) ∩ A_{R} ⊂ Nbhd ( ⋃_{j=1}^{M′} L′_{j}, O(ϵR)) ∩ A_{R}

The proof uses the following lemma:

Lemma 3.6 (Quasi-flats are unions of polyhedra). Suppose F ∈ F, x ∈ Ω_{F} and R > λ′ρ (where ρ is as in Step 2 and λ′ = λ′(κ)). Let B_{R} denote B(ϕ(x), R). Then there exist m = m(κ) and λ = λ(κ) and convex polyhedra P_{1}, …, P_{m} such that

hd (ϕ̃(F) ∩ B_{R}, ⋃_{j=1}^{m} P_{j} ∩ B_{R}) < λϵR

Remark. If instead of the assumption R ≫ ρ we made the assumption “R sufficiently large depending on F ”, Lemma 3.6 would follow from [EF, Lemma 8.3] (restated as Lemma 3.11 below). For the case of non-graded quasi-isometries, intersecting with a ball is not necessary. Thus Lemma 3.6 can be viewed as a strengthening of [EF, Lemma 8.3] in the sense that a the Hausdorff distance from a quasi-flat to some explicitly given set is bounded only in terms of the quasi-isometry constants.

Proof of Lemma 3.6. By Step 2 we have a (κ, 2κϵ, ρ) graded quasi-isometry centered at x, ϕ̃: F → Nbhd(⋃_{j=1}^{M} F′_{j}, N).

Fix η′′ ≫ η ≫ η′ ≫ 1. In the proof of this lemma, all implicit constants depend only on κ and X, and the dependence on η′′, η and η′ is shown explicitly.

Let λ_{0} be as in Lemma B.1 part (i). For each pair (i, j) there is an interval I_{ij} = [d(F′_{i}, F′_{j}), λ_{0}(1+ d(F′_{i}, F′_{j}))] for which Lemma B.1 cannot be applied. Since the number of flats M is O(1) (i.e depends only κ and X), we may choose r_{0} = O(η′′ϵR) such that r_{0} ≫ ϵR and [r_{0}, η′′r_{0}] does not intersect any I_{ij}. For example, r_{0} = r_{0}(η′′ϵR) can be constructed as follows: Denote by I the union of the half-open intervals (\frac{1}{η′′} d(F′_{i}, F′_{j}), λ_{0}(1 + d(F′_{i}, F′_{j})]. Then let r_{0}(t), t ∈ (0, ∞) be a piecewise linear function (with slope 1) continuous from the right, and with jumps such that the range of r_{0}(t) is (0, ∞) \ I. Note that t ≤ r_{0}(t) ≤ λ_{1}(η′′)^{M(M−1)/2}t where λ_{1} depends only on λ_{0} and M.

For r_{0} < r < η′′r_{0}, let P_{ij}(r) be as in Lemma B.1 part (i) with F_{1} = F′_{i}, F_{2} = F′_{j} if r_{0} > λ_{0}(1 + d(F′_{i}, F′_{j})) and let P_{ij}(r) = ∅ if d(F′_{i}, F′_{j}) > η′′r_{0}. Note that P_{ii}(r) = F′_{i}.

For r ∈ [r_{0}, ηr_{0}], s ∈ [r_{0}, ηr_{0}], s < r and σ ⊂ {1, …, M} let

P_{i,σ}(s, r) = ⋂_{j∈σ} P_{ij}(s) ⋂_{j∉σ} P_{ij}(r)^{c}

Note that

F′_{i} = ⋃_{σ⊂{1,...,M}} P_{i,σ}(s, r) \ ⋃_{j=1}^{M}(P_{ij}(r) \ P_{ij}(s))

From its definition P_{ij}(r) \ P_{ij}(s) is contained in a union of neighborhoods of hyperplanes, each of width O(ηr_{0}). Hence, for each i,

hd (F′_{i}, ⋃_{σ⊂{1,...,M}} P_{i,σ}(s, r)) = O(ηr_{0}) (17)

For any a < b, let

F′_{i,σ}(a, b) = {x ∈ F′_{i} : d(x, F′_{j}) ≤ a, ∀j ∈ σ  and  d(x, F_{j}) > b, ∀j ∉ σ} (18)

Note that F′_{i,σ} = ∅ if i ∉ σ. Let λ_{1}, λ_{2} be as in Lemma B.1 part (i). Then, by Lemma B.1,

⋂_{j∈σ} P_{ij}(λ_{1}a) ⋂_{j∉σ} P_{ij}(λ_{2}b)^{c} ⊂ F′_{i,σ}(a, b) ⊂ ⋂_{j∈σ} P_{ij}(λ_{2}a) ⋂_{j∉σ} P_{ij}(λ_{1}b)^{c}

Hence,

P_{i,σ}(λ_{1}a, λ_{2}b) ⊂ F′_{i,σ}(a, b) ⊂ P_{i,σ}(λ_{2}a, λ_{1}b) (19)

The hyperplanes bounding P_{i,σ}(s, r) subdivide F′_{i} into convex polyhedra; for each pair (s, r) there is a certain combinatorial pattern (i.e which hyperplanes bound which convex polyhedra). The possible number of patterns is bounded depending only on X, and it is easy to see that the set of (s, r) which give rise to a fixed pattern is a convex subset of ℝ^{2}. Thus we have a decomposition of the square [r_{0}, ηr_{0}]^{2} into a finite number of disjoint convex sets. Hence, if η ≫ η′ is sufficiently large, one of the sets must contain a box of side length ηr_{0}/η′ i.e. there exists (s_{1}, r_{1}) ∈ [r_{0}, ηr_{0}]^{2} such that the combinatorial pattern is the same for all (s, r) ∈ [s_{1}/η′, s_{1}] × [r_{1}/η′, r_{1}]. Without loss of generality, we may assume that r_{1} > s_{1}. Hence,

hd (P_{i,σ}(s, r), P_{i,σ}(s_{1}, r_{1})) = O(r_{1}) for all (s, r) ∈ [s_{1}/η′, s_{1}] × [r_{1}/η′, r_{1}]. (20)

Suppose a ∈ [2s_{1}/(λ_{1}η′), s_{1}/(2λ_{2})] and b ∈ [2r_{1}/(λ_{1}η′), r_{1}/(2λ_{2})] Then by (20) and (19)

hd(F′_{i,σ}(a, b), P_{i,σ}(s_{1}, r_{1})) = O(r_{1}) = O(ηr_{0}) (21)

By (18), for any i, j ∈ σ, F′_{i,σ}(a, b) ⊂ Nbhd(F′_{j,σ}(2a, b − a), a). Hence, by (21),

hd (P_{i,σ}(s_{1}, r_{1}), P_{j,σ}(s_{1}, r_{1})) = O(r_{1}) = O(ηr_{0}) for all i, j ∈ σ (22)

Write P_{σ} = P_{i,σ}(s_{1}, r_{1}) where i = min_{j∈σ} j. Hence, combining (22) and (17),

hd ( ⋃_{j=1}^{M} F′_{j}, ⋃_{σ⊂{1,...,M}} P_{σ}) = O(ηr_{0}) (23)

Hence (after changing ϕ̃ by O(ηr_{0})) we may view ϕ̃ as an (κ, C′) quasi-isometry ϕ̃ : B → ⋃ P_{σ}, where C′ = O(ηr_{0}) and B is a ball containing ϕ̃^{−1}(B_{R}). We would like to apply Lemma A.3; we now verify that for any r ≫ ηr_{0} the estimate (46) is satisfied for (the convex pieces of) the P_{σ}, with C′ in place of C. First suppose σ and τ are not disjoint; choose j ∈ σ ∩ τ. By (22), the Hausdorff distance from P_{σ} to P_{j,σ}(s_{1}, r_{1}) is O(ηr_{0}), and the Hausdorff distance from P_{τ} to P_{j,τ} (s_{1}, r_{1}) is also O(ηr_{0}). But P_{j,σ}(s_{1}, r_{1}) and P_{j,τ} (s_{1}, r_{1}) are disjoint polyhedra lying in the flat F′_{j}. Then, the required estimate clearly holds for any r ≫ C′. If σ and τ are disjoint, then pick j ∈ τ \ σ. Since

Int(P_{i,σ}(s_{1}, r_{1}), r) ⊂ P_{i,σ}(s_{1}, r_{1} + r) ⊂ F′_{i,σ}(λ_{1}^{−1}s_{1}, λ_{2}^{−1}(r_{1} + r))

for any x ∈ Int(P_{i,σ}, r), and since j ∉ σ, we have d(x, F′_{j}) ≥ λ_{2}^{−1}(r_{1} + r) ≥ λ_{2}^{−1}r. Hence, for any y ∈ B(x, 4κη_{1}η_{2}C′), d(y, F′_{j}) ≥ λ_{2}^{−1}r − 4κη_{1}η_{2}C′ ≻ r if r ≫ C′. But, by (19), for any z ∈ P_{τ}, d(z, F_{j}) = O(ηr_{0}) = O(C′). Hence, y ∉ P_{τ}. Thus (46) holds. Now the lemma follows from Lemma A.3.

Lemma 3.7 (Closeness sets of polyhedra). Suppose F_{1}, F_{2} ∈ F are flats, x ∈ Ω_{F_{1}} ∩ Ω_{F_{2}} and F_{2} ∈ Trans_{F_{1}}(x). Choose any R such that R ≫ ρ, and write B_{R} for B(ϕ̃(x), R). Let P_{1}, P_{2} ⊂ X be polyhedra and suppose there exist subsets U_{i} ∈ F_{i} such that hd(ϕ̃(U_{i}), P_{i} ∩ B_{R}) = O(ϵR). Then, for any r with ϵR ≪ r ≪ R:

(a) The injectivity radius of P_{1} ∩ Nbhd(P_{2}, r) ∩ B_{R} is O(r).

(b) The exists a hyperplane L ⊂ P_{1} such that

P_{1} ∩ Nbhd(P_{2}, r) ∩ B_{R} ⊂ Nbhd(L, O(r)) ∩ B_{R}.

Proof. Let d_{0} = d(F′_{1}, F′_{2}) where the F′_{j} are flats containing P_{j}. Let d(r) = r when 0 < r < d_{0}, and d(r) = λ_{0}(1 + d_{0}) + r − d_{0} when r ≥ d_{0}. Then r ≤ d(r) ≤ λ_{0}(r + 1) and the range of d(r) does not contain the interval [d_{0}, λ_{0}(1 + d_{0})) where Lemma B.1 cannot be applied. If ϵR ≪ r ≪ R then d(r) = O(r) and by Lemma B.1 part (iii) there exists a polyhedron Q ⊂ P_{1} such that for hd(Q, P_{1} ∩ Nbhd(P_{2}, d(r))) = O(r).

By assumption, ϕ̃ has a coarse inverse ϕ^{∗} : (P_{1} ∪ P_{2}) ∩ B_{R} → U_{1} ∪ U_{2}, which is a (κ, C′) quasiisometric embedding with C′ = O(ϵR). By Lemma A.4,

(24) ϕ^{∗}(P_{1} ∩ Nbhd(P_{2}, d(r) ∩ B_{R}) ⊂ Nbhd(U_{1}, O(r)) ∩ Nbhd(U_{2}, O(r)) ⊂

⊂ Nbhd(F_{1}, O(r)) ∩ Nbhd(F_{2}, O(r))

By Lemma 3.1, the injectivity radius of Nbhd(F_{1}, O(r)) ∩ Nbhd(F_{2}, O(r)) is O(r). Hence, since ϕ^{∗} is a quasi-isometry, the injectivity radius of P_{1} ∩ Nbhd(P_{2}, d(r)) ∩ B_{R} and hence of Q is O(r). Now the assertion (b) follows from Lemma B.9 and the fact that d(r) ≥ r.

Proof of Lemma 3.5: Choose R_{0} sufficiently large so that for any y ∈ F outside B_{R_{0}}, there exists a point x ∈ Ω_{F}^{∗} with d(x, y) = O(ϵd(y, x_{0})), and suppose R > 4κR_{0}.

If L does not intersect B = B(x_{0}, 4κR) then ϕ̃(L) does not intersect B(ϕ(x_{0}), 2R) and there is nothing to prove. If L does intersect then B we can replace L by a parallel hyperplane L_{1} such that hd(L ∩ B, L_{1} ∩ B) = O(ϵR), and L_{1} contains a point x ∈ Ω_{F}^{∗}. Choose F_{0} ∈ Trans_{F} (x) such that F ∩ F_{0} = L_{1}.

By Lemma 3.6, there exist finite unions of polyhedra, P_{1}, …, P_{M} and Q_{1}, …, Q_{M} such that hd(ϕ̃(F) ∩ A_{R}, ⋃ P_{i} ∩ A_{R}) = O(ϵR) and hd(ϕ̃(F_{0}) ∩ A_{R}, ⋃ Q_{j} ∩ A_{R}) = O(ϵR). By Lemma A.4,

(25) ϕ̃(L_{1}) ∩ A_{R} ⊂ Nbhd(⋃_{i=1}^{M} P_{i}, O(ϵR)) ∩ Nbhd( ⋃_{j=1}^{M} Q_{j}, O(ϵR)) ∩ A_{R} ⊂

⊂ ⋃_{i=1}^{M} ⋃_{j=1}^{M} Nbhd(P_{i}, O(ϵR)) ∩ Nbhd(Q_{j}, O(ϵR)) ∩ A_{R}

But by Lemma 3.7 (b) with U_{1} = ϕ̃^{−1}(P_{i} ∩ B_{4R}), U_{2} = ϕ̃^{−1}(Q_{j} ∩ B_{4R}), for every i, j there exists a hyperplane L′_{ij} ⊂ P_{i} such that

Nbhd(P_{i}, O(ϵR)) ∩ Nbhd(Q_{j}, O(ϵR)) ∩ A_{R} ⊂ Nbhd(L′_{ij}, O(ϵR)) ∩ A_{R}

This implies that the lemma holds for L_{1}. Hence it also holds for L.

The same argument as in the proof of Lemma 3.5 also shows the following variant of Lemma 3.5:

Lemma 3.8. Suppose F is a flat such that ϕ̃(F) is close to a single flat F′ (in the sense of Step 2 (c)). Suppose x ∈ Ω_{F}^{∗}, and L is a hyperplane in F passing through x. Suppose R ≫ ρ, where ρ is as in Step 2. Let A_{R} = A(R, 2R; ϕ(x)) ⊂ X, Then there exists M′ > 0 (depending only on κ, X) such that there exist at most M′ hyperplanes L′_{1} ⋯ L′_{M′} ⊂ F′ with

ϕ̃(L) ∩ A_{R} ⊂ Nbhd ( ⋃_{j=1}^{M′} L′_{j}, O(ϵR)) ∩ A_{R}

3.2. A certain class of quasi-isometries ℝ^{n} → ℝ^{n}.

Lemma 3.9 (Complexes). Let S_{1}, …, S_{r} be r fixed hyperplanes passing through the origin in ℝ^{n}; their intersections form a cellular complex Y, whose maximal cells have dimension n − 1. Suppose f is a 1 − 1 cellular map from Y to Y (i.e f maps cells to cells, and preserves adjacency relations). Then either f sends each hyperplane to a hyperplane, or there exist distinct hyperplanes L and M such that f(L) separates ℝ^{n} into two connected components H_{1} and H_{2}, and f(M) lies entirely in the closure of H_{1}.

Proof. We may assume that f is a continuous map Y → Y which respects the simplicial complex structure. By the Jordan curve theorem, f(L) separates ℝ^{n} into two connected components H_{1} and H_{2}. Suppose f(L) is not a hyperplane. Then there exist two adjacent cells C_{1} and C_{2} in L such that f(C_{1}) and f(C_{2}) do not lie in the same hyperplane. Since C_{1} and C_{2} are adjacent, there exists a subspace C of codimension 2 containing C_{1} ∩C_{2}. Pick x_{0} ∈ C_{1} ∩C_{2} away from the origin, and let H_{0} be a 2 dimensional plane passing through f(x_{0}) and orthogonal to f(C). Denote by D_{0} = D_{0}(ϵR) a disk in H_{0} centered at x_{0} and of radius ϵR where R is the distance from x_{0} to the origin and ϵ > 0 is small.

Then for any hyperplane M ⊃ C, f(M) ∩ D_{0} is “broken line”, i.e. a one-dimensional piecewise straight curve, with the (possible) turn at f(x_{0}). For M = L, there is indeed a “turn” at f(x_{0}), i.e f(L) ∩ D_{0} consists of two half-lines intersecting at f(x_{0}). Note that f(L) ∩ D_{0} divides D_{0} into two open connected components D_{1} ⊂ H_{1} and D_{2} ⊂ H_{2}, and the number of half-lines in D_{1} and D_{2} differs by at least 2. For definiteness suppose the number of half-lines in D_{1} is larger than in D_{2}. Then for at least one hyperplane M containing C, f(M) ∩ D_{0} is contained entirely in D̄_{1}. We now claim that f(M) ⊂ H̄_{1}. Let M_{+} and M_{−} be the two open half-hyperplanes into which L divides M, so that M = M_{+} ∪ M_{−} ∪ (M ∩ L). Note that f^{−1}(D_{1}) intersects both M_{+} and M_{−} nontrivially. Hence, any point x ∈ M \ (M ∩ L) can be connected to a point in f^{−1}(D_{1}) by a path γ ⊂ M which does not cross M ∩ L. But then f(γ) is a path connecting f(x) to a point in f(M) ∩ D_{0} ⊂ H̄_{1}. Since f is one-to-one, this path does not cross f(L), hence f(x) ∈ H̄_{1}.

Lemma 3.10 (No Turns). Suppose q is a (continuous) (κ, C) quasi-isometry from a subset of ℝ^{n} to a subset of ℝ^{n}. Suppose B′ is a simply connected open subset of Euclidean space (with the Euclidean metric), contained in the image of q, and let B = q^{−1}(B′). Let S_{1}, …, S_{r} be r fixed hyperplanes passing through the origin. We call any hyperplane parallel to one of the S_{i} a “marked” hyperplane. Suppose there exist constants m and C′ > C such that for every marked hyperplane L in B there exist m marked hyperplanes L′_{1}, …, L′_{m} in B′ such that q(L) ⊂ Nbhd(L′_{1} ∪ … L′_{m}, C′). Then there exists β = β′(κ, X)C′ such that for every L intersecting B, and every subset B′_{1} ⊂ B satisfying d(B′_{1}, ∂B) ≫ β, q(L) ∩ B′_{1} is within β of a single marked hyperplane L′ in B′. Conversely, for every marked hyperplane L′ intersecting B′_{1} there exists a hyperplane L in B such that q^{−1}(L′ ∩ B′_{1}) is within β of L.

Proof. Since we will consider only marked hyperplanes, we will call them simply hyperplanes. Let r denote the number of hyperplanes passing through a point, and let s = 2^{m}r. Pick constants η′_{j} and λ′_{jk}, 0 ≤ j ≤ s, 0 ≤ k ≤ s, such that 1 = η′_{0} ≪ ⋯ ≪ η′_{s−1} ≪ η′_{s} = β′, and η′_{j} = λ′_{j0} ≪ λ′_{j1} ⋯ ≪ λ′_{js} = η′_{j+1}. Write η_{j} = η′_{j}C′, and λ_{jk} = λ′_{jk}C′. Suppose there is a hyperplane L for which the conclusion of the lemma is false with the choice of β = η_{s}.

For each j suppose L_{1}, …, L_{b} are hyperplanes such that q(L) ⊂ Nbhd(⋃_{k=1}^{b} L_{k}, η_{j}). The union of the L_{k} has the structure of a cellular complex Y_{j} in ℝ^{n} of dimension n − 1; we shall refer to cells of dimension n − 1 as simply “cells”. We may compose q with nearest point projection onto Y_{j} to get a (κ, η_{j}) quasi-isometry q_{j} : L → Y_{j}. Using the standard “connect-the-dots” construction, we may assume that q_{j} is continuous.

By Lemma A.3 (applied to q composed with the nearest point projection onto Y_{j}) there exists a parameter ρ_{j}, η_{j} ≪ ρ_{j} ≪ η_{j+1}, such that

hd (q(L) ∩ B′_{1}, ⋃_{Q∈I} Q ∩ B′_{1}) ≤ ρ_{j} (26)

where I is a certain subset of the cells of Y_{j}. We define d(Y_{j}) to be the minimum possible cardinality of the subset I, such that (26) holds (with fixed ρ_{j}). Then clearly d(Y_{j}) is less then the total number of cells, which is bounded by 2^{b}. Let d_{j} be the minimum of d(Y_{j}) over all possible Y_{j}. Note that d_{j+1} ≤ d_{j}. Since d_{1} ≤ 2^{m} and s > 2^{m}, we claim that there exists j such that d_{j} = d_{j+1}.

Now fix such a j, and let d = d_{j} = d_{j+1}. Let Q′_{1}, …, Q′_{d} denote the sells in I. We claim that there exists a point p on q(L) such that within η_{j} of p there exists at least two cells say Q′_{1} and Q′_{2} which do not lie on the same hyperplane (indeed if this was false, q(L) would be within η_{j} of one hyperplane). The cells Q′_{1} and Q′_{2} have diameter at least η_{j+1} − η_{j} (otherwise d_{j+1} < d_{j}).

Let L_{1}, … L_{r} be the hyperplanes passing through q^{−1}(p). By assumption q(L_{j}) ∩ B′ is within the C′ neighborhood of at most m hyperplanes S_{j1}, ..., S_{jm}. Denote by L_{jk} hyperplanes parallel to S_{jk} so that all these hyperplanes pass though p.

If the hyperplane S_{ji} intersects B(p, λ_{jk}) then the C′-neighborhood of S_{ji} will be within 2λ_{jk}- neighborhood of L_{ji} since λ_{jk} ≫ C′. Consider the ball B(p, λ_{j1}). If not all hyperplanes S_{ji} intersecting B(p, λ_{j1}) intersect B(p, λ_{j0}) then we consider B(p, λ_{j2}). Again if not all hyperplanes intersecting B(p, λ_{j2}) intersect B(p, λ_{j1}) then consider B(p, λ_{j3}) and etc. Doing this for each 1 ≤ l ≤ r and since s > mr we get that there exists an integer k such that for each l, 1 ≤ l ≤ r, q(L_{l}) ∩ B(p, λ_{j,k+1}) is within a distance 2λ_{jk} of a union of hyperplanes passing through p.

We now compose q with nearest-point projection to get a map (which we call q̂) from Y = ⋃_{i=1}^{r} L_{i} ∩B(q^{−1}(p), λ_{j,k+1}) to Y′ = ⋃_{i=1}^{r} L′_{i} ∩B(p, λ_{j,k+1}), where the L′_{i} are the hyperplanes passing through p. Note that q̂ is a (κ, 2λ_{jk}) quasi-isometry.

Denote by Z the n − 2-skeleton of Y, i.e the union of all intersections L_{i} ∩ L_{m}, 1 ≤ i, m ≤ r, and by Z′ the union of all intersections L′_{i} ∩ L′_{m}, 1 ≤ i, m ≤ r. Let B′_{p} denote the ball B(p, \frac{1}{2}λ_{j,k+1}) and B_{p} = B(q^{−1}(p), \frac{1}{2}λ_{j,k+1}). By (26), for each hyperplane L_{i} there exists a union of cells D_{i} ⊂ Y′ such that hd(q(L_{i}) ∩ B′_{p}, D_{i}) ≤ λ^{(4)} where λ_{jk} ≪ λ^{(4)} ≪ λ_{j,k+1}. Note that since q̂ is a (κ, 2λ_{jk}) quasi-isometric embedding, and the diameters of the cells in Y′ are ≻ λ_{j,k+1} ≫ λ^{(4)}, for i ≠ m the cells in D_{i} and D_{m} are distinct. Thus, D_{i} ∩ D_{m} ⊂ Z′. Applying Lemma A.4 part (ii) with U = L_{i} ∩ q̂^{−1}(B′_{p}), V = L_{m} ∩ q̂^{−1}(B′_{p}), U′ = L_{i} ∩ B′_{p}, V′ = L_{m} ∩ B′_{p}, and letting i, m vary, we get that exists a constant λ^{(3)}, with λ^{(4)} ≪ λ^{(3)} ≪ λ_{j,k+1}, such that

q̂(Z) ∩ B′_{p} ⊂ Nbhd(Z′ ∩ B′_{p}, λ^{(3)}) (27)

Let Q be any cell of Y. By a volume preservation argument, there exists a cell Q′ of Y′ such that q̂(Q) intersects Int(Q′, λ^{(2)}), where λ^{(3)} ≪ λ^{(2)} ≪ λ_{j,k+1}. In view of (27) and the fact that q̂ is a (κ, 2λ_{jk}) quasi-isometric embedding, q̂^{−1}(Int(Q′, λ^{(2)}) ∩ B′_{p}) ⊂ Int(Q, λ^{(3)}). Thus, we may apply Corollary A.2 with U = Q and V = Q′ to show that Int(Q′, λ^{(2)}) ∩ B′_{p} ⊂ Nbhd(q(Q), λ^{(1)}) ∩ B′_{p}, where λ^{(2)} ≪ λ^{(1)} ≪ λ_{j,k+1}. Since q̂ is a quasi-isometric embedding, the images of distinct cells must consist of distinct cells. Hence, since the number of cells in Y′ is equal to the number of cells in Y, for any cell Q_{i} ∈ Y there exists a unique cell Q′_{i} ∈ Y′ such that hd (q̂(Q_{i}) ∩ B′_{p}, Q′_{i} ∩ B′_{p}) ≤ λ^{(1)}. The adjacency relations are preserved since by Lemma A.4 part (ii), for any cells Q_{l}, Q_{m} of Y, hd(q(Q_{l} ∩ Q_{m}) ∩ B′_{p}, Q′_{l} ∩ Q′_{m} ∩ B′_{p}) ≤ λ^{(0)}, where λ^{(1)} ≪ λ^{(0)} ≪ λ_{j,k+1}. Hence, by a volume preservation argument, dim(Q_{l} ∩ Q_{m}) = dim(Q′_{l} ∩ Q′_{m}).

Thus q̂ induces a one-to-one map of complexes f : Y → Y′. Since we know that q̂(L) is not contained in a single hyperplane, by Lemma 3.9, there exists a hyperplane M ∈ Y such that f(M) does not cross f(L). Pick x_{1}, x_{2} ∈ M on opposite sides of L such that λ^{(0)} ≪ d(x_{i}, q̂^{−1}(p)) ≪ λ_{j,k+1}, and d(x_{i}, L) ≫ λ^{(0)}, i = 1, 2. Note that q(x) and q̂(x) differ by O(λ_{jk}) ≪ λ^{(0)} on Y, and q is onto B′_{p} by Lemma A.1. Then d(q(x_{i}), f(L)) ≫ λ^{(0)}. Since f(L) is a finite union of disjoint cells, and the points q̂(x_{1}) ∈ q̂(M) and q̂(x_{2}) ∈ q̂(M) are on the same side of f(L), there exists a path γ ⊂ B′_{p} connecting q(x_{1}) and q(x_{2}) with d(γ, f(L)) ≫ λ^{(0)}.

Then γ_{0} = q^{−1}(γ) ⊂ q^{−1}(B′_{p}) is a path connecting x_{1} and x_{2} with d(γ_{0}, L) ≫ λ^{(0)}. This is a contradiction since x_{1} and x_{2} are on opposite sides on L.

To prove the converse let L′ be a hyperplane in B′_{1}, and pick a point x′ ∈ L′. Let x = ϕ^{−1}(x′) and consider all planes L_{1}, ..., L_{r} passing through x. As it was proven above for each q(L_{j}), 1 ≤ j ≤ r, there exists L′_{j} such that q(L_{j}) ∩ B′_{1} ⊂ Nbhd(L′_{j}, β) ∩ B′_{1}. Since x′ ∈ q(L_{j}) we can by taking a parallel hyperplane and changing β to 2β assume that L′_{j} passes through x′. Since q is (κ, 2β) quasi-isometry we have that distinct L_{j} correspond distinct L′_{j}. Therefore there is L such that q(L) ∩ B′_{1} ⊂ Nbhd(L′, 2β) ∩ B′_{1}. Modifying q on L by composing it with the nearest point projection on L′ and using Lemma A.1, we get L′ ∩B′_{1} ⊂ Nbhd(q(L), β′). Therefore, since q is a quasi-isometry, q^{−1}(L′ ∩ B′_{1}) ⊂ Nbhd(L, β_{1}).

3.3. Hyperplanes and Weyl chambers.

Lemma 3.11 (ϕ̃(F) is close to cone over limit set). Suppose F ∈ F is a flat, and let ϕ̃ be the graded quasi-isometric embedding of Step 2. Then,

ϕ̃(F) ∼_{λϵ} V

where λ is a constant depending only on κ and X and V is the union of Weyl chambers passing through the origin:

V = ⋃_{k∈L(ϕ)} π(kA_{+}) ⊂ X

where the limit set L(ϕ) is defined in [EF, Definition 5.1].

Note. Unlike Lemma 3.6 we do not claim a bound of the type ϕ̃(F) ∼_{λϵ,ϕ(x),λρ} V.

Proof. Argue as in the proof of [EF, Lemma 8.3].

Lemma 3.12 (Walls into Hyperplanes). Suppose F ⊂ F^{∗} is a flat, and let V be as in Lemma 3.11. Let L′ be a codimension 1 wall such that outside a sufficiently large ball, L′ is contained completely in a single chamber ℭ′ ⊂ V. Then there exists a hyperplane L ⊂ F such that, outside a sufficiently large ball, L′ ⊂ ϕ̃(L)[λϵ] for some constant λ depending only on κ and X.

Conversely, if L is a wall in F such that outside a sufficiently large ball ϕ̃(L) is contained completely in ℭ′[λϵ] where ℭ′ is a chamber ℭ′ in V, then there exists a hyperplane L′ such that outside a sufficiently large ball, ϕ̃(L) ∩ ℭ′ ⊂ L′[λϵ].

Proof. Let A′(R_{j}, 4R_{j}) be an annular domain in the chamber ℭ′ = π(kA_{+}) centered at e, where R_{j} = 2^{j}R_{1} with R_{1} = 2κR. Denote by B′_{j} the part of ℭ′ ∩ A′(R_{j}, 4R_{j}) such that

d(B′_{j}, ∂ℭ′) ≫ β′(κ, X)C′

where β′(κ, X) and C′ = O(ϵR_{j}) are the same as in Lemma 3.5 and Lemma 3.10 and let B_{j} = ϕ̃^{−1}(B′_{j}). Let L′ be a codimension 1 wall contained in ℭ′. Denote by L′_{j} a parallel translate of L′ such that L′_{j} ∩B′_{j} is not empty. Note that we can choose L′_{j} so that d(L′_{j} ∩A′(R_{j}, 4R_{j}), ∂ℭ′) ≫ β′ϵR_{j}, and d(L′, L′_{j}) ≺ ϵR_{j}. Combining Lemma 3.5 with the ”converse” statement of Lemma 3.10 (with B_{1} = B_{j} and B′_{1} = B′_{j} defined as above) we get a hyperplane L_{j} ⊂ F such that

ϕ̃(L_{j} ∩ B_{j}) ⊂ L′_{j}[λϵ] ∩ B′_{j} ⊂ L′[λ′ϵ] ∩ B′_{j}

where λ′ is independent of ϵ and R_{j}. Then by the local packing lemma (Lemma A.1),

ϕ̃(L_{j} ∩ B_{j})[λ′′ϵ] ⊃ L′_{j} ∩ B′_{j} (28)

Let U′_{j} be a ball in L′_{j} ∩ A′(2R_{j}, 3R_{j}) of radius ≻ R_{j}. Since L′_{j} and L′_{j+1} are parallel there exists a ball U′_{j+1} ∈ L′_{j+1} such that hd(U′_{j}, U′_{j+1}) = O(ϵR_{j}). By (28) there exist subsets U_{j} ∈ B_{j} ∩ L_{j} and U_{j+1} ∈ B_{j} ∩ L_{j+1} so that hd(ϕ̃(U_{j}), U′_{j}) = O(ϵR_{j}) and hd(ϕ̃(U_{j+1}), U′_{j+1}) = O(ϵR_{j}), Thus, U_{j} ⊂ L_{j}, U_{j+1} ⊂ L_{j+1} and hd(U_{j}, U_{j+1}) = O(ϵR_{j}), Also, since ϕ̃ is a quasi-isometric embedding, U_{j} contains a ball of radius ≻ R_{j}. This implies that L_{j} and L_{j+1} are parallel, and d(L_{j}, L_{j+1}) ≤ λϵR_{j}. Hence

d(L_{0}, L_{j}) ≤ ∑_{k=1}^{j} d(L_{k−1}, L_{k}) ≤ ∑_{k=1}^{j} λϵ2^{k}R_{1} ≤ 2λϵR_{j}

and

ϕ̃(L_{0}) ∩ B′_{j} ⊂ ϕ̃(L_{j})[λ_{1}ϵ] ∩ B′_{j} ⊂ L′[λ_{2}ϵ].

Analogously, let L be a wall in F such that ϕ̃(L) ⊂ ℭ′[λϵ]. Let A_{j} = A(R_{j}, 4R_{j}) denote an annular domain in F centered at π(x). Let B_{j} ⊂ A_{j} denote the part of A_{j} such that ϕ̃(B_{j}) ⊂ ℭ′ and d(ϕ̃(B_{j}), ∂ℭ′) ≫ β′(κ, X)C. Let B′_{j} = ϕ̃(B_{j}).

Denote by L_{j} a hyperplane parallel to L such that d(L, L_{j}) ≺ ϵR_{j} and L_{j} ∩ B_{j} is not empty. By Lemma 3.5 and Lemma 3.10 there exists a hyperplane L′_{j} such that ϕ̃(L_{j} ∩ B_{j}) ⊂ L′_{j}[λϵ] ∩ B′_{j}. Since L_{j} ∩ B_{j} ∩ B_{j+1} is not empty and contains balls of radius ≻ R_{j} we get as above that L′_{j} and L′_{j+1} are parallel hyperplanes and d(L′_{j}, L′_{j+1}) = O(ϵR_{j}). Therefore d(L′_{0}, L′_{j}) ≺ ϵR_{j} and ϕ̃(L ∩ A_{j}) ⊂ ϕ̃(L_{j} ∩ B_{j})[λϵ] ⊂ L′_{j}[λ′ϵ] ∩ B′_{j} ⊂ L′_{0}[λ′′ϵ] ∩ B′_{j} ⊂ L′_{0}[λ′′ϵ]. Since j is arbitrary we have ϕ̃(L) ∩ ℭ′ ⊂ L′_{0}[λ′′ϵ].

Using Lemma 3.8 instead of Lemma 3.5 in the above proof we get the following lemma:

Lemma 3.13 (Hyperplanes to hyperplanes). Let F ⊂ F^{∗} be a flat and suppose ϕ̃(F) is close (in the sense of Step 2 (c)) to a single flat F′. Suppose x ∈ Ω_{F}^{∗}, and let L ⊃ F be a hyperplane passing through x. Then, there exists a constant λ depending only on κ and X and a hyperplane L′ ⊂ F′ such that

ϕ̃(L) ∼_{λϵ,ϕ(x),λρ} L′

Lemma 3.14 (Chambers to Chambers). Suppose F ⊂ F^{∗}, and let V be as in Lemma 3.11. Let ℭ be a chamber in F. Then there exists a chamber ℭ′ in V such that ϕ̃(ℭ) ∼_{λϵ} ℭ′. The chamber ℭ′ is unique up to equivalence.

Proof. Let ℭ′ be an arbitrary chamber in V. Taking a parallel translate of ℭ′ in V (and denoting it again by ℭ′) we can assume that ϕ̃ packs ℭ′, i.e ϕ̃(F) ⊃ ℭ′ \ ∂ℭ′[λϵ] (see [EF, Lemma 8.3] or Corollary A.2) By Lemma 3.12, for each wall L′_{i} in ∂ℭ′ there exists a hyperplane L_{i} in F such that outside a sufficiently large ball, L′_{i} ⊂ ϕ̃(L_{i})[λϵ]. By replacing L_{i} by a parallel translate, we may assume that all L_{i} pass through the point x ∈ F, and still, outside a sufficiently large ball, L′_{i} ⊂ ϕ̃(L_{i})[λϵ]. Let P_{j} be the polyhedra bounded by the L_{i}. By Corollary A.2 we may assume that (outside a sufficiently large ball), ϕ̃^{−1}(ℭ′) packs one of these polyhedra, which we denote P, i.e ϕ̃^{−1}(ℭ′) ⊃ P \ ∂P[λϵ]. Since the diameter of ℭ′ ∩ A(R, 2R) is ≻ R, and hd(ϕ̃^{−1}(ℭ′) ∩ A(R, 2R), P ∩ A(R, 2R)) = O(ϵR) (recall ϵ ≪ 1), P must contain a Weyl chamber ℭ. Hence ϕ̃(ℭ) ⊂ ϕ̃(P) ⊂ ℭ′[λϵ].

Now consider a wall L_{i} bounding ℭ. Since ϕ̃(L_{i}) ⊂ ℭ′[λϵ], arguing as in the proof of Lemma 3.12, we may conclude that ϕ̃(L_{i}) is contained in L′_{i}[λϵ] for some hyperplane L′_{i} belonging to a flat F′ ⊃ ℭ′. We may replace the L′_{i} by parallel translates all passing through the same point; still, outside a sufficiently large ball, ϕ̃(L_{i}) ⊂ L′_{i}[λϵ]. By the local packing lemma, ϕ̃ restricted to ℭ[λϵ] packs one of the polyhedra bounded by the L′_{i}, say P′. Since ϕ̃(ℭ) ⊂ ℭ′[λϵ], L′_{i} ⊂ ℭ′[2λϵ], hence P′ ⊂ ℭ′[λ′ϵ]. Also since the injectivity radius of P′ ∩ A(R, 2R) ≻ R, P′ contains a Weyl chamber ℭ′_{1} ⊂ ℭ′[λϵ]. By Lemma B.7 this can only happen if ℭ′_{1} is equivalent to ℭ′. Thus, outside of a large ball, ℭ′ ⊂ ϕ̃(ℭ)[λϵ].

Since ϕ̃^{−1}(ℭ′) ⊃ P \ ∂P[λϵ] we have ϕ̃^{−1}(ℭ′)[λϵ] ⊃ P ⊃ ℭ. The uniqueness of ℭ follows from the last assertion of Lemma B.7.

Therefore we have a map from chambers in V to equivalence classes of chambers in F. This map is one-to-one and onto: the first property follows immediately from the fact that ϕ̃^{−1} is a graded quasi-isometry, and the second from the fact that F ⊂ ϕ̃^{−1}(V)[λϵ] outside of a large ball.

Proof of Proposition 3.3. By Lemma 3.14 one has a well defined map ϕ_{0} from the equivalence classes of chambers in F (i.e F (∞) ⊂ X̂) to equivalence classes of chambers in V, i.e. L(ϕ) ⊂ X̂.

From the proof of Lemma 3.14 it is clear that neighboring chambers in F map to neighboring chambers (here neighboring means sharing a codimension 1 wall). This implies that if ℭ_{1}, ℭ_{2} are chambers in F, and k_{1}, k_{2} the corresponding points in F (∞), then then d_{c}(ϕ_{0}(k_{1}), ϕ_{0}(k_{2})) ≤ d_{c}(k_{1}, k_{2}). (Recall d_{c} is the combinatorial distance in the Tits building T (X), i.e the length of the minimal gallery connecting the two chambers). Indeed there exists a minimal gallery γ connecting k_{1} and k_{2} and lying in F; then ϕ_{0}(γ) is a gallery connecting ϕ_{0}(k_{1}) and ϕ_{0}(k_{2}) and ℓ(ϕ_{0}(γ)) ≤ ℓ(γ).

From Lemma 3.11, and the proof of [EF, Theorem 1.1] (see [EF, §7]) there exist two limit points a′, b′ ∈ L(ϕ) such that a′ ∉ b′S, i.e. d_{c}(a′, b′) is maximal. Let F′ denote the unique flat such that a′, b′ ⊂ F′(∞) (see [EF, §7.1]). Let a, b be points in F (∞) such that ϕ_{0}(a) = a′ and ϕ_{0}(b) = b′. Then d_{c}(a, b) ≥ d_{c}(a′, b′), and since the latter is maximal, d_{c}(a, b) is maximal. Let γ denote any minimal gallery connecting a and b; then ϕ_{0}(γ) is a minimal gallery connecting a′ and b′; since F′(∞) is an apartment and apartments are convex (see [B, Proposition, §IV.4, p. 88]), ϕ(γ) ⊂ F′(∞). Since any chamber is contained in such a minimal gallery (see [B, Lemma, §IV.5, p. 92]), this implies that ϕ_{0}(F (∞)) ⊂ F′(∞), i.e L(ϕ) ⊂ F′(∞). Now part (i) follows easily from Theorem 1.5 and [EF, Corollary 7.4].

Part (ii) follows immediately from combining part (i) with Lemma 3.14.

Proof of Proposition 3.4. Part (a) is immediate from Proposition 3.3 and Lemma 3.13. To get part (b) one may argue as follows: Let Y ⊂ F denote the union of the hyperplanes passing though x, let x′ ∈ F′ denote the nearest point in F′ to ϕ(x), and let Y′ ⊂ F′ denote the union of the hyperplanes passing through x′. By part (a), ϕ̃(Y) ∼_{λϵ,x′,λρ} Y′. Note that F \ Y is the union of the interiors of the chambers passing through x. For any chamber ℭ ⊂ F based at x we may pick a point u ∈ ℭ such that d(u, x) ≫ ρ and d(u, Y) ≻ d(u, x). Then d(ϕ̃(u), Y′) ≻ d(ϕ̃(u), x′). Now part (b) follows from the graded version of Corollary A.2.

4. The boundary map

Definition 4.1 (The boundary map ϕ_{0} : U → X̂). Let U ⊂ X̂ denote the set of equivalence classes of Weyl chambers of flats in F^{∗}. Let ϕ_{0} : U → X̂ be the map defined on these Weyl chambers by part (ii) of Step 3.

We note that ϕ_{0} is well defined. Indeed, let ℭ_{1} ⊂ F_{1} and ℭ_{2} ⊂ F_{2} be equivalent Weyl chambers, i.e. hd(ℭ_{1}, ℭ_{2}) < c < ∞, and the flats F_{1} and F_{2} belong to F^{∗}. Pick x_{0} ∈ F_{1}. By Step 2 part (c) there exists a (κ, λϵ, C′) graded q.i. embedding ϕ̃ based at x_{0} from F_{1} ∪ F_{2} to X, and λ depends only on κ and X. For any x_{1} ∈ ℭ_{1} there exists x_{2} ∈ ℭ_{2} such that d(x_{1}, x_{2}) ≤ c. Then since ϕ̃ is a graded q.i. embedding, d(ϕ̃(x_{1}), ϕ̃(x_{2})) ≺ d(x_{1}, x_{2}) + ϵd(x_{1}, x_{0}) + ϵd(x_{2}, x_{0}) ≺ ϵd(ϕ̃(x_{1}), ϕ̃(x_{0})), where we assumed that d(x_{1}, x_{0}) ≫ max(c/ϵ, C′). Since x_{1} ∈ ℭ_{1} is arbitrary, this shows that ϕ̃(ℭ_{1}) ⊂ ϕ̃(ℭ_{2})[λϵ], outside of a large ball. Analogously one has ϕ̃(ℭ_{2}) ⊂ ϕ̃(ℭ_{1})[λϵ]. Therefore, ϕ̃(ℭ_{1}) ∼_{λϵ} ϕ̃(ℭ_{2}). Hence if we denote ϕ_{0}(ℭ_{i}) by ℭ′_{i} then ℭ′_{1} ∼_{λϵ} ℭ′_{2}. Then ℭ′_{1} is equivalent to ℭ′_{2} by the last assertion of Lemma B.7. This shows that ϕ_{0} is well defined.

Recall that the cells of maximal dimension in the Tits building T (X) are equivalence classes of Weyl chambers, i.e points in K/M. For σ ⊂ Δ and k ∈ K, kK_{σ} can be viewed as a cell in T (X) of codimension |σ| (here for a subset σ of Δ, |σ| is the cardinality of σ).

Definition 4.2 (Tits Map defined on a subset). Let E be a subset of X̂, and let ψ : E → X̂ be a map. We say that ψ is a Tits map defined on E if for all k ∈ E and any σ ⊂ Δ there exists τ ⊂ Δ with |τ| = |σ| such that ψ(kK_{σ} ∩ E) ⊂ ψ(k)K_{τ}.

In other words, on its domain of definition, ψ sends adjacent simplices in T (X) into adjacent simplices, and preserves the dimension of the intersection. Note that we do not require ψ to be continuous.

Step 4 (ϕ_{0} preserves partial ordering in T (X)). The map ϕ_{0} : U → X̂ defined in Definition 4.1 is a Tits map defined almost everywhere.

The proof uses the following:

Lemma 4.3 (Walls to walls). Suppose F ∈ F^{∗} is a flat and let F′ be as in Step 3. Let L ⊂ F be a σ-wall. Then there exists a τ-wall L′ in F′ such that |σ| = |τ|, and ϕ̃(L) ∼_{λϵ} L′. The constant λ depends only on κ and X.

Proof of Lemma 4.3. The wall L may be viewed as the intersection of a collection of Weyl chambers ℭ_{1}, …, ℭ_{r} ⊂ F. By Step 3 there exist Weyl chambers ℭ′_{i} ⊂ F′ such that ϕ̃(ℭ_{i}) ∼_{λϵ} ℭ′_{i}. We may assume all the ℭ′_{i} are based at the same point x′ ∈ F′ (otherwise replace them by equivalent Weyl chambers). Let L′ = ∩_{i=1}^{r}ℭ′_{i}. From the geometry of the root system (i.e all angles bounded from below), it is clear that (the graded version of) Lemma A.4 part (ii) can be applied repeatedly. Thus, ϕ̃(L) ∼_{λϵ} L′. The assertion that |σ| = |τ| follows easily by volume considerations, or the packing lemma (cf. Lemma A.1).

Proof of Step 4. Suppose ℭ_{1}, ℭ_{2} ∈ U, and let k_{i} ∈ K be such that π(k_{i}A^{+}) is equivalent to ℭ_{i} for i = 1, 2. By assumption k_{2} ∈ k_{1}K_{σ}. This implies that there exist σ-walls L_{i} ⊂ ∂ℭ_{i} such that hd(L_{1}, L_{2}) < ∞. Let ℭ′_{i} = ϕ_{0}(ℭ_{i}).

By Lemma 4.3 there exists τ ⊂ Δ with |τ| = |σ| and a τ-wall L′_{1} such that ϕ̃(L_{1}) ∼_{λϵ} L′_{1}. Since L_{1} and L_{2} are equivalent, this also implies that ϕ̃(L_{2}) ∼_{λϵ} L′_{1}. Hence, outside a sufficiently large ball, L′_{1} ⊂ ℭ′_{1}[λ′ϵ] ∩ ℭ′_{2}[λ′ϵ].

Let k′_{1}, k′_{2}, k′ ∈ K be such that hd(π(k′_{i}A^{+}), ℭ′_{i}) < ∞ and hd(π(k′A_{τ}^{+}), L′_{1}) < ∞. Then, outside a sufficiently large ball, π(k′A_{τ}^{+}) ⊂ π(k′_{1}A^{+})[λ′′ϵ]∩π(k′_{2}A^{+})[λ′′ϵ]. Then by Lemma B.7, k′_{i} ⊂ k′K_{τ}.

5. Uniform continuity of the boundary map

Step 5 (ϕ_{0} is bi-Hölder). There exists a set U_{2} ⊂ U_{1} ⊂ K/M of full (K-invariant) measure such that for x, y ∈ U_{2}, the map ϕ_{0} satisfies the Hölder type inequalities

c^{−1}d_{K}(x, y)^{λ} ≤ d_{K}(ϕ_{0}(x), ϕ_{0}(y)) ≤ cd_{K}(x, y)^{1/λ} x, y ∈ U_{2}

Here d_{K} is the left K-invariant metric on X̂ ≅ K/M, and λ and c do not depend on x, y.

Remark. The identification of X̂ with K/M and thus d_{K} depends on the choice of origin. However, since the different possible metrics are conjugate by diffeomorphisms and thus equivalent, the notion of bi-Hölder is independent of choices. For chambers ℭ_{1}, ℭ_{2} ⊂ X, we let d_{K}(ℭ_{1}, ℭ_{2}) = d_{K}(𝔇_{1}, 𝔇_{2}), where 𝔇_{i} is equivalent to ℭ_{i} and based at e.

The motivation for Step 5 is the following well known:

Lemma 5.1. Let Y be a rank 1 symmetric space, and let q : Y → Y be a quasi-isometry. Then the induced boundary map q_{0} : Ŷ → Ŷ is bi-Holder. (Here Ŷ is the boundary at infinity, i.e. the equivalence classes of geodesic rays).

Proof of Lemma 5.1. Without loss of generality, q(e) = e. Let γ_{1} = π(k_{1}A^{+}) and γ_{2} = π(k_{2}A^{+}) be geodesic rays starting at e, where k_{1}, k_{2} ∈ K. Write θ = d_{K}(k_{1}, k_{2}); then θ is the angle between γ_{1} and γ_{2}. We parameterize the γ_{i} by arc length. Pick a sufficiently large Q > 0, and let t = sup{s : s > 0 and d(γ_{1}(s), γ_{2}(s)) < Q}. By Lemma B.5, c_{1}| log θ| < t < c_{2}| log θ| where c_{1}, c_{2} are independent of θ.

By Mostow’s lemma, there exist geodesic rays γ′_{i} and a constant N depending only on Y and the q.i. constants of q such that hd(q(γ_{i}), γ′_{i}) < N. Without loss of generality, the γ′_{i} also start at e, and Q ≫ N. Let t′ = sup{s : s > 0 and d(γ′_{1}(s), γ′_{2}(s)) < 2κQ}, and t′′ = sup{s : s > 0 and d(γ_{1}(s), γ_{2}(s)) < Q/(2κ)}. Since q is a quasi-isometry, and Q ≫ N, t ≺ t′′ ≤ t′ ≺ t. Hence by Lemma B.5, c′_{1}| log θ′| ≤ | log θ| ≤ c′_{2}| log θ′| where c′_{1} and c′_{2} are independent of θ and θ′ is the angle between γ′_{1} and γ′_{2}. After exponentiating, we get the desired estimate.

The proof of Lemma 5.1 carries over easily to the higher rank case – cf. Lemma 5.4. However, the presence of holes may create difficulties, since the entire argument in the proof of Lemma 5.1 may take place in a hole, e.g. when trying to prove continuity across the endpoint of a horosphere. In the higher rank case, this difficulty can be overcome as follows:

Suppose we consider two Weyl chambers ℭ_{1} and ℭ_{2} based at e which share a codimension 1 wall L_{0}. Let F_{1} be a flat containing ℭ_{1} and F_{2} a flat containing ℭ_{2}; then F_{1} and F_{2} intersect in a hyperplane L which contains L_{0}. Note that F_{1} and F_{2} are contained in a totally geodesic submanifold Z of X which is a metric product Z = L × Y, where Y is a rank 1 symmetric space. In fact, F_{1} ∪ F_{2} = L × (γ_{1} ∪ γ_{2}), where γ_{1} and γ_{2} are two geodesics in Y passing through the same point. The angle d_{K}(ℭ_{1}, ℭ_{2}) is the same as the angle between the geodesics γ_{1} and γ_{2} and can be measured above any point of L.

The strategy of the proof of Step 5 is the following: suppose the common base point of ℭ_{1} and ℭ_{2} is in or near a hole (in F_{1} or F_{2}). By the Moore ergodicity theorem, applied to the subgroups A_{σ} of A, we may assume that L is uniformly distributed; hence there is a point p of L_{0} which is away from the holes in F_{1} and F_{2}. We then apply the argument in the proof of Lemma 5.1 to p × (γ_{1} ∪ γ_{2}) ⊂ p × Y ≅ Y.

For this argument, we need results on images of hyperplanes similar to Step 3 part (i) (Proposition 3.4 (a) is not sufficient). This, together with some related issues is done in §5.1. The program sketched in the previous paragraph is carried out in §5.3. The assumption that ℭ_{1} and ℭ_{2} share a codimension 1 wall is removed in §5.4.

5.1. Mapping hyperplanes.

Lemma 5.2 (Hyperplanes to hyperplanes, Uniform Version). There exists a constant N = N(κ, ϵ, C), a family of hyperplanes L, and for each L ∈ L nonempty subsets Ω_{L} ⊂ L, such that the following holds:

(a) L is a Γ-invariant i.e. if γ ∈ Γ and L ∈ L then γL ∈ L.

(b) L has full measure, i.e. there exists a set E ⊂ Γ\G with μ(E) = 1 such that for any y ∈ p^{−1}(E) and any α ∈ Σ, π(yA_{α}) ∈ L.

(c) For every L ∈ L, there exist a hyperplane L′ such that ϕ(Ω_{L}) lies within a distance N of L′.

(d) For every L ∈ L, and every x ∈ L,

(29)  lim inf_{r→∞} \frac{|B(x, r) ∩ Ω_{L}|}{|B(x, r) ∩ L|} ≥ 1 − |Σ|ϵ

(e) For every y ∈ p^{−1}(E), π(yA) ∈ F^{∗} and for any α ∈ Σ

lim inf_{r→∞} \frac{|Ω_{F} ∩ B(y, r) ∩ L_{α}|}{|B(y, r) ∩ L_{α}|} ≥ 1 − |Σ|ϵ

Proof. Let Ω^{∗} and E^{∗} be as in the proof of Lemma 3.2. Let χ_{Ω^{∗}} be the characteristic function of Ω^{∗}. By the ergodic theorem Theorem 1.4, there exists a set E ⊂ E^{∗} ⊂ Γ\G with μ(E) = 1 such that for all x ∈ E, and all α ∈ Σ,

(30)  lim_{r→∞} \frac{1}{|B_{r}|} ∫_{B_{r}} χ_{Ω^{∗}}(x exp v) dν_{α}(v) = μ(Ω^{∗}) ≥ 1 − |Σ|ϵ

where ν_{α} is Lebesgue measure on the Lie algebra of A_{α} ≅ ℝ^{n−1}, and here B_{r} is the ball of radius r is this Lie algebra. Let L = {π(yA_{α}) : y ∈ p^{−1}(E)}. Then (a) and (b) hold by construction. For L = π(zA_{α}) ∈ L, let Ω_{z,L} = {π(z exp v) : p(z exp v) ∈ Ω^{∗} and v ∈ A_{α}}, and Ω_{L} = ⋃_{z with L=π(zA_{α})} Ω_{z,L}. Then, (d) holds (already for Ω_{z,L}) by (30). Also (e) holds since Ω_{z,L} ⊂ Ω_{F} and (30).

It remains to prove (c). In fact we will show that (c) holds already for Ω_{z,L}; Ω_{L} is introduced only to simplify notation.

Suppose x ∈ Ω_{z,L}. Let F_{1} = π(zA). Then x ∈ Ω_{F_{1}}, and there exists F_{2} ∈ Trans_{F_{1}}(x) such that F_{1} ∩ F_{2} = L and x ∈ Ω_{F_{2}}. Let F′_{1}, F′_{2} be as in Step 3.

Since Ω_{z,L} ⊂ L∩Ω_{F_{1}} ∩Ω_{F_{2}}, by Step 3 (i), ϕ(Ω_{z,L}) ⊂ Nbhd(F′_{1}, N)∩Nbhd(F′_{2}, N). Choose r ≫ N, and let P′ be as in Lemma B.1 part (i) re. the flats F′_{1} and F′_{2}. Then,

ϕ(Ω_{z,L}) ⊂ Nbhd(P′, O(r)) (31)

Let ϕ̃ : F_{1} ∪ F_{2} → X be as in Step 2 part (c). Note that since (d) holds for Ω_{z,L}, there exists λ depending only on X such that Ω_{z,L}[λϵ] = L. Then (31) implies that (outside a sufficiently large ball),

ϕ̃(L) ⊂ P′[λ′ϵ]. (32)

Conversely, choose any x ∈ P′ with ϵd(x, z) ≫ r, and let ϕ̃^{∗} : F′_{1} ∪ F′_{2} → F_{1} ∪ F_{2} be the coarse inverse of ϕ̃. Then, since ϕ̃^{∗} is a graded quasi-isometry, ϕ̃^{∗}(x)(F′_{i}) = F_{i}, and x ∈ Nbhd(F′_{1}, O(r)) ∩ Nbhd(F′_{2}, O(r)), we have ϕ̃^{∗}(x) ∈ F_{1}[λ_{1}ϵ] ∩ F_{2}[λ_{1}ϵ]. Hence by Lemma 3.1, ϕ̃^{∗}(x) ∈ L[λ′_{1}ϵ]. Hence, combining with (32), we get ϕ̃(L) ∼_{λ′ϵ} P′.

By Proposition 3.4, there exists a hyperplane L′ such that ϕ̃(L) ∼_{λϵ} L′. Hence P′ ∼_{λ′′ϵ} L′. This implies that P is a strip between two hyperplanes parallel to L′.

In view of (31), to show (c) we need only show that the width of the strip is O(r). This can be checked as follows: Choose R so that r ≪ R ≪ r/ϵ. Then, by Lemma 3.7 part (a), the injectivity radius of P′ ∩ B_{R} is O(r). Since P′ is a strip, this shows that the width of the strip is O(r).

5.2. Choice of origin. Choose x_{0} ∈ Γ\G. Let U′_{x_{0}} ⊂ K/M denote the set of k ∈ K such that π(x_{0}k) ∈ E where E is as in Lemma 5.2. Note that for almost all choices of x_{0},

m_{K}(U′_{x_{0}}) = 1, (33)

(where m_{K} is the K-invariant measure on K/M). We choose x_{0} so that (33) holds, and denote U′_{x_{0}} by U′. For simplicity of notation we set x_{0} = e (this can always be accomplished by replacing Γ with a suitable conjugate).

Let U_{1} denote the set of points k in K/M such that k ∈ U′ and for all σ ⊂ Δ, m_{K}(U′ ∩ kK_{σ}) = m_{K}(kK_{σ}). By Fubini’s theorem, m_{K}(U_{1}) = 1.

We normalize ϕ so that ϕ(e) = e.

5.3. Angles between flats. The following lemma is the main estimate used in the proof of Step 5.

Lemma 5.3 (Angle between flats is bi-Hölder). Suppose α ∈ Δ, k ∈ U_{1}, and x_{1}, x_{2} ∈ kK_{α} ∩ U′. Then,

C^{−1}d_{K}(x_{1}, x_{2})^{1/λ} ≤ d_{K}(ϕ_{0}(x_{1}), ϕ_{0}(x_{2})) ≤ Cd_{K}(x_{1}, x_{2})^{λ}

where C = C(k) and λ is independent of x_{1}, x_{2} and k.

The proof uses the following:

Lemma 5.4. Suppose F_{1}, F_{2} ∈ F^{∗} are flats, and ℭ_{1} ⊂ F_{1} and ℭ_{2} ⊂ F_{2} are Weyl chambers both based at the point p ∈ F_{1} ∩ F_{2}. Suppose ℭ_{1} and ℭ_{2} share a codimension 1 wall, and p ∈ Ω_{F_{1}}^{∗} ∩ Ω_{F_{2}}^{∗}. Suppose p′ ∈ B(ϕ(p), O(N)). Let g_{0} ∈ G be such that g_{0}·p = e, and let g′_{0} ∈ G be such that g′_{0}·p′ = e. Then,

C^{−1}d_{K}(g_{0}ℭ_{1}, g_{0}ℭ_{2})^{1/λ} ≤ d_{K}(g′_{0} · ϕ_{0}(ℭ_{1}), g′_{0} · ϕ_{0}(ℭ_{2})) ≤ Cd_{K}(g_{0}ℭ_{1}, g_{0}ℭ_{2})^{λ} (34)

where C and λ are independent of ℭ_{1}, ℭ_{2}, F_{1}, F_{2}, g_{0}, g′_{0} and p.

Remarks. The assumption that ℭ_{1} and ℭ_{2} share a codimension 1 wall is not necessary. Also, if p = e, then we can let g_{0} = 1, g′_{0} = 1, and the estimate of Step 5 follows immediately from (34). However, in general, we may not assume that for almost all k ∈ U_{1}, e ∈ Ω_{π(kA)}^{∗}, so Lemma 5.4 cannot be applied directly.

Proof of Lemma 5.4. By Proposition 3.4 part (b), there exist Weyl chambers ℭ′′_{1}, ℭ′′_{2} such that ϕ̃(ℭ_{i}) ∼_{λϵ,ϕ(x),ρ} ℭ′′_{i}. By Step 3, since p ∈ Ω_{F_{i}}, d(p′, F′_{i}) = O(N). By Lemma B.8 for i = 1, 2 we may replace ℭ′′_{i} by an equivalent chamber ℭ′_{i} such that ℭ′_{1} and ℭ′_{2} are both based at p′, and still ϕ̃(ℭ_{i}) ∼_{λϵ,p′,ρ′} ℭ′_{i} for any ρ′ ≫ max(ρ, N). For i = 1, 2, let 𝔇_{i} = g_{0}ℭ_{i}, 𝔇′_{i} = g′_{0}ℭ′_{i}. Then all four of these Weyl chambers are based at e, hence there exist k_{i}, k′_{i} ∈ K such that 𝔇_{i} = π(k_{i}A^{+}), 𝔇′_{i} = π(k′_{i}A^{+}), i = 1, 2. By Step 4, ℭ′_{1} and ℭ′_{2} (hence also 𝔇′_{1} and 𝔇′_{2}) share a codimension 1 wall. Let θ = d_{K}(k_{1}, k_{2}), θ′ = d_{K}(k′_{1}, k′_{2}). Hence (34) is equivalent to C^{−1}θ^{1/λ} ≤ θ′ ≤ Cθ^{λ}.

Let ψ̃ = g′_{0} ∘ ϕ̃ ∘ g_{0}^{−1}. Then, ψ̃ : 𝔇_{1} ∪ 𝔇_{2} → Nbhd(𝔇′_{1}, N) ∪ Nbhd(𝔇′_{2}, N) is a graded quasiisometric embedding, and ψ̃(𝔇_{i}) ∼_{λϵ,e,ρ′} 𝔇′_{i}, i = 1, 2. It follows from Proposition 3.4 (cf. the proof of Lemma 4.3) that

ψ̃(∂𝔇_{i}) ∼_{λϵ,e,ρ′} ∂𝔇′_{i}. (35)

Let Q ≫ 1 and 0 < ν_{3} < 1 be as in Lemma B.6. In the following paragraph, the implicit constants do not depend on ν_{3} and Q. Without loss of generality, θ is sufficiently small so that ν_{3}| log θ| ≫ max(Q/ϵ, N/ϵ, ρ′/ϵ). By Lemma B.6, there exist points z_{i} ∈ 𝔇_{i} such that d(z_{1}, z_{2}) ≤ Q, d(z_{i}, e) = O(| log θ|) and d(z_{i}, ∂𝔇_{i}) ≥ ν_{3}| log θ|. Then, since ψ̃ is a graded q.i. embedding and by (35), d(ψ̃(z_{i}), ∂𝔇′_{i}) ≻ ν_{3}| log θ|. Also,

d(ψ̃(z_{1}), ψ̃(z_{2})) = O(d(z_{1}, z_{2})) + O(ϵd(z_{1}, e)) + O(ϵd(z_{2}, e)) ≪ ν_{3}| log θ|

Let z′_{i} denote the nearest point projection of ψ̃(z_{i}) onto 𝔇′_{i}. Then, there exists r ≻ ν_{3}| log θ| (with the implied constant depending only on κ, X) such that d(z′_{1}, z′_{2}) ≤ λ_{2}r, d(z′_{i}, ∂𝔇′_{i}) ≥ ν′r, where λ_{2}, ν′ are as in Lemma B.4. Hence, by Lemma B.4, θ′ ≤ ce^{−νr}. Hence, θ′ ≤ cθ^{λ}, where λ depends only on κ, X (and ν_{3}, but ν_{3} and Q depend only on X).

The opposite inequality follows from applying the same argument to the map ψ̃^{∗} : 𝔇′_{1} ∪ 𝔇′_{2} → 𝔇_{1} ∪ 𝔇_{2} which is the coarse inverse of ψ̃ composed with nearest point projection.

Proof of Lemma 5.3. We write x_{1} = kk_{1}, x_{2} = kk_{2}, where k_{i} ∈ K_{α}. Let L = π(kA_{α}). Since k ∈ U_{1}, L ∈ L, it follows from Lemma 5.2 that there exists a hyperplane L′ such that ϕ(Ω_{L}) ⊂ Nbhd(L′, N). Pick a point x_{0} in Ω_{L}. Then, in particular, ϕ(x_{0}) ∈ Nbhd(L′, N). Let x′_{0} ∈ G be such that L′ = π(x′_{0}A_{β}) for some β ∈ Σ, and π(x′_{0}) ∈ L is the closest point to ϕ(x_{0}). Let g′_{1} = x_{0}^{′−1}, so that g′_{1}π(x′_{0}) = e. Note that g′_{1}, x′_{0} and x_{0} are independent of x_{1} and x_{2}.

Since kk_{i} ∈ U′, by Lemma 5.2 parts (d) and (e) lim inf_{r→∞} |Ω_{F_{1}} ∩ Ω_{F_{2}} ∩ Ω_{L} ∩ B(x_{0}, r)| ≥ (1 − 3|Σ|ϵ)|B(x_{0}, r) ∩ L|. Hence there exists a point p ∈ L ∩ Ω_{F_{1}} ∩ Ω_{F_{2}} ∩ Ω_{L}.

Since p ∈ L, p = π(ka) for some a ∈ A_{α}. For i = 1, 2 let ℭ_{i} denote the Weyl chambers π(kak_{i}A^{+}). Then the ℭ_{i} are based at p. Since k_{i} ∈ K_{α}, a ∈ A_{α} and K_{α} and A_{α} commute, ℭ_{i} is equivalent to π(kk_{i}A^{+}). Hence d_{K}(ℭ_{1}, ℭ_{2}) = d_{K}(k_{1}, k_{2}), and

ϕ_{0}(x_{i}) = ϕ_{0}(π(x_{i}A^{+})) = ϕ_{0}(ℭ_{i}). (36)

Let g_{0} = (ka)^{−1}, so that g_{0}p = e. Then g_{0}ℭ_{i} = π(k_{i}A^{+}), hence

d_{K}(g_{0}ℭ_{1}, g_{0}ℭ_{2}) = d_{K}(k_{1}, k_{2}) = d_{K}(x_{1}, x_{2}). (37)

Now let p′ be the closest point in L′ to ϕ(p). We have d(p′, ϕ(p)) ≤ N. Also, since p′ ∈ L′, there exists a′ ∈ A_{β} such that p′ = π(x′_{0}a′). Let g′_{0} = (x′_{0}a′)^{−1} so that g′_{0}p′ = e.

For i = 1, 2, let ℭ′_{i} denote the Weyl chamber in the equivalence class ϕ_{0}(ℭ_{i}) based at p′. Let L_{+} denote the common codimension 1 wall of ℭ_{1} and ℭ_{2}. By Step 4, ℭ′_{1} and ℭ′_{2} share a common codimension 1 wall H_{+}, and by Lemma 4.3, ϕ̃(L_{+}) ∼_{λϵ} H_{+}. But ϕ̃(L_{+}) ⊂ ϕ̃(L) ∼_{λϵ} L′, hence outside a large ball, H_{+} ⊂ L′[λϵ]. Since H_{+} is based at p′ and p′ ∈ L′, this implies that H_{+} ⊂ L′. Hence ℭ′_{1} and ℭ′_{2} share a codimension 1 wall contained in L′.

Hence ℭ′_{i} = π(x′_{0}a′k′_{i}A^{+}), where k′_{i} ∈ K_{β}. Let 𝔇′_{i} = π(x′_{0}k′_{i}A^{+}), so that 𝔇′_{i} is based at π(x′_{0}). Since a′ ∈ A_{β}, k′_{i} ∈ K_{β} and A_{β} and K_{β} commute, ℭ′_{i} is equivalent to 𝔇′_{i}, so we may write ϕ_{0}(ℭ_{i}) = 𝔇′_{i}. Also note that g′_{0}ℭ′_{i} = π(k′_{i}A^{+}) = g′_{1}𝔇′_{i}, so we may write

g′_{0} · ϕ_{0}(ℭ_{i}) = g′_{0}ℭ′_{i} = g′_{1}𝔇′_{i} = g′_{1} · ϕ_{0}(ℭ_{i}) (38)

By Lemma 5.4,

C^{−1}d_{K}(g_{0}ℭ_{1}, g_{0}ℭ_{2})^{1/λ} ≤ d_{K}(g′_{0} · ϕ_{0}(ℭ_{1}), g′_{0} · ϕ_{0}(ℭ_{2})) ≤ Cd_{K}(g_{0}ℭ_{1}, g_{0}ℭ_{2})^{λ} (39)

Combining (36), (37), (38) and (39), we get

C^{−1}d_{K}(x_{1}, x_{2})^{1/λ} ≤ d_{K}(g′_{1} · ϕ_{0}(x_{1}), g′_{1} · ϕ_{0}(x_{2})) ≤ Cd_{K}(x_{1}, x_{2})^{λ} (40)

Since g′_{1} acts as a diffeomorphism of the boundary, and is independent of x_{1}, x_{2}, this implies

c′′_{2}d_{K}(x_{1}, x_{2})^{1/λ} ≤ d_{K}(ϕ_{0}(x_{1}), ϕ_{0}(x_{2})) ≤ c′′_{1}d_{K}(x_{1}, x_{2})^{λ}

where c′′_{1}, c′′_{2} depend on g′_{1}, hence L.

5.4. Differentiability and T (X). In this section we complete the proof of Step 5. In view of Lemma 5.3 and Step 4 it is enough to prove the following:

Proposition 5.5. Let U_{1} ⊂ X̂ be a subset with m_{K}(U_{1}) = 1, and let ϕ_{0} : U_{1} → X̂ be a Tits map defined on U_{1}. Suppose ϕ_{0} satisfies the conclusion of Lemma 5.3. Then, there exists a subset U_{2} ⊂ U_{1} with m_{K}(U_{2}) = 1 such that the restriction of ϕ_{0} to U_{2} is bi-Holder.

The proof of Proposition 5.5 does not involve either quasi-isometries or ergodic theory; instead it deals with some extra structure of the building T (X). Lemma 5.3 does not immediately imply the statement of Step 5 because of the restriction to Weyl chambers with a common codimension 1 wall L, and because the constant C depends on L. The idea of the proof is to use Step 4 to show that finitely many applications of Lemma 5.3 imply the statement for ϕ_{0}.

Lemma 5.6 (Image of uniformly distributed set under ϕ_{0} is dense). Let E be a subset of full measure in K/M. Then, ϕ_{0}(E ∩ U_{1}) is dense in K/M.

Proof. Without loss of generality, we may assume that E ⊂ U_{1}. Let E_{1} = E ∩ {k ∈ K/M : m_{K_{α}}(kK_{α} ∩ E) = 1, ∀α ∈ Δ}, where m_{K_{α}} is normalized Haar measure on K_{α}/M, and we denote the push forward measure on kK_{α}/M also by m_{K_{α}}. Then, by Fubini’s theorem, m_{K}(E_{1}) = 1.

Suppose k ∈ E_{1}. Then we claim that ϕ_{0}(E) is dense in ⋃_{α∈Δ} k′K_{α}, where k′ = ϕ_{0}(k). Indeed by Step 4 for each α ∈ Δ there exists β ∈ Δ such that ϕ_{0}(kK_{α} ∩ E) ⊂ k′K_{β}, and by Lemma 5.3 the associated map ψ_{α} = ℓ_{k′}^{−1} ∘ ϕ_{0} ∘ ℓ_{k} : K_{α}/M → K_{β}/M (where ℓ_{k} denotes left translation by k) is bi-Hölder on its domain of definition (i.e. k^{−1}U_{1}∩K_{α}/M). Hence ψ_{α} extends to an injective continuous map \overline{ψ_{α}} : K_{α}/M → K_{β}/M. Choose α so that K_{α}/M has maximal dimension: then by invariance of domain (see e.g. [Mu, Theorem 36.5]) \overline{ψ_{α}} is a homeomorphism, and dim(K_{β}/M) = dim(K_{α}/M). Since \overline{ψ_{α}} is a homeomorphism, the image of the full measure (hence dense) set E ∩ kK_{α}/M is dense. Proceeding in this way from maximal to minimal dimension of K_{α}/M, we see that for any α ∈ Δ, ϕ_{0}(E ∩ kK_{α}/M) is dense in ϕ_{0}(k)K_{β}/M. This implies the claim.

Similarly, for j ≥ 2, let E_{j} = E_{j−1} ∩ {k ∈ K/M : m_{K_{α}}(kK_{α} ∩ E_{j−1}) = 1}. Then for any j, m_{K}(E_{j}) = 1, and for any k ∈ E_{j}, ϕ_{0}(E_{j−1} ∩ k ∪_{α∈Δ} K_{α}/M) is dense in ⋃_{α∈Δ} ϕ_{0}(k)K_{α}

To finish the proof of the lemma, note that the maximal combinatorial distance in the Tits building is ℓ < ∞. Hence for every two elements x, y ∈ K/M there exists a gallery of length ℓ connecting x and y, i.e. a sequence of elements x = x_{0}, x_{1}, …, x_{ℓ} = y, with x_{i+1} ∈ x_{i}K_{α_{i}}, with α_{i} ∈ Δ. We fix x ∈ ϕ_{0}(E_{ℓ}), and let y denote an arbitrary point in K/M. By the claim, there exists x′_{1} ∈ ϕ_{0}(E_{ℓ−1}) arbitrarily close to x_{1}. By the same argument, there exists x′_{2} ∈ ϕ_{0}(E_{ℓ−2}) arbitrarily close to x_{2}, etc. Thus there exists a point x′_{ℓ} ∈ ϕ_{0}(E) arbitrarily close to y. This proves the lemma.

Lemma 5.7 (The function f_{α}). Suppose t ∈ K_{α}, s ∈ S, and y = ts ∉ S.

(a) tM is uniquely determined by y; we denote it by f_{α}(y). Also for m ∈ M, f_{α}(ym) = f_{α}(y). There is a well defined function f_{α} : S^{c}/M → K_{α}/M.

(b) There exists a Zariski closed proper subset S_{α} of K/M such that the restriction of f_{α} to S_{α}^{c} is real analytic.

(c) f_{α}(my) = mf_{α}(y) for m ∈ M and y ∈ S^{c}.

Proof. Since s ∈ S, s ∈ K ∩ Bw′B for some w′ ∈ W, where w′ = w_{α_{1}} … w_{α_{r}} and r < ℓ, the maximal combinatorial distance (see the “algebraic preliminaries”). Then s = k_{α_{1}} … k_{α_{r}} where k_{α_{i}} ∈ K ∩ Bw_{α_{i}}B = K_{α_{i}} . Consider the path (of length r + 1) x_{0}, …, x_{r+1} where x_{0} = eM, x_{1} = tM, x_{i} = tk_{1} … k_{i−1}M, 1 ≤ i ≤ r + 1, so that x_{r+1} = yM. Thus r + 1 ≥ d_{c}(yM, eM). But since y ∉ S, d_{c}(y, e) = ℓ, hence r = ℓ − 1. Hence (x_{0}, …, x_{r+1}) is a minimal gallery connecting the chambers eM and yM.

Since apartments in the Tits building are convex (see [B, Proposition, §IV.2, p. 88]), any minimal gallery connecting two chambers always lies in an apartment containing the two chambers. The apartment A containing eM and yM is unique since y ∉ S. Thus the chamber x_{1} = tM is uniquely determined as the chamber in A which is adjacent to the chamber eM and obtained from it by reflection in the wall associated to w_{α}. Hence tM ∈ K_{α}/M is determined uniquely by yM ∈ S^{c}/M. This proves (a).

Since r = ℓ − 1, s lies in the interior of a maximal non-open Bruhat cell. Let K′_{α} ⊂ K_{α} be a submanifold containing t transversal to tM, so that near tM, K_{α}/M is locally diffeomorphic to K′_{α}. Then the product map Ψ_{α} : K′_{α} × S/M → K/M is real analytic; by part (a) it is one-to-one near (t, s), and f_{α}(Ψ_{α}(t′, s′M)) = t′M. Since Ψ_{α} is one-to-one near (t, s), we can have DΨ_{α} = 0 only on a (possibly empty) proper Zariski closed subset V_{α}; then by the implicit function theorem, f_{α} is differentiable away from the proper Zariski closed subset Ψ_{α}(V_{α}). It is easy to check that Ψ_{α}(V_{α}) does not in fact depend on the choice of K′_{α}. This proves (b).

To show (c) note that if m ∈ M, and t = f_{α}(y), i.e. t ∈ K_{α}/M and (eM, tM, tk_{1}M, …, tk_{1} … k_{ℓ−1}M = yM) is a minimal gallery connecting eM and yM, then (meM, mtM, mtk_{1}M, …, mtk_{1} … k_{ℓ−1}M = myM) is a minimal gallery connecting meM = eM to myM. Hence f_{α}(my) = mf_{α}(y) for m ∈ M.

For k ∈ K, α ∈ Δ, we write f_{k,α}(y) = f_{α}(k^{−1}y).

Lemma 5.8 (Coordinates on K/M).

(i) For any x, y ∈ K/M with x ≠ y, there exists k ∈ K and α ∈ Δ such that f_{k,α}(x) ≠ f_{k,α}(y). (ii) For any x ∈ K/M there exist α_{1}, …, α_{m} ∈ Δ and an open full measure subset V of ∏_{i=1}^{m} K,

invariant under right-multiplication by ∏_{i=1}^{m} M, such that for any (k_{1}, …, k_{m}) ∈ V there exists an open neighborhood U of x such that the map Φ : U → K_{α_{1}}/M × … K_{α_{m}}/M given by Φ(y) = (f_{k_{1},α_{1}}(y), …, f_{k_{m},α_{m}}(y)) is a diffeomorphism onto its image.

Proof. To prove (i), we may assume without loss of generality that x = e. Suppose f_{α}(k^{−1}) = f_{α}(k^{−1}y) for all α and all k such that both sides are defined. Now suppose s ∈ S_{ℓ−1} where S_{ℓ−1} is the subset of S with d_{c}(s, e) = ℓ − 1. Then there exists α ∈ Δ such that s forms a minimal gallery together with almost all elements t ∈ K_{α}; then f_{α}(ts) is defined. Writing k^{−1} = ts, we get t = f_{α}(k^{−1}) = f_{α}(k^{−1}y) = f_{α}(tsy), i.e sy ∈ S. Note that f_{α}(tsy) is defined for almost all t if s in a dense subset S′_{ℓ−1} of S_{ℓ−1}. Hence we have sy ∈ S for all s ∈ S′_{ℓ−1}. However, S is closed and S′_{ℓ−1} is dense in S. Hence, Sy ⊂ S. But this is clearly impossible unless y = e. Indeed, let A be an apartment containing e and y^{−1}, and let w denote the chamber in A opposite e. Then there exists a minimal gallery connecting e and w and passing though y^{−1} (see [B, Lemma, §IV.5, p. 92]). Thus w = y^{−1}s for some s ∈ S, hence w^{−1} = s^{−1}y. Since S^{−1} = S, s^{−1} ∈ S. However w^{−1} ∉ S hence we have a contradiction. This proves (i). Note that (i) implies that for any x ≠ y there exists α ∈ Δ such that the set {k ∈ K : f_{k,α}(x) ≠ f_{k,α}(y)} is open and dense K.

Suppose (ii) does not hold. Then there exists a point y ∈ K/M and a vector X ∈ T_{y}(K/M) such that for all all α ∈ Δ and all k with k^{−1}y ∈ S_{α}^{c}, Df_{k,α}(X) = 0. Since for any k_{0} ∈ K, f_{k_{0}k,α}(k_{0}y) = f_{k,α}(y), the vector k_{0}X ∈ T_{k_{0}y}(K/M) satisfies the same condition, i.e Df_{k,α}(k_{0}X) = 0 for all α ∈ Δ and all k ∈ K with k^{−1}(k_{0}y) ∈ S_{α}^{c}. We thus define a vector field on K/M by choosing the vector k_{0}X in T_{k_{0}y}(K/M). Let γ : [0, 1] → K/M be a flow line for this vector field; then f_{k,α}(γ(0)) = f_{k,α}(γ(1)) for all α ∈ Δ and k ∈ K such that both sides are defined. This contradicts (i).

Proof of Proposition 5.5. For k ∈ U_{1}, ϕ_{0}(kK_{α}) ⊂ ϕ_{0}(k)K_{β}, for some β ∈ Δ.

Suppose x ∈ U_{1} ∩ S^{c}. Let t = f_{k,α}(x), i.e k^{−1}x = ts, s ∈ S. Write s = k_{1} … k_{ℓ−1} where k_{i} ∈ K_{α_{i}} for some α_{i} ∈ Δ. Then the gallery

(eM, tM, tk_{1}M, tk_{1}k_{2}M, …, tk_{1} … k_{ℓ−1}M)

is a minimal gallery connecting e and k^{−1}x. Hence

(kM, ktM, ktk_{1}M, ktk_{1}k_{2}M, …, ktk_{1} … k_{ℓ−1}M)

is a minimal gallery connecting k and x.

Suppose this gallery lies completely in U_{1} (this will happen for x ∈ H_{k,α} where H_{k,α} ⊂ K/M has full measure). Then

(ϕ_{0}(k)M, ϕ_{0}(kt)M, ϕ_{0}(ktk_{1})M, ϕ_{0}(ktk_{1}k_{2})M, …, ϕ_{0}(ktk_{1} … k_{ℓ−1})M)

is also a gallery connecting ϕ_{0}(k) and ϕ_{0}(x). If ϕ_{0}(x) ∉ ϕ_{0}(k)S, then this gallery is minimal, and (since ϕ_{0}(kt) ⊂ ϕ_{0}(k)K_{β}), f_{ϕ_{0}(k),β}(ϕ_{0}(x)) = ϕ_{0}(k)^{−1}ϕ_{0}(kt). Hence

f_{ϕ_{0}(k),β}(ϕ_{0}(x)) = ϕ_{0}(k)^{−1}ϕ_{0}(kf_{k,α}(x)), for x ∈ H_{k,α} (41)

Choose x ∈ U_{1}. By Lemma 5.8 for 1 ≤ i ≤ m there exists subsets E_{i} ⊂ K of full measure such that if for 1 ≤ i ≤ m, k_{i} ∈ E_{i} then condition (ii) of Lemma 5.8 holds for (k_{1}, …, k_{m}) and a neighborhood U_{x} of x. Now let V be as in Lemma 5.8 part (ii) but for the point ϕ_{0}(x) instead of x. By Lemma 5.6 the set ∏_{i=1}^{m} ϕ_{0}(E_{i}) is dense in ∏_{i=1}^{m} K/M. Since V is open there exist k_{i} ∈ E_{i} such that (ϕ_{0}(k_{1}), …, ϕ_{0}(k_{m})) ∈ V. Thus we may assume that the condition of part (ii) of Lemma 5.8 holds for x, the k_{i}, and for some neighborhood of U_{x} of x, and also for ϕ_{0}(x), the ϕ_{0}(k_{i}), and some neighborhood U_{ϕ_{0}(x)} of ϕ_{0}(x). Then for y ∈ U_{x} ∩ ϕ_{0}^{−1}(U_{ϕ_{0}(x)}) ∩_{i=1}^{m} H_{k_{i},α_{i}}, using (41)

ϕ_{0}(y) = Φ^{−1}(ϕ_{0}(k_{1})^{−1}ϕ_{0}(k_{1}f_{k_{1},α_{1}}(y)), …, ϕ_{0}(k_{m})^{−1}ϕ_{0}(k_{m}f_{k_{m},α_{m}}(y))) (42)

where Φ is the diffeomorphism from Lemma 5.8 at the point ϕ_{0}(x). By Lemma 5.3, the restriction of ϕ_{0} to k_{i}K_{α_{i}} ∩ U_{1} is bi-Hölder. Then by (42), ϕ_{0}, restricted to U_{x} ∩ ϕ_{0}^{−1}(U_{ϕ_{0}(x)}) ∩_{i=1}^{m} H_{k_{i},α_{i}} is bi-Hölder, hence uniformly continuous. Then, by compactness, there exists a full measure subset U_{2} ⊂ U_{1} such that ϕ_{0} restricted to U_{2} is uniformly continuous. Hence ϕ_{0}|_{U_{2}} extends to a continuous map \overline{ϕ_{0}} defined on all of K/M. It is clear that \overline{ϕ_{0}} is locally injective; it remains to show that \overline{ϕ_{0}} is injective.

For every x, y ∈ K, there exists k, α so that (i) of Lemma 5.8 holds; for fixed k and α the set of (x, y) which satisfies this condition is open. Hence, using the compactness of K × K there exists a finite collection of k_{i} ∈ U_{1} such that if f_{k_{i},α_{i}}(x) = f_{k_{i},α_{i}}(y) for all i for which these functions are defined, then x = y.

Suppose \overline{ϕ_{0}} is not injective. Then, since \overline{ϕ_{0}} is locally injective, there exist disjoint neighborhoods U and V such that \overline{ϕ_{0}}(U) = \overline{ϕ_{0}}(V). Then, after passing to smaller neighborhoods if needed, there exists i such that f_{k_{i},α_{i}}(U) and f_{k_{i},α_{i}}(V) are disjoint. Since ϕ_{0} restricted to k_{i}K_{α_{i}} ∩ U_{1} is bi-Hölder, \overline{ϕ_{0}} restricted to k_{i}K_{α_{i}} is injective. Hence \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(U)) and \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(V)) are disjoint.

The fact that \overline{ϕ_{0}} is locally onto implies that there exist x ∈ U, y ∈ V such that z = \overline{ϕ_{0}}(x) = \overline{ϕ_{0}}(y). By perturbing k_{i} we may make sure that f_{ϕ_{0}(k_{i}),β}(z) makes sense for all β ∈ Δ, and still \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(x)) ≠ \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(y)). But by (41),

f_{\overline{ϕ_{0}}(k_{i}),β_{i}}(z) = \overline{ϕ_{0}}(k_{i})^{−1}\overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(x)) = \overline{ϕ_{0}}(k_{i})^{−1}\overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(y))

This contradicts the fact that \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(x)) ≠ \overline{ϕ_{0}}(k_{i}f_{k_{i},α_{i}}(y)).

6. The boundary map and the Tits building

In view of Step 5, there exists a continuous bijection \overline{ϕ_{0}} : X̂ → X̂ such that ϕ_{0} agrees with \overline{ϕ_{0}} almost everywhere.

Step 6 (ϕ_{0} is induced by an isometry). If G has no rank one factors, then there exists an element g ∈ Isom(X) such that for almost all x ∈ X̂, ϕ_{0}(x) = g · x, where · denotes action on the boundary. If G has rank one factors but rank(G) ≥ 2, then ϕ_{0} is a factor preserving map (possibly composed with a permutation of isomorphic factors).

Note. If we identify X̂ with K/M, we may push forward the measure m_{K} to obtain a measure ν on X̂. Even though ν depends on the identification (i.e. on the choice of basepoint), the measure class of ν is independent of all choices. By “almost all x ∈ X̂” we mean “ν-almost-everywhere”.

Proposition 6.1 (ϕ̄_{0} induces a Tits isomorphism T (X) → T (X)). There exists an order preserving bijection ϕ_{G} : T (X) → T (X) such that the restriction of ϕ_{G} to X̂ agrees with \overline{ϕ_{0}} (and hence with ϕ_{0} almost everywhere).

Proof of Proposition 6.1. The only properties of \overline{ϕ_{0}} used in the proof are that it is a uniformly continuous injective map which is Tits almost everywhere.

In view of our realization of T (X), it is enough to show that for any σ ⊂ Δ, there exists τ ⊂ Δ with |τ| = |σ| such that for any k ∈ K,

\overline{ϕ_{0}}(kK_{σ}) = \overline{ϕ_{0}}(k)K_{τ} (43)

Indeed, if (43) holds, we can define ϕ_{G}(kK_{σ}) = \overline{ϕ_{0}}(k)K_{τ}; since ϕ_{0} is uniformly continuous and injective, ϕ_{G} is an order preserving injection. The fact that ϕ_{G} is a bijection follows from (43) and invariance of domain (see e.g. [Mu, Theorem 36.5]).

We now prove (43). In particular, we need to show that τ in (43) (and Step 4) depends only on σ and not on k ∈ K/K_{σ}. Let π_{σ} denote the projection K/M → K/K_{σ}. Then π_{σ} is clearly continuous. Let

E ⊂ U_{1} = {k ∈ U_{1} : m_{K}(kK_{θ} ∩ U_{1}) = m_{K}(kK_{θ}) for all θ ⊂ Δ},

Then m_{K}(E) = 1, and thus in particular E is dense. For each τ with |τ| = |σ|, let

F_{τ} = \{z ∈ K/K_{σ} : \overline{ϕ_{0}}(π_{σ}^{−1}(z)) ⊂ k′K_{τ} for some k′ = k′(z)\} (44)

Suppose for now that σ is such that dim K/K_{σ} is maximal, i.e dim K/K_{θ} ≤ dim K/K_{σ} for all θ ⊂ Δ with |θ| = |σ|. Since \overline{ϕ_{0}} is continuous and bijective, invariance of domain implies that for every τ with F_{τ} non-empty, equality holds in (44), i.e \overline{ϕ_{0}}(π_{σ}^{−1}(z)) = k′K_{τ}. This also implies that the distinct F_{τ} are disjoint.

By Step 4, for each k ∈ U_{1} there exists τ ⊂ Δ with |τ| = |σ| and k′ ∈ K such that ϕ_{0}(kK_{σ} ∩U′) ⊂ k′K_{τ} Thus E ⊂ ⋃_{|τ|=|σ|} π_{σ}^{−1}(F_{τ}) ∩ U_{1}, and if x ∈ E and π_{σ}(x) ∈ F_{τ} then ϕ_{0}(xK_{σ} ∩ U′) ⊂ k′K_{τ} for some k′ ∈ K.

We now claim that

\overline{π_{σ}(E) ∩ F_{τ}} ⊂ F_{τ} . (45)

Suppose π_{σ}(x) ∈ F̄_{τ}. Let y ∈ K be any point such that π_{σ}(x) = π_{σ}(y). Let x_{j} and y_{j} be two sequences in E ∩ π_{σ}^{−1}(F_{τ}) converging to x and y respectively. Since x_{j} ∈ E and lim π_{σ}(x_{j}) = lim π_{σ}(y_{j}), there exists y′_{j} ∈ E ∩ x_{j}K_{σ} such that d(y′_{j}, y_{j}) → 0. Since π_{σ}(x_{j}) = π_{σ}(y′_{j}) ∈ F_{τ}, π_{τ} (\overline{ϕ_{0}}(y′_{j})) = π_{τ} (\overline{ϕ_{0}}(x_{j})). Since \overline{ϕ_{0}} and π_{τ} are both continuous, π_{τ} (\overline{ϕ_{0}}(x)) = lim π_{τ} (\overline{ϕ_{0}}(x_{j})) = lim π_{τ} (\overline{ϕ_{0}}(y′_{j})) = lim π_{τ} (\overline{ϕ_{0}}(y_{j})) = π_{τ} (\overline{ϕ_{0}}(y)). Thus, since y ∈ π_{σ}^{−1}(π_{σ}(x)) is arbitrary, π_{σ}(x) ∈ F_{τ}. This proves (45).

We now claim that each (non-empty) F_{τ} is open. Indeed, suppose x ∈ F_{τ}. Let B_{j} be a sequence of balls shrinking to x. If all sufficiently small B_{j} contained points in F_{τ′} ∩ π_{σ}(E), where τ′ ≠ τ, then by (45) we would have x ∈ F_{τ′}, which would contradicts the disjointness of the distinct F_{τ}. Hence for some n, B_{n} ∩ π_{σ}(E) ⊂ F_{τ}. Since π_{σ}(E) is dense, this implies B_{n} ⊂ F_{τ}, hence F_{τ} is open. Now by connectedness, one of the F_{τ} is all of K/K_{σ}, and the rest are empty. Hence (43) holds for σ.

Note that if (43) holds, then σ and τ determine each other. Thus to prove (43) for all σ (without the assumption that K/K_{σ} has maximal dimension), one can proceed by reverse induction on the dimension of K/K_{σ}.

Proof of Step 6. It is enough to prove that for all x in K/M ≅ X̂, \overline{ϕ_{0}}(x) = g · x for some g ∈ Isom(X). By Proposition 6.1, \overline{ϕ_{0}} induces a bijection ϕ_{G} of the Tits building associated to X into itself; the map on the cells of maximal dimension (i.e chambers) is \overline{ϕ_{0}}, hence a homeomorphism. Thus, ϕ_{G} is an order preserving bijection of the Tits building onto itself which is also a homeomorphism of the Furstenberg boundary into itself. By the theorem of Tits ([Ti], see also [Mo, Corollary 16.2 and §17]) any such map is induced by an isometry as long as G has no rank one factors, and is factor preserving (up to a permutation) even if G has rank one factors.

7. The quasi-isometry ϕ is close to an isometry

In §7, §8 and §9 we assume that G has no rank one factors; we will return to the case where G has rank one factors in §10.

Step 7 (ϕ is close to an isometry). There exists an isometry g and a number C > 0 such that for all γ ∈ Γ, d(ϕ(γ), π(gγ)) < C.

We first note a corollary of Step 6:

Corollary 7.1. Let g ∈ Isom(X) be as in Step 6. Then there exists a subset E_{0} ⊂ G with the following properties:

(a) E_{0} has full measure, i.e for any compact U ⊂ G, m(E_{0} ∩ U) = m(U) where m(·) denotes Haar measure on G.

(b) Let F_{0} ⊂ F^{∗} denote the family of flats {π(xA) : x ∈ E_{0}}. Then, for any F ∈ F_{0} and any x ∈ Ω_{F}, we have d(ϕ(x), gF) ≤ N.

Proof of Corollary 7.1. Let

E_{0} = {x ∈ p^{−1}(E^{∗}) such that for every chamber ℭ ⊂ π(xA), ϕ_{0}(ℭ) = g · ℭ.}

where E^{∗} is as in the proof of Lemma 3.2. Then, part (a) holds since the condition ϕ_{0}(ℭ) = g · ℭ holds a.e. by Step 6. The condition (b) holds by Step 3, since we may identify F′ as gF.

We also note the following structural lemma, analogous to Lemma 3.1:

Lemma 7.2. For every δ > 0 there exists a subset Q ⊂ K/M × K/M with m_{K}(Q) > 1 − δ such that for (k_{1}, k_{2}) ∈ Q and k ∈ K we have (kk_{1}, kk_{2}) ∈ Q and for any h ∈ G the following hold:

(a) The flats F_{1} = π(hk_{1}A) and F_{2} = π(hk_{2}A) satisfy F_{1} ∩ F_{2} = {π(h)}.

(b) The flats F_{1} and F_{2} are “almost orthogonal”, i.e. there exists constants λ_{1}, λ_{2} depending only on X and δ, such that if r > λ_{2} then

Nbhd(F_{1}, r) ∩ Nbhd(F_{2}, r) ⊂ B(π(h), λ_{1}r)

Proof. Without loss of generality, we may assume h = 1. Let R = ⋃_{w∈W} ⋃_{σ⊂Δ} K_{σ}w. We choose Q of the form {k ∈ K : d_{K}(k, R) ≥ ν}, where ν = ν(δ, X). Suppose x_{1} ∈ F_{1}, x_{2} ∈ F_{2}. Then we may write x_{i} = π(k′_{i}a_{i}) where a_{i} ∈ A^{+} and k′_{i} ∈ k_{i}W. Note that for every σ ⊂ Δ, d_{K}(k′_{1}, k′_{2}K_{σ}) > θ where θ depends only on ν and X.

For i = 1, 2 there exists α_{i} ∈ Δ such that α_{i}(x_{i}) ≻ d(x_{i}, e). Let σ = {α_{1}} ∪ {α_{2}}. Then, by Lemma B.3, if s = min_{i=1,2} d(x_{i}, e) is sufficiently large, then d(x_{1}, x_{2}) ≻ s. This implies (b).

Proof of Step 7. Let D ⊂ G denote a fundamental domain for Γ\G. For u ∈ D let S_{u} = {k ∈ K : p(uk) ∈ Ω }, where Ω is as in Step 2 (e). By Step 2 (e) and Fubini’s theorem there exists a compact subset U of D of positive measure such that for any u ∈ U, m_{K}(S_{u}) ≥ 1 − 3|Σ|ϵ. Then, in view of Lemma 7.2, for any u ∈ U, m_{K}(Q ∩ (S_{u} × S_{u})) > 0.

Now γ ∈ Γ be arbitrary. Let E_{0} be as in Corollary 7.1. In view of Corollary 7.1 (a), we may choose u ∈ U and (k_{1}, k_{2}) ∈ Q ∩ (S_{u} × S_{u}) so that γuk_{i} ∈ E_{0} for i = 1, 2. Let h = γu and for i = 1, 2 let F_{i} = π(hk_{i}A). Note that since U is compact, d(π(γ), π(h)) < c where c depends only on U (and thus only on X).

Since p(hk_{i}) ∈ Ω, π(h) ∈ Ω_{F_{1}} ∩ Ω_{F_{2}}. Hence, by Corollary 7.1 (b),

ϕ(π(h)) ⊂ Nbhd(gF_{1}, N) ∩ Nbhd(gF_{2}, N)

Since g acts by isometries, and F_{1}, F_{2} are almost orthogonal, the flats gF_{1} and gF_{2} are also almost orthogonal, and intersect at π(gh) = gπ(h) ∈ X. Hence, by Lemma 7.2,

ϕ(π(h)) ⊂ Nbhd(gF_{1}, N) ∩ Nbhd(gF_{2}, N) ⊂ Nbhd(gF_{1} ∩ gF_{2}, N′) = Nbhd(π(gh), N′)

where N′ depends only on N and X. Since d(π(γ), π(h)) < c, ϕ is a quasi-isometry and g is an isometry, for any γ ∈ Γ, d(ϕ(γ), π(gγ)) is bounded independently of γ.

Remark: We actually proved that d(ϕ(γ), π(gγ)) is bounded by a constant depending only on κ, C, X, and Γ.

8. The quasi-isometry ϕ is close to a commensuration

Step 8 (ϕ is close to a commensuration). The element g of Step 7 belongs to Comm(Γ).

Lemma 8.1. Suppose g ∈ Isom(X) is such that the orbit p(gΓ) ⊂ Γ\Isom(X) is bounded. Then p(gΓ) is finite, and g ∈ Comm(Γ).

Proof. This lemma, and its equivalence to the statement of Step 8 are due to Nimish Shah, who proved a much more general statement in [Sh]. The proof uses ideas from the theory of unipotent flows. Recall that a finite index subgroup of Γ is generated by unipotent elements. By [Sh], \overline{p(g)Γ} = p(F) for a closed subgroup F of Isom(X) containing Γ. By Borel density, F^{0} is normal in G, and p(F^{0}) is closed. Hence F^{0} = G or F^{0} = {1}. Since p(F^{0}) is compact, F^{0} ≠ G. Hence F^{0} is the identity, and thus p(gΓ) is finite.

Proof of Step 8. This follows immediately from Lemma 8.1. Indeed ϕ : Γ → Γ is a quasi-isometry, and by Step 7 there exists g ∈ Isom(X) and c < ∞ such that for all γ ∈ Γ, d(π(gγ), ϕ(γ)) < c. Since ϕ(γ) ∈ Γ, we get d(p(gγ), p(Γ)) < c. Hence the orbit p(gΓ) is bounded in Γ\Isom(X). By Lemma 8.1 this implies g ∈ Comm(Γ).

This completes the proof of Theorem 0.1.

9. Proof of Corollary 0.2

Proof of Corollary 0.2. This is by now standard; the relevant argument appeared in [CC], [S1], [FS], and [S2]. We give only a short outline for completeness. Suppose Λ is finitely generated, and we have a (κ, c) quasi-isometry q : Λ → Γ. For each λ ∈ Λ, left multiplication by λ is an isometry ℓ_{λ} : Λ → Λ; so q ∘ ℓ_{λ} ∘ q^{−1} is a quasi-isometry from Γ to Γ. Thus we have a map Ψ : Λ → QI(Γ) sending λ to q ∘ ℓ_{λ} ∘ q^{−1}, which one can verify to be a homomorphism. By Theorem 0.1 there is an isomorphism i of QI(Γ) with Comm(Γ). Let Φ = i ∘ Ψ. Thus Φ(Λ) is a subgroup of Comm(Γ).

The fact that Λ is finitely generated and the definition of Φ imply that for any r ≫ 1, the set {λ ∈ Λ : d(Φ(λ)1, 1) < r} is finite. Hence the kernel of Φ is finite, and also Φ(Λ) is discrete.

To show Φ(Λ) is a lattice, one may argue as follows. By the construction, uniformly for γ ∈ Γ, d(γ, Ψ(q^{−1}(γ))1) < ∞.

Note that for any λ ∈ Λ, Ψ(λ) is a (κ, C) quasi-isometry Γ → Γ, with constants κ and C which are independent of λ. Hence, since our bound in Theorem 0.1 depended only on Γ and the quasi-isometry constants, d(Ψ(λ), Φ(λ)) < ∞ uniformly for λ ∈ Λ. Hence, uniformly for γ ∈ Γ, d(γ, Φ(q^{−1}(γ))1) < ∞, i.e. every point γ ∈ Γ is within a finite distance from a point in Φ(Λ). Hence Φ(Λ) has finite co-volume and is thus a lattice.

10. Rank One Factors

In this section we prove the following:

Proposition 10.1 (ϕ extends to all of G and is factor preserving). Let G be a semisiple Lie group with finite center and rank at least 2; suppose G = G_{1} × G_{2} × … × G_{n} where each G_{i} is simple. Let Γ be an irreducible nonuniform lattice in G, and let ϕ be a quasi-isometry of Γ to itself, in the word metric. Then for each i there exists a quasi-isometry ψ_{i} : G_{i} → G_{i} (defined on all of G_{i}) such that ϕ is a bounded distance from the product map ψ_{1} × ψ_{2} × … × ψ_{n} (possibly composed with an isometry permuting isomorphic factors). If G_{i} is any factor with rank G_{i} ≥ 2 then ψ_{i} can be taken to be an isometry.

This is an intermideate result which is intended to be combined with results from [S1], [S2] and [FS] to give a generalization of Theorem 0.1 and Corollary 0.2 to the case where rank G ≥ 2 but G has rank one factors; see [Fa] for an outline of the argument. The proof of Proposition 10.1 is a variation on the argument in §7. We begin with the following:

Lemma 10.2. Let G be as in Proposition 10.1. Fix i, 1 ≤ i ≤ n, and let π_{i} denote the projection G → G_{i}/K_{i}, where K_{i} is the maximal compact subgroup of G_{i}. Let C be a compact subset of Γ\G. Suppose x ∈ p^{−1}(C), y ∈ p^{−1}(C), and π_{i}(x) = π_{i}(y). Then there exist flats F_{1}, F_{2}, H_{1}, H_{2} and points x_{0}, y_{0} ∈ G such that the following hold:

(a) d(π(x), π(x_{0})) < c_{0} and d(π(y), π(y_{0})) < c_{0} where c_{0} is independent of x and y.

(b) {π(x_{0})} = F_{1} ∩ F_{2} and {π(y_{0})} = H_{1} ∩ H_{2}.

(c) F_{1} and F_{2} are almost orthogonal (see Lemma 7.2); so are H_{1} and H_{2}.

(d) π(x_{0}) ∈ Ω_{F_{1}} ∩ Ω_{F_{2}} and π(y_{0}) ∈ Ω_{H_{1}} ∩ Ω_{H_{2}}, where sets of the type Ω_{F} are as in Step 3.

(e) For every chamber ℭ contained in F_{1} or F_{2} or H_{1} or H_{2}, ϕ_{0}(ℭ) = \overline{ϕ_{0}}(ℭ).

(f) π_{i}(F_{1}) = π_{i}(H_{1}) and π_{i}(F_{2}) = π_{i}(H_{2}).

Proof of Lemma 10.2. Let E_{0} be as in the proof of Corollary 7.1, i.e.

E_{0} = \{x ∈ p^{−1}(E^{∗}) such that for every chamber ℭ ⊂ π(xA), ϕ_{0}(ℭ) = \overline{ϕ_{0}}(ℭ)\}

where E^{∗} is as in the proof of Lemma 3.2. Then E_{0} has full measure. Let E′_{0} = {x ∈ G such that xk ∈ E_{0} for almost all k ∈ K}. Then E′_{0} also has full measure.

Let D be a fundamental domain for Γ\G. Let U ⊂ D be as in the proof of Step 7. Since x, y ∈ p^{−1}(C), C and U are compact, and E′_{0} has full measure, there exists z ∈ G with d(π(z), e) < c_{0}, xz ∈ p^{−1}(U) ∩ E′_{0} and yz ∈ p^{−1}(U) ∩ E′_{0}. Let x_{0} = xz and let y_{0} = yz.

Let Q ⊂ K/M ×K/M be as in Lemma 7.2. For u ∈ U let S_{u} ⊂ K/M be as in the proof of Step 7. By an argument identical to the one in §7, m_{K}(Q ∩ (S_{p(x_{0})} × S_{p(x_{0})}) ∩ (S_{p(y_{0})} × S_{p(y_{0})})) > 0. Hence there exist (k_{1}, k_{2}) ∈ K/M × K/M such that (k_{1}, k_{2}) ∈ Q ∩ (S_{p(x_{0})} × S_{p(x_{0})}) ∩ (S_{p(y_{0})} × S_{p(y_{0})}) and for j = 1, 2, x_{0}k_{j} ∈ E_{0} and y_{0}k_{j} ∈ E_{0}. By construction, the flats F_{j} = π(x_{0}k_{j}A) and H_{j} = π(y_{0}k_{j}A) satisfy the conditions of the lemma.

Lemma 10.3. Let G be as in Proposition 10.1. Fix i, 1 ≤ i ≤ n, and let π_{i}(·) and C be as in Lemma 10.2. Then if x ∈ p^{−1}(C), y ∈ p^{−1}(C), and π_{i}(x) = π_{i}(y), then d(π_{i}(ϕ(x)), π_{i}(ϕ(y))) < C, where C does not depend on x and y.

Proof of Lemma 10.3. Let x_{0}, y_{0}, F_{1}, F_{2}, H_{1} and H_{2} be as in Lemma 10.2. By Step 3, for j = 1, 2 there exist flats F′_{j}, H′_{j} such that ϕ(Ω_{F_{j}}) ⊂ Nbhd(F′_{j}, N) and ϕ(Ω_{H_{j}}) ⊂ Nbhd(H′_{j}, N), Hence ϕ(x_{0}) ⊂ Nbhd(F′_{1}, N) ∩ Nbhd(F′_{2}, N) and ϕ(y_{0}) ⊂ Nbhd(H′_{1}, N) ∩ Nbhd(H′_{2}, N).

Note that since F_{1}, F_{2} and H_{1}, H_{2} are almost orthogonal, there exist points x′, y′ and a constant C′ independent of x and y such that Nbhd(F′_{1}, N)∩Nbhd(F′_{2}, N) ⊂ Nbhd(x′, C′) and Nbhd(H′_{1}, N)∩ Nbhd(H′_{2}, N) ⊂ Nbhd(y′, C′). Thus d(ϕ(x_{0}), x′) < C′′, and d(ϕ(y_{0}), y′) < C′′, where C′′ does not depend on x and y.

Since π_{i}(F_{1}) = π_{i}(H_{1}) and π_{i}(F_{2}) = π_{i}(H_{2}), by Step 6 we have π_{i}(F′_{1}) = π_{i}(H′_{1}) and π_{i}(F′_{2}) = π_{i}(H′_{2}). Hence d(π_{i}(ϕ(x)), π_{i}(ϕ(y))) < C as required.

Proof of Proposition 10.1. Since Γ is irreducible, π_{i}(p^{−1}(C)) = G_{i}/K_{i}. If u ∈ G_{i}/K_{i}, choose x ∈ π_{i}^{−1}(u) ∩ p^{−1}(C), and let ψ_{i}(u) = π_{i}(ϕ(x)). In view of Lemma 10.3, this (up to a fixed additive constant) does not not depend on the choice of x. To see that ψ_{i} is a quasi-isometry, we may argue as follows: suppose u_{1}, u_{2} ∈ G_{i}/K_{i}. It follows, e.g. from the Moore ergodicity theorem applied to the action of the Cartan subgroup of ∏_{j≠i} G_{j} on Γ\G that there exist points x_{1}, x_{2} ∈ p^{−1}(C) such that π_{i}(x_{1}) = u_{1}, π_{i}(x_{2}) = u_{2}, and π_{j}(x_{1}) = π_{j}(x_{2}) for j ≠ i. Note that d(x_{1}, x_{2}) = d(u_{1}, u_{2}), and by Lemma 10.3, |d(ψ_{i}(u_{1}), ψ_{i}(u_{2})) − d(ϕ(x_{1}), ϕ(x_{2}))| < C′′ where C′′ is fixed. Since ϕ is a quasi-isometry, this implies that ψ_{i} also is.

A. Quasi-isometries

A.1. The packing lemma and its corollaries. The following lemma is a coarse version of invariance of domain. The proof given in [EF, §8.2] is a variation of an argument due to Geoff Mess^{1}. See also [FS, Corollary 5.3].

Lemma A.1 (Local Packing). Let q : B → ℝ^{n} be a continuous map which is a (κ, C)-quasi-isometric embedding from the ball B = B(x, r_{1}) ⊂ ℝ^{n} into ℝ^{n}. Then there exist constants η_{1} > 1, η_{2} > 1 depending only on κ such that for any r < r_{1},

Nbhd(q(B), η_{1}C) ⊃ B(q(x), r/η_{2})

(i.e. q(B) essentially contains the ball B(q(x), r/η_{2})).

Corollary A.2 (Packing of metric interiors). Let U, V be subsets of ℝ^{n} and let q : U → ℝ^{n} be a (κ, C) quasi-isometric embedding. Suppose there exists R > 0 with Int(V, R) path connected and q^{−1}(Int(V, R)) a non-empty subset of Int(U, 4η_{1}η_{2}C). Then Int(V, R) ⊂ Nbhd(q(U), η_{1}C).

1Thanks to Shmuel Weinberger for telling us about this argument.

Proof. Let W = Nbhd(q(U), η_{1}C). Take any y ∈ q(U) ∩ Int(V, R). Let r_{1} = 4η_{1}η_{2}C. By assumption, q^{−1}(y) ∈ Int(U, r_{1}), hence by Lemma A.1, W ⊃ B(y, 3η_{1}C). Since W ∩ Int(V, R) is non-empty and Int(V, R) is connected, this implies that W ⊃ Int(V, R).

Lemma A.3 (Packing of complexes). Let P_{1}, ⋯, P_{m} be isometrically embedded d-dimensional convex Euclidean polyhedra in X. Let B be a ball in X, and suppose q : ℝ^{d} → X is a (κ, C) quasiisometric embedding such that

q(ℝ^{d}) ∩ B ⊂ ⋃_{i=1}^{m} P_{i}

Suppose the P_{i} “come close only near their boundaries”, i.e. there exists a constant r > (4κη_{1}η_{2}+1)C (where η_{1}, η_{2} are as in Lemma A.1) such that for all i ≠ j,

Int(P_{i}, r) ∩ Nbhd(P_{j}, (4κη_{1}η_{2} + 1)C) = ∅ (46)

Suppose there is a constant λ′′ such that for all i,

hd(Int(P_{i}, r), P_{i}) ≤ λ′′r. (47)

Then there exists a subset σ of {1, …, m} and a constant λ′ (depending only on λ′′, κ, m and the number faces of the P_{i}) such that

hd (q(ℝ^{d}) ∩ B′, ⋃_{i∈σ} P_{i} ∩ B′) ≤ λ′r

where B′ = Int(B, λ′r).

Proof. Let V_{j} = P_{j} ∩ B, and let σ = {j ∈ [1, m] : q(ℝ^{d}) ∩ Int(V_{j}, r) ≠ ∅}. By convexity V_{j} is connected for every j ∈ σ.

For j ∈ σ, let U_{j} = q^{−1}(V_{j}). Now suppose x ∈ q^{−1}(Int(V_{j}, r)). Then

q(B(x, 4η_{1}η_{2}C)) ⊂ B(q(x), (4κη_{1}η_{2} + 1)C) ∩ q(ℝ^{d}) ∩ B ⊂ V_{j}

by (46). Hence q^{−1}(Int(V_{j}, r)) ⊂ Int(U_{j}, 4η_{1}η_{2}C), and thus by Corollary A.2, Int(V_{j}, r) ⊂ Nbhd(q(U_{j}), η_{1}C). Since q(U_{j}) ⊂ V_{j} and using (47), hd(q(U_{j}), V_{j}) ≤ λ′′r. Hence

⋃_{j∈σ} V_{j} ⊂ Nbhd(q(ℝ^{d}), λ′′r) (48)

Suppose ρ ≫ C, and x ∈ q(ℝ^{d}) ∩ Int(B, ρ). Then, since q(B(q^{−1}(x), ρ/(2κ))) ⊂ q(ℝ^{d}) ∩ B(x, ρ) and using again Lemma A.1, there exists a constant ν depending only on κ, and d such that

Vol_{d}(q(ℝ^{d}) ∩ B(x, ρ)) ≥ νρ^{d} (49)

where Vol_{d} denotes the d-dimensional volume.

Now suppose there exists x ∈ q(ℝ^{d}) such that x ∈ B′, but d(x, ⋃_{j∈σ} V_{j}) ≥ ρ. Then, q(ℝ^{d}) ∩ B(x, ρ) ⊂ ⋃_{j∉σ} V_{j}. But then, since q(ℝ^{d})∩Int(V_{j}, r) = ∅ for j ∉ σ, Vol_{d}(q(ℝ^{d})∩B(x, ρ)) ≤ λ′′rρ^{d−1}, where λ′′ depends only on m, d, and the number of faces of the P_{i}. When ρ > (λ′′/ν)r, this contradicts (49). Let λ′ = λ′′/ν. Then,

q(ℝ^{d}) ∩ B′ ⊂ Nbhd( ⋃_{j∈σ} V_{j}, λ′r) (50)

The estimates (48) and (50) imply the lemma.

A.2. Closeness sets and intersections.

Lemma A.4 (Quasi-isometries preserve closeness sets). Suppose X and Y are metric spaces, q : X → Y is a (κ, C) quasi-isometric embedding, U and V are subsets of X, and U′, V′ are subsets of Y.

(i) Suppose q(U) ⊂ U′, q(V) ⊂ Nbhd(V′, C′), and r > C′ > C. Then

q(U ∩ Nbhd(V, r)) ⊂ U′ ∩ Nbhd(V′, 3κr)

(ii) Suppose hd(q(U), U′) < C′, hd(q(V).V′) < C′ and r > 3κC′ > 3κC. Suppose also that there exists a constant λ such that

(51) U′ ∩ Nbhd(V′, 3κr) ⊂ Nbhd(U′ ∩ V′, λr) and

U ∩ Nbhd(V, 3κr) ⊂ Nbhd(U ∩ V, λr)

Then,

hd (q(U ∩ V), U′ ∩ V′) < λ′r

where λ′ depends only on κ and λ.

Proof. Suppose u ∈ U ∩ Nbhd(V, r), then there exists v ∈ V with d(u, v) ≤ r. Since r > C, by (1), d(q(u), q(v)) ≤ 2κr. By assumption, there exists v′ ∈ V′ such that d(q(v), v′) ≤ C′. Then, d(q(u), v′) ≤ 2κr + C′ ≤ 3κr, hence q(u) ∈ U′ ∩ Nbhd(V′, 3κr).

To get (ii) we first modify q by O(C′) so that q(U) ⊂ U′. Then by part (i), q(U ∩ V) ⊂ q(U ∩ Nbhd(V, r)) ⊂ U′ ∩ Nbhd(V′, 3κr). Hence, q(U ∩ V) ⊂ Nbhd(U′ ∩ V′, λr). Applying the same argument to the coarse inverse of q shows that U′ ∩ V′ ⊂ Nbhd(q(U ∩ V), λ′r).

B. Geometric Estimates

B.1. Geometry of neighboring flats. In this subsection we prove the following lemma:

Lemma B.1 (Geometry of neighboring flats). (cf. [Mo, Theorem 7.8].)

(i) There exist constants λ_{0}, λ_{1}, λ_{2} depending only on X such that the following holds: For each pair of flats F_{1}, F_{2} ⊂ X there exist constants c_{α} ≥ 0 such that if we denote

P(r) = {x ∈ F_{1} : c_{α}e^{α(x)} ≤ e^{r}, ∀α ∈ Σ} (52)

then for r > λ_{0}(1 + d(F_{1}, F_{2})), we have P(λ_{1}r) ⊂ F_{1} ∩ Nbhd(F_{2}, r) ⊂ P(λ_{2}r). (Note that if c_{α} = 0, then the inequality (52) is always satisfied). In particular, there exists a convex polyhedron P ⊂ F_{1} such that hd(F_{1} ∩ Nbhd(F_{2}, r), P) = O(r).

(ii) Let L_{1} be a hyperplane in F_{1}. Then there exists a hyperplane L_{2} ⊂ F_{2} such that for all

r > λ_{0}(1 + d(F_{1}, F_{2})), we have F_{2} ∩ Nbhd(L_{1}, r) ⊂ Nbhd(L_{2}, λr) where λ depends only on X. (iii) Let P_{1} be a (possibly unbounded) polyhedron in F_{1}, and r > λ_{0}(1 + d(F_{1}, F_{2})). Then there

exists a polyhedron P_{2} ⊂ F_{2} such that hd(F_{2} ∩ Nbhd(P_{1}, r), P_{2}) ≤ λr, where λ depends only on X.

The proof of Lemma B.1 uses the following:

Lemma B.2 (Gaussian Elimination). Let MAN denote the Langlands decomposition of a Borel subgroup B where M is compact. Let W = N_{K}(A)/M denote the Weyl group associated to (G, A), let w_{0} denote the longest element of W and let N̄ denote w_{0}Nw_{0}^{−1}.

For every compact set C ⊂ G, there exist compact sets U_{N} ⊂ N, U_{A} ⊂ A, and UN̄ ⊂ N̄ such that C ⊂ W UN̄MU_{A}U_{N}. The set U_{N} can be chosen so that a^{−1}U_{N}a ⊆ U_{N} for a ∈ \overline{A_{+}}.

Proof of Lemma B.2. Note that for w ∈ W, w^{−1}BwB ⊂ w_{0}^{−1}Bw_{0}B. Then, for w ∈ W, BwB = w(w^{−1}Bw)B ⊂ w(w_{0}^{−1}Bw_{0})B. Hence, using the Bruhat decomposition,

G = ⋃_{w∈W} w(w_{0}^{−1}Bw_{0})B. (53)

Now for g in (w_{0}^{−1}Bw_{0})B, we can write g = n̄man, where n ∈ N̄, m ∈ M, a ∈ A, n ∈ N, and the functions g → n̄(g) etc. are clearly continuous (e.g. one can see this by embedding in SL(n, ℝ)). Thus these functions are uniformly continuous on any compact set in the interior of (w_{0}^{−1}Bw_{0})B. This, together with (53) implies the lemma.

We remark that the SL(n, ℝ) case of this lemma is the well known “complete pivoting” algorithm for inverting a matrix. In this case N is the set of upper triangular matrices with 1’s along the main diagonal, N̄ is the set of lower triangular matrices with 1’s along the main diagonal, A is the set of diagonal matrices of determinant 1 and M is the finite group of diagonal matrices with ±1 along the diagonal (and determinant 1). Thus (after a suitable permutation corresponding to W) one must factor a matrix into a product of an upper triangular and a lower triangular matrix; this is done by Gaussian elimination, except that one always performs row exchanges in order to divide by as large a number as possible.

Proof of Lemma B.1. Without loss of generality, we may assume F_{2} = π(A), and that there exists a point π(g) on F_{1} such that F_{1} = π(gA), and d(π(g), e) < d(F_{1}, F_{2}) + 1. In the sequel we abuse notation by writing for g_{1}, g_{2} ∈ G, d(g_{1}, g_{2}) instead of d(π(g_{1}), π(g_{2})).

We first prove the following inequality: there exists w ∈ W and a constant λ depending only on X such that for any a ∈ A^{+}

λ^{−1}d(wa, ga) − O(d(g, 1)) ≤ d(F_{2}, π(ga)) ≤ λd(wa, ga) + O(d(g, 1)) (54)

(all the implied constants depend only on X). This is proved as follows. Using the Iwasawa decomposition, we may write g = ka_{1}n_{1} where k ∈ K, a_{1} ∈ A, n_{1} ∈ N. Then

d(g, 1) = d(a_{1}n_{1}, 1) = d(n_{1}, a_{1}^{−1}). (55)

But d(n_{1}, a_{1}^{−1}) ≥ d(n_{1}, A) ≥ λ^{−1}d(n_{1}, 1) where the last inequality uses [EF, Lemma A.3] (or [Mo, Lemma 7.6]). Hence

d(n_{1}, 1) ≤ λd(g, 1). (56)

Then from (55) and (56),

d(a_{1}, 1) = d(1, a_{1}^{−1}) ≤ d(1, n_{1}) + d(n_{1}, a_{1}^{−1}) ≤ (λ + 1)d(g, 1) (57)

Then using Lemma B.2 we may write k = wn̄ma_{2}n_{2} where w ∈ W, n̄ ∈ N̄, m ∈ M, a_{2} ∈ A, n_{2} ∈ N, where d(a_{2}, 1) = O(1) and d(n_{2}, 1) = O(1). Then, g = wn̄ma_{3}n_{3}, where a_{3} = a_{2}a_{1} ∈ A and n_{3} = (a_{1}^{−1}n_{2}a_{1})n_{1} ∈ N. By (57) and the triangle inequality, d(a_{3}, 1) = O(d(g, 1)). Also by (57),

(56) and the triangle inequality, d(n_{3}, 1) = O(d(g, 1)). Thus,

(58)  d(A, ga) = d(A, wn̄ma_{3}n_{3}a)
= d(A, wa(a^{−1}n̄a)(ma_{3}a^{−1}n_{3}a))
= d(a^{−1}w^{−1}A, (a^{−1}n̄a)(ma_{3}a^{−1}n_{3}a))
= d(A, (a^{−1}n̄a)(ma_{3}a^{−1}n_{3}a))  since wAw^{−1} = A and a ∈ A
= d(A, a^{−1}n̄a) + O(d(ma_{3}a^{−1}n_{3}a, 1))  by the triangle inequality
= d(A, a^{−1}n̄a) + O(d(g, 1))

where in the last line we used d(n_{3}, 1) = O(d(g, 1)), d(a_{3}, 1) = O(d(g, 1)) and the fact that conjugation by a is non-expanding on N. Similarly,

d(wa, ga) = d(wa, wn̄ma_{3}n_{3}a) = d(wa, wa(a^{−1}n̄a)(ma_{3}a^{−1}n_{3}a))

= d(1, a^{−1}n̄a) + O(d(g, 1)) (59)

But by [EF, Lemma A.3] (or [Mo, Lemma 7.6]), for n̄′ ∈ N̄, λ^{−1}d(1, n̄′) ≤ d(A, n̄′) ≤ λd(1, n̄′). This, together with (58) and (59) implies (54).

Recall that W acts transitively on the Weyl chambers of A based at e. Hence any chamber based at g of F_{1} is of the form ℭ_{w} = π(gw\overline{A_{+}}w^{−1}) for some w ∈ W. Then by (54), for each chamber ℭ_{w} of F_{1} based at π(g), there exists w′ ∈ W such that

d(w′a, gwa) = O(r),  when  π(gwa) ∈ ℭ_{w} ∩ Nbhd(F_{2}, r).

Thus on ℭ_{w} ∩Nbhd(F_{2}, r), the map ψ_{w′} : F_{1} → F_{2} given by ψ_{w′}(π(gwa)) = π(w′a) moves each point by at most O(r). Putting these together we get a map ψ : F_{1} ∩ Nbhd(F_{2}, r) → F_{2} which is “coarsely the identity”, i.e it moves each point by at most O(r). This implies that the w′ can be chosen in a consistent way, i.e. there exists w′′ ∈ W such that

(60)  d(ψ(π(ga)), π(ga)) = d(w′′a, ga) = O(r),  when  π(ga) ∈ F_{1} ∩ Nbhd(F_{2}, r).

The equation (60) implies (ii), since we may take L_{2} = ψ(L_{1}).

We now prove (i). In view of (60),

(61) d(w′′a, ga) = O(r) if and only if d(A, ga) = O(r)

First we show that for each w ∈ W and each α ∈ wΣ^{+}, there exists a constant c_{α,w} ≥ 0 such that if P_{w}(r) = {x ∈ ℭ_{w} : c_{α,w}e^{α(x)} ≤ e^{r}} for all α ∈ wΣ^{+} then

P_{w}(λ′′_{1}r) ⊂ ℭ_{w} ∩ Nbhd(F_{2}, r) ⊂ P_{w}(λ′′_{2}r) (62)

To prove (62) we may assume w = 1 (to handle the other Weyl chambers in F_{1}, we may replace g by gw for a suitable w ∈ W). We use (58), for a ∈ \overline{A_{+}}. Write n̄ = exp u, where u ∈ Lie(N̄). Then we may write u = ∑_{α∈Σ^{+}} u_{α}X_{−α} where the X_{−α} are the bases for the root spaces, and u_{α} ∈ ℝ. (For notational simplicity, if the multiplicity m_{α} of α is greater then 1, we think of α as repeated m_{α} times in Σ). Then a^{−1}n̄a = exp(∑_{α∈Σ^{+}} u_{α}e^{α(a)}X_{−α}).

By [EF, Lemmas A.2 and A.3] (or by [Mo, Lemma 7.6]), for u ∈ Lie(N̄), if either ∥u∥ ≫ 1 or d(π(exp u), π(A)) ≫ 1, d(π(exp u), π(A)) is comparable to log ∥u∥, where ∥ · ∥ is the Euclidean norm on Lie(N̄). In our case ∥u∥ ≫ 1 since r ≫ 1. Hence

λ_{1} log( ∑_{α∈Σ^{+}} u_{α}^{2}e^{2α(a)}) ≤ d(e, π(a^{−1}n̄a)) ≤ λ_{2} log( ∑_{α∈Σ^{+}} u_{α}^{2}e^{2α(a)})

We write a = exp x for x ∈ Lie(A) = ℝ^{n}. Then by definition (see [EF, §2]), α(a) = α(x), and the function α on ℝ^{n} is linear. Then

max_{α∈Σ^{+}} λ_{1} log(u_{α}^{2}e^{α(2x)}) ≤ d(e, π(a^{−1}n̄a)) ≤ max_{α∈Σ^{+}} λ_{2} log(Cu_{α}^{2}e^{α(2x)})

Exponentiating this, and using (58) we get:

max_{α∈Σ^{+}} u_{α}^{2λ_{1}}e^{α(2xλ_{1})−δ} ≤ exp d(F_{2}, π(ga)) ≤ C^{λ_{2}} max_{α∈Σ^{+}} u_{α}^{2λ_{2}}e^{α(2xλ_{2})+δ}

Since ℭ ∩ Nbhd(F_{2}, r) is the set {g exp x ∈ ℭ | d(F_{2}, π(g exp x)) ≤ r}, this implies (62).

Pick r_{0} ≫ 1 (depending only on X), and let Q = ⋃_{w∈W} P_{w}(r_{0}). Then, by the structure of P_{w}(r), hd(F_{1} ∩ Nbhd(F_{2}, r), Q) = O(r). We now show that there exists a convex polyhedron P such that hd(P, Q) = O(r). This would imply (i). First suppose G = SL(n + 1, ℝ). In view of (61) (changing w′′ to w) it is enough to show that for some polyhedron P,

hd({π(ga) ∈ F_{1} : d(π(ga), π(wa)) ≤ r}, P) = O(r). (63)

But,

d(π(ga), π(wa)) = d(π(a^{−1}w^{−1}ga), e)

Let g_{ij} denote the entry of w^{−1}g at row i column j. Then, by [EF, Lemma A.2], λ^{−1} max_{ij} |(a_{i}/a_{j})g_{ij}|^{2} ≤ d(π(a^{−1}w^{−1}ga), e) ≤ λ max_{ij} |(a_{i}/a_{j})g_{ij}|^{2} where λ depends only on X. But a_{i}/a_{j} = e^{α(log a)} for some root α of SL(n + 1, ℝ). This implies (63) for the SL(n + 1, ℝ) case. For the general case, note that we may take a linear representation of G, i.e an embedding G in SL(m, ℝ) for some m. Then the result for SL(m, ℝ) implies (63) but a priori P may be bounded by generalized hyperplanes of the form ω(x) ≤ c_{ω}, where the ω’s are the weights of the representation. Still, there are finitely many weights, and each ω is given by an expression of the form ω = ∑_{α} c_{α}α, where the implied constants depend only on X. This, together with the fact that hd(P, Q) = O(r) where Q is a union of (non-generalized) polyhedra implies that P may also be taken to be bounded by non-generalized hyperplanes.

To get (iii) we can take P_{2} = ψ(P_{1})∩P(r) where P(r) is as in part (i) with F_{1} and F_{2} interchanged.

B.2. Angles between Weyl Chambers. Let Θ(·) denote the function defined in [EF, §2.1].

Lemma B.3 (Hyperbolic geometry of X). There exist constants ν, λ_{2} depending only on X such that the following holds: Suppose x, y ∈ X satisfy α(x) ≥ r, α(y) ≥ r for all α ∈ σ, where σ ⊆ Δ is non-empty. Then if d_{K}(Θ(x), Θ(y)K_{σ}) ≥ e^{−νr} then d(x, y) ≥ λ_{2}r.

Proof. This is stated and proved in [EF] as Lemma A.4 in the case where G = SL(n, ℝ). This case implies the general case by an embedding argument, as in the proof of [EF, Lemma 4.1].

Lemma B.4 (Angles between Weyl chambers). There exist constants ν, ν′, λ_{2} depending only on X such that the following holds: Suppose k_{1}, k_{2} ∈ K, ℭ_{i} = π(k_{i}A^{+}), and there exist points z_{i} ∈ ℭ_{i} such that d(z_{1}, z_{2}) ≤ λ_{2}r, and d(z_{i}, ∂ℭ_{i}) ≥ ν′r. Then d_{K}(k_{1}, k_{2}) ≤ e^{−νr}.

Proof. This follows from Lemma B.3 with σ = Δ. The condition that α(z_{i}) ≥ r for all α ∈ Δ is implied by d(z_{i}, ∂ℭ_{i}) ≻ r.

Lemma B.5 (hyperbolic geometry). Let γ_{1}(t) and γ_{2}(t) be two geodesic rays, parameterized by arc length, emanating from the point e in a space of Y negative curvature (bounded away from zero). Pick a sufficiently large number Q, and let t denote the largest value of t′ such that d(γ_{1}(t′), γ_{2}(t′)) ≤ Q. Then (for sufficiently large t),

c_{1} log(1/θ) < t < c_{2} log(1/θ)

where θ is the angle between γ_{1} and γ_{2}, and c_{1} and c_{2} depend only on Q and Y.

Proof. Standard.

Lemma B.5 implies the following partial converse to Lemma B.4:

Lemma B.6. Suppose α ∈ Δ, k_{2} ∈ k_{1}K_{α}, and let θ = d_{K}(k_{1}, k_{2}). Then for any sufficiently large Q (depending only on X) there exist z_{i} ∈ π(k_{i}A^{+}) such that d(z_{1}, z_{2}) ≤ Q, d(z_{i}, e) ≤ λ| log θ| and d(z_{i}, ∂π(k_{i}A^{+})) ≥ ν_{Q}| log θ|. The constant ν_{Q} depends only on Q and X, and the constant λ depends only on X.

Remark. This lemma holds even without the assumption that k_{2} ∈ k_{1}K_{α}.

Proof. Let t = t(Q, θ) be as in Lemma B.5, relative to the geodesics γ_{i}(t) = k_{i} exp(tH_{α}), where H_{α} ∈ 𝔞 is a unit vector orthogonal to the hyperplane α = 0. Then H_{α} belongs to the Lie algebra of M_{α}, and thus the γ_{i} lie in X_{α} = M_{α}/(K ∩ M_{α}). Since X_{α} has rank one, Lemma B.5 applies, and (for θ ≪ 1),

c′′_{Q}| log θ| ≤ t ≤ c′_{Q}| log θ|

where c′_{Q} and c′′_{Q} depend only on Q and X. Pick a generic positive unit vector v in the Lie algebra of A_{α} (i.e. the hyperplane α = 0), and choose ν > 0 such that v + νH_{α} ∈ 𝔞^{+}. Then the points z_{i} = exp(k_{i}t(v + νH_{α})) satisfy the conditions of the lemma.

B.3. Equivalence of Weyl chambers.

Lemma B.7 (Containment of chambers). Suppose ℭ_{1} and ℭ_{2} are Weyl chambers, L is a σ-wall, and suppose that outside a sufficiently large ball, L ⊂ ℭ_{1}[λϵ] ∩ ℭ_{2}[λϵ]. Let k_{1}, k_{2} ∈ K/M be such that 𝔇_{i} = π(k_{i}A^{+}) is equivalent to ℭ_{i}. Then k_{2} ∈ k_{1}K_{σ}, and 𝔇_{1} and 𝔇_{2} share a σ-wall. In particular, if ℭ_{2} ⊂ ℭ_{1}[λϵ], then k_{1} = k_{2}.

Proof. After replacing the ℭ_{i} by equivalent Weyl chambers, we may assume that ℭ_{i} = π(k_{i}A^{+}). Also, after replacing L by a wall passing through e and a finite Hausdorff distance away, we may assume L = π(kA_{σ}^{+}) for some k ∈ K. For any r > 0 there exists a_{r} ∈ \overline{A^{+}} such that α(a_{r}) = r for α ∈ σ, α(a_{r}) = 0 for α ∉ σ. Then π(ka_{r}) ∈ L. Since L ⊂ π(k_{i}A^{+})[λϵ], there exist a_{i} ∈ A^{+} such that d(π(k_{i}a_{i}), π(ka_{r})) = O(ϵr). By [EF, Lemma 3.1], d(x, y) ≻ |α(x) − α(y)| for all x, y ∈ X, α ∈ Δ. Hence we may assume α(a_{i}) ≥ r/2 for all α ∈ σ. By Lemma B.3 if k_{i} ∉ kK_{σ}, for sufficiently large r, d(π(k_{i}a_{i}), π(ka_{r})) ≻ r. This is a contradiction.

The following lemma is well known

Lemma B.8 (Equivalent Chambers). Let ℭ be a Weyl chamber in X based at p. Then there exists an equivalent Weyl chamber ℭ′ based at e such that hd(ℭ, ℭ′) = O(d(p, e)).

Proof. We may assume ℭ = π(gA^{+}) for g ∈ G. Write g = kan using the Iwasawa decomposition. Let ℭ′ = π(kA^{+}). Then, for a_{1} ∈ A^{+},

d(π(ga_{1}), π(ka_{1})) = d(π(ka_{1}(a_{1}^{−1}ana_{1})), π(ka_{1})) =
= d(π(a_{1}^{−1}ana_{1}), e) ≺ d(π(an), e) = d(ge, e).

Thus, hd(ℭ, ℭ′) = O(d(ge, e)).

B.4. Convex Polyhedra in ℝ^{n}.

Lemma B.9 (Convex Polyhedra). Let P be a (possibly unbounded) convex polyhedron in ℝ^{n}. For every hyperplane L of ∂P, let d_{L} denote the maximum distance between a point of P and L. Let d_{P} be the smallest of d_{L}. Then the injectivity radius R_{P} of P satisfies

R_{P} ≥ \frac{1}{n + 1}d_{P}

Proof. Let B(c, R) be a ball of maximal radius R inscribed in P. Suppose that there are exactly n + 1 hyperplanes L_{k}, k = 1, ..., n + 1 of ∂P that are tangent to B(c, R) and form a simplex Q. Let d_{k}, k = 1, ..., n + 1, be the distance between L_{k} and the opposite vertex of Q and let d_{Q} be the smallest of d_{k}. Since Q ⊃ P we have d_{Q} ≥ d_{P}. We claim that R ≥ \frac{1}{n+1}d_{Q}. Indeed, connecting c with all vertices of Q we get n + 1 pyramids. The volume of k-th pyramid is \frac{1}{n}RA_{k} where A_{k} is the area of Q ∩ L_{k}. Therefore for the volume of Q we get V_{Q} = \frac{1}{n}R ∑_{k=1}^{n+1} A_{k}. Also we have V_{Q} = \frac{1}{n}d_{k}A_{k}, k = 1, ..., n + 1. Therefore V_{Q} = \frac{1}{n+1} ∑_{k=1}^{n+1} \frac{1}{n}d_{k}A_{k} ≥ \frac{1}{n(n+1)}d_{Q} ∑_{k=1}^{n+1} A_{k}. Hence R ≥ \frac{1}{n+1}d_{Q} ≥ \frac{1}{n+1}d_{P}.

In general case if the number of hyperplanes L of ∂P tangent to B(c, R) is ≥ (n+1) we can achieve, by slightly perturbing these hyperplanes (and hence changing P), that exactly n + 1 hyperplane are tangent to B(c, R) and they form a simplex. After getting the inequality R ≥ \frac{1}{n+1}d_{P} for the new polyhedron we roll back all changed hyperplanes and get by taking the limit that the inequality holds for the original polyhedron P too.

If the number of hyperplane in P tangent to B(c, R) is less then n + 1 then there must be a subspace S in ℝ^{n} of some dimension m ≥ 1 that is parallel to all such hyperplanes. Then in the orthogonal complement to S we repeat the estimates above and get R ≥ \frac{1}{n−m+1}d_{P}.

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