A rigidity theorem for the solvable Baumslag-Solitar groups

Benson Farb and Lee Mosher with an appendix by Daryl Cooper

September 20, 1996

1 Introduction

In [Gr1], [Gr2] Gromov proposes and studies the problem of classifying finitely generated groups up to quasi-isometry. On the one hand this has motivated an industry of producing quasi-isometry invariants. On the other hand there are rigidity results which give a complete quasi-isometry classification within certain classes of groups. So far most of the progress in rigidity has been made for groups which arise in geometry. For example, in a series of papers by several authors (see, e.g. [Tu], [CC], [Me], [Pa], [Sch2], [FS], [Ch], [Sch1], [KlL], [EF], [Es], or [Fa] for a summary), a battery of new techniques related to the proof of the Mostow Rigidity Theorem has led to a nearly complete quasi-isometry classification of lattices in semisimple Lie groups. Quasi-isometric rigidity results for other groups arising in geometry and topology include [Ri], [BG], [KaL].

In this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory.

The solvable Baumslag-Solitar groups. The solvable Baumslag-Solitar groups BS(1, n) are given by the presentation

BS(1, n) = ⟨a, b | aba^{−1} = b^{n}⟩.

∗The first author is supported in part by an NSF postdoctoral fellowship, the second author by NSF grant DMS 95–04946, and the third author by NSF grant DMS 95–10505.

The work of the authors at MSRI was supported by NSF grant DMS–9022140.

These groups have served as a proving ground for many new ideas in combinatorial and geometric group theory. For example, they are the simplest groups to have an exponential isoperimetric function, and they were the first known groups to be asynchronously automatic but not automatic (see [ECH^{+}]).

The rigidity machinery for lattices in Lie groups does not directly apply to the group BS(1, n) when n ≥ 2, for although BS(1, n) has a faithful, indiscrete representation into P SL_{2}(R) given by

a ↦ ( \begin{matrix}n^{1/2} & 0 \\ 0 & n^{−1/2} \end{matrix} ) b ↦ ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

it does not seem to be a lattice in any connected Lie group (see §9). Furthermore, BS(1, n), n ≥ 2 is not a 3-manifold group ([JS], [Kr]), and is not a nonpositively curved group in any sense of the word (see §9).

Our main result is the following (for definitions see below):

Theorem 7.1 (Quasi-isometric iff commensurable). Given integers m, n ≥ 2, the groups BS(1, m) and BS(1, n) are quasi-isometric if and only if they are commensurable. This happens if and only if there exist integers r, j, k > 0 such that m = r^{j} and n = r^{k}.

We heard about this question from S. Weinberger [We] and S. Gersten [Ge]. As far as we know, Theorem 7.1 is the first quasi-isometric rigidity result for non-nilpotent solvable groups. Note that BS(1, 1) ≈ Z ⊕ Z is not quasi-isometric to BS(1, n), n ≥ 2, because they have different isoperimetric functions (see [ECH^{+}]).

Geometry, boundaries, methods. As does any finitely presented group, BS(1, n) acts properly discontinuously and cocompactly by isometries on a certain metric 2-complex, in this case denoted X_{n}. In Section 3 we construct X_{n} with an explicit metric (see Figure 1). The space X_{n} admits a beautiful geometry which has aspects of both positive and negative curvature. By a basic observation of Svarc and Milnor (see §2), X_{n} is quasi-isometric to BS(1, n), so we can use X_{n} to study quasi-isometries of BS(1, n).

A key object of study in our proof of Theorem 7.1 is the upper boundary ∂^{u}X_{n} of X_{n} (see §6), which is the space of hyperbolic planes in X_{n}. This is to be compared/contrasted to the space of flats, or Furstenberg boundary, studied in Mostow’s proof [Mo], for example the boundary at infinity of a hyperbolic space. It turns out that the upper boundary ∂^{u}X_{n} is a certain Figure 1: A piece of the 2-complex X_{2} associated to BS(1, 2), together with the fibration π: X_{2} → T_{2} (see §3). For each edge E ⊂ T_{2}, the inverse image π^{−1}(E) is isometric to the “horostrip” 1 ≤ y ≤ 2 in the upper half plane model of H^{2}. The geodesic curvature vector on a horocycle induces an orientation on each edge of T_{2}. For each coherently oriented proper line L ⊂ T_{2}, the set H_{L} = π^{−1}(L) is an isometrically embedded hyperbolic plane in X_{2}. The point inverse images form a family of horocycles based at a point ∞_{L} ∈ ∂H_{L}. The figure shows the “upper boundary” ∂^{u}X_{2} (see §5), the set of hyperbolic planes in X_{2}. Also shown is the “lower boundary” ∂_{ℓ}X_{n} (see §6), a line obtained by identifying all of the lines ∂H_{L} − {∞_{L}}.

self-similar Cantor set, which can be thought of as the n-adic rationals Q_{n} with the usual metric.

In the core of this paper, Sections 4, 5, and 6, we use topological and geometric arguments to show that a quasi-isometry f : X_{n} → X_{m} induces a bilipschitz homeomorphism f^{u} : ∂^{u}X_{n} → ∂^{u}X_{m}. Homeomorphisms of self-similar Cantor sets have been studied by Cooper and Pignataro [CP]. In an appendix to our paper, Cooper provides an extension of their work which we use to finish the proof of Theorem 7.1. The idea is that bilipschitz homeomorphisms of self-similar Cantor sets are locally measure linear (for Hausdorff measure in the appropriate dimension). It is the local similarity groups of the self-similar Cantor set ∂^{u}X_{n} which are quasi-isometry invariants of X_{n}, and which allow detection of the prime factors (with multiplicities) of n.

The quasi-isometry group. For any metric space X, one can form the group QI(X) of all self quasi-isometries of X modulo those that lie a bounded distance (in the sup norm) from the identity. Modding-out by this equivalence relation makes QI(X) into a group. In proving Theorem 7.1, we actually compute the quasi-isometry group of BS(1, n), namely:

QI(BS(1, n)) ≈ Bilip(R) × Bilip(Q_{n})

where Bilip(Y) denotes the group of bilipschitz homeomorphisms of a metric space Y.

Acknowledgements

We’d like to thank Alex Eskin, Curt McMullen, and Bill Thurston for useful conversations, and Jen Taback for corrections. We especially thank Curt McMullen and Bill Thurston for critical help on Proposition 5.3. Thanks also to Steve Gersten and to Shmuel Weinberger, both of whom asked about quasi-isometries of Baumslag-Solitar groups. Finally, we’d like to express our thanks and appreciation to Daryl Cooper for telling us about homeomorphisms of self-similar Cantor sets, and for writing the appendix to this paper.

2 Preliminaries on quasi-isometries

A (coarse) quasi-isometry between metric spaces is a map f : X → Y such that, for some constants K, C, C′ > 0:

1. K^{−1}d_{X}(x_{1}, x_{2}) − C ≤ d_{Y} (f(x_{1}), f(x_{2})) ≤ Kd_{X}(x_{1}, x_{2}) + C for all x_{1}, x_{2} ∈ X.

2. The C′-neighborhood of f(X) is all of Y.

The map f is called a (K, C)-quasi-isometry, or a K-quasi-isometry for short. There is always a coarse inverse of f, a quasi-isometry g: Y → X such that f ∘ g and g ∘ f are a bounded distance from the identity maps in the sup norm; these bounds, and the quasi-isometry constants for g, depend only on the quasi-isometry constants of f.

A map satisfying (1) but not necessarily (2) is called a quasi-isometric embedding of X into Y.

It is a fundamental observation of Svarc and Milnor (see [Mi], and also [Ca]) 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. More generally, let X be a proper metric space and let Γ be a group of isometries acting properly discontinuously on X. If Γ\X is compact, then Γ is finitely generated and is quasi-isometric to X.

Two groups are (abstractly) commensurable if they have finite index subgroups which are isomorphic. Clearly commensurable groups are quasiisometric.

Furthermore, by the now-standard “connect-the-dots” technique (see, e.g. [FS], 5.4), we can change a quasi-isometry (and it’s coarse inverse) by a bounded amount in the sup norm so that it is a lipschitz map, while still being a quasi-isometry. This may require enlarging the constants, but their new values depend only on the old values.

By Rademacher’s Theorem (see e.g. [EG], 3.1.2), lipschitz maps are differentiable almost everywhere, enabling one to define the length of a lipschitz path γ : [0, 1] → X when X is a piecewise Riemannian simplicial complex. More precisely, if the derivative D_{x}γ exists at a point x ∈ [0, 1], it sends a unit vector at x to a vector in T_{γ(x)}(X). We can compute the length of this vector using the metric on X. This defines a function L(x) almost everywhere on [0, 1], and we can then define the length of γ to be ℓ(γ) = ∫_{[0,1]} L(x)dx. This integral exists because L(x) is a bounded measurable function defined almost everywhere on [0, 1], as ∥D_{x}γ∥ is bounded by the lipschitz constant of γ.

Hence it makes sense to talk about how a Lipschitz quasi-isometry between piecewise Riemannian complexes distorts path length ℓ_{X}. We also could have defined and used an obvious coarse notion of path length, which agrees with the above notion up to constant multiplicative and additive factors.

Lemma 2.1 (bounded stretch). Let q : X → Y be a lipschitz K-quasi-isometric embedding between piecewise Riemannian complexes. Then there exists K′, depending only on K and not on q, so that if γ is lipschitz path in X, then

\frac{1}{K′} ℓ_{X}(γ) − K′ ≤ ℓ_{Y} (q ∘ γ) ≤ K′ ℓ_{X}(γ) + K′

Proof. The second inequality follows easily from the fact that q is lipschitz. The first inequality follows from applying the same reasoning to a lipschitz map which is a bounded distance from a coarse inverse of q. ⋄

3 The 2-complex X_{n}

For the rest of the paper let H^{2} = {(x, y) | x ∈ R, y > 0} denote the hyperbolic plane in the upper half plane model, with metric (dx^{2} + dy^{2})/y^{2}, and with ∂H^{2} = R ∪ {∞}.

In this section we construct a metric 2-complex X_{n} on which BS(1, n) acts properly discontinuously and cocompactly by isometries. This 2-complex is a well-known object in combinatorial group theory (see e.g. [ECH^{+}]). We shall put an equivariant metric on this complex, which is designed so that X_{n} contains many isometric copies of H^{2}.

To describe X_{n} topologically, consider the “horobrick” H_{n} ⊂ H^{2} defined by 0 ≤ x ≤ n, 1 ≤ y ≤ n. The left and right sides of H_{n} are both geodesic segments of hyperbolic length log(n); orient these segments upward and label each with a. The top of H_{n} is a horocyclic segment of hyperbolic length 1; orient this segment to the right and label it with b. The bottom of H_{n} is a horocyclic segment of hyperbolic length n; divide the bottom into n equal subsegments of length 1, orient each subsegment to the right, and label with b. Now form a piecewise hyperbolic complex C_{n} by identifying the a segments isometrically, and identifying the b segments isometrically. The subdivision of ∂H_{n} induces a cell structure on C_{n}, from which one easily reads off the presentation π_{1}(C_{n}) = BS(1, n) = ⟨a, b | aba^{−1} = b^{n}⟩. Let X_{n} be the universal cover of C_{n}, and lift the metric on C_{n} to give a piecewise hyperbolic metric on X_{n}. The group BS(1, n) acts on X_{n} as the deck transformation group of the universal covering map X_{n} → C_{n}.

To describe further features of X_{n}, let s be the “horostrip” in H^{2} given by 1 ≤ y ≤ n. The parabolic isometry τ: (x, y) ↦ (x + n, y) acts on s, and the quotient s/τ is a metric cylinder having one boundary circle of length 1 and the other boundary circle of length n. We alternatively construct C_{n} by gluing the long boundary circle of s/τ to the short boundary circle, via a degree n locally isometric convering map, induced by the path isometry taking the horocycle y = 1 to the horocycle y = n via the formula (x, 1) ↦ (nx, n).

Define a horocycle in C_{n} to be the image, under the map s → s/τ → C_{n}, of a horocycle {y = t} in s. Define a horocycle in X_{n} to be a connected lift of a horocycle in C_{n}. Collapsing each horocycle of X_{n} to a point defines a fibration π: X_{n} → T_{n}, where T_{n} is an infinite, regular, (n + 1)-valent tree (see Figure 1). The fibration of X_{n} by horocycles is preserved by the action of BS(1, n) on X_{n}, and hence there is an induced action of BS(1, n) on the tree T_{n}. The tree T_{n} is the usual Bass-Serre tree (see [Se]) associated to the HNN extension BS(1, n) = Z∗_{ϕ} where ϕ(k) = nk.

The horocycles on the horostrip s have a transverse orientation pointing towards ∞. This induces a transverse orientation on each horocycle of X_{n}, which in turn induces an orientation on each edge of T_{n}. Each vertex of T_{n} has one incoming and n outgoing edges (see Figure 1).

A proper line in T_{n} is the image of a proper embedding R → T_{n}. There are two types of proper lines L:

• L is coherently oriented, i.e. the orientation on L induced by T_{n} agrees with a global orientation on L.

• L is not coherently oriented. Since each vertex has exactly one incoming edge, there is a unique point p ∈ L with the property that each component of L − p is coherently oriented, and these two orientations point away from p.

The inverse image P = π^{−1}(L) of a proper line L is a proper plane in X_{n}. There are two types of proper planes, depending on the orientation type of L:

• If L is coherently oriented then P is an isometrically embedded hyperbolic plane in X_{n}.

• If L is not coherently oriented then P is an isometrically embedded doubled horoball in X_{n}, that is, the union S_{1}∪S_{2} of horoballs with common horocircle boundary ∂S_{1} = ∂S_{2} called the seam. The horocircles are identified via an isometry in the path metric.

Given a hyperbolic plane P ⊂ X_{n}, when we speak of a horocycle in P we shall always mean one of the form π^{−1}(x) for some point x ∈ T_{n}. These form a concentric family of horocycles, based at a point in ∂P which we denote ∞. The line at infinity of P will mean ∂P − ∞.

Note that a doubled horoball has concentrated positive curvature along its seam. In particular X_{n} is not a nonpositively curved space. Indeed it is easy to see that BS(1, n) has no properly discontinuous cocompact isometric action on any CAT(0) space, because BS(1, n) has exponential isoperimetric function.

4 Quasi-hyperbolic planes in X_{n}

Our first “rigidity” result is a study of subsets of X_{n} which are quasiisometric to hyperbolic planes. Let d_{H} denote Hausdorff distance between two subsets of a metric space, i.e. d_{H}(C, C′) is the infimum of numbers δ ∈ [0, ∞] such that C ⊂ N_{δ}(C′) and C′ ⊂ N_{δ}(C).

Proposition 4.1 (Quasi-hyperbolic planes). For all K ≥ 1, C ≥ 0 there exists A ≥ 0, such that if ϕ : H^{2} → X_{n} is a (K, C)-quasi-isometric embedding, then there exists a hyperbolic plane Q ⊂ X_{n} such that d_{H}(ϕ(H^{2}), Q) ≤ A.

Proposition 4.1 may be thought of as an analogy to X_{n} of the fact that quasi-geodesics in a hyperbolic space lie close to geodesics, although Proposition 4.1 seems much harder to prove.

Applying the “connect-the-dots” argument discussed above, for the rest of the proof we may replace ϕ by a lipschitz quasi-isometric embedding which is close to ϕ in the sup norm.

Coarse topology

In proving Proposition 4.1 we will use two coarse topological theorems. The first is due to R. Schwartz, and both were first applied to quasi-isometric rigidity problems in [FS].

Recall that a metric space (M, d) is uniformly contractible if there is a function α : R^{+} → R^{+} having the following property: If a continuous map of a finite simplicial complex Δ → M is contained in an r-ball, then it is contractible in an α(r)-ball. The function α is not supposed to depend on the dimension of Δ. Clearly any contractible space admitting a cocompact group of isometries is uniformly contractible.

Proposition 4.2 (Coarse Separation Theorem). Suppose ϕ: H^{2} → Y is a lipschitz K-quasi-isometric embedding of H^{2} into a uniformly contractible Riemannian manifold Y diffeomorphic to R^{3}. Then Y − ϕ(H^{2}) has at least two components.

Remark. This is an immediate consequence of the general Coarse Separation Theorem given as Theorem 5.2 of [FS], or more precisely from the lipschitz version of that theorem given in Section 5 of [FS]. To apply Theorem 5.2 of [FS], one uses the fact that H^{2} is uniformly contractible, and has an “expanding sphere”, namely the r-spheres for r > 0.

The following is a metrical version of invariance of domain:

Proposition 4.3 (Packing Theorem). Suppose that J is diffeomorphic to R^{n} and is uniformly contractible, and ϕ: J → J is a K-quasi-isometric embedding. Then nbhd_{K′}(ϕ(J)) = J for some K′ depending only on the pair (K, J).

A short proof of the packing theorem is given as Lemma 8.2 in [EF]. A different proof (assuming more hypotheses) is given in Section 5 of [FS].

Proof of Proposition 4.1. Let X = X_{n}, and let π : X → T be the natural projection to the tree T = T_{n}. The proof of Proposition 4.1 is divided into several steps.

Step 1. There exists a proper line L ⊂ T such that the plane Q = π^{−1}(L) is contained in ϕ(H^{2}).

We will adjust the given setup to a situation in which Coarse Separation (Proposition 4.2) can be applied.

To start with, let α: T → R^{2} be any proper embedding. For example, if S is a cusped hyperbolic surface of sufficiently large topological type, then S has a spine Σ which is a rose of valence at least as large as the valence of all vertices in T, and then there is a proper embedding of T into Σ̃ ⊂ S̃ ≈ H^{2} ≈ R^{2}.

Choose a homeomorphism β: X → T × R. The map (α × Id) ∘ β gives a proper topological embedding

X ⟶^{β} T × R ⟶^{α×Id} R^{2} × R = R^{3}

Regarding X now as a subset of R^{3}, for each component C of R^{3}−X the frontier ∂C is a topological plane in X. Metrically the plane ∂C is either a hyperbolic plane or a doubled horoball in X, and in either case ∂C is totally geodesic in X, in that the induced metric from X and path metric on ∂C are the same. In fact, the embedding α: T → R^{2} can be chosen so that each ∂C is a doubled horoball.

Choose a homeomorphism from \overline{C} = C ∪ ∂C to ∂C × [0, ∞). Using this product decomposition of \overline{C}, we may define the metric on \overline{C} to be the product of the metric on ∂C (as a totally geodesic subset of X) and the usual metric on [0, ∞). This defines a path metric on \overline{C} for each component C of R^{3} −X. These path metrics agree where they meet along X, so pasting them together gives a geodesic metric d on all of R^{3}. In this metric, the embedding X ↪ (R^{3}, d) is totally geodesic, meaning that for any geodesic in (R^{3}, d) whose endpoints lie in X, the entire geodesic lies in X.

The metric space (R^{3}, d) is uniformly contractible: any ball B ⊂ (R^{3}, d) with B ∩ X ≠ ∅ admits a deformation retraction B × [0, 1] → B such that B × 1 → B ∩ X; and furthermore X is uniformly contractible because it admits a cocompact isometric group action. If B ∩ X = ∅ then B admits such a deformation into a (uniformly contractible) copy ∂C×{x}, x ∈ (0, ∞) of ∂C.

Consider the lipschitz quasi-isometric embedding ϕ: H^{2} → X. Composing this with the isometric embedding X → (R^{3}, d), we may regard ϕ as a lipschitz quasi-isometric embedding H^{2} ⟶^{ϕ} (R^{3}, d). Applying Coarse Separation (Proposition 4.2), we know R^{3} − ϕ(H^{2}) has at least two components. Since ϕ(H^{2}) ⊂ X, it follows that there are at least two components C, C′ of R^{3} − X which are separated by ϕ(H^{2}). There exists a finite sequence of components C = C_{0}, …, C_{J} = C′ of R^{3} − X such that C_{j−1}, C_{j} are adjacent, meaning that \overline{C}_{j−1} ∩ \overline{C}_{j} = π^{−1}(E_{j}) for some closed edge E_{j} of T. Without loss of generality, we may therefore choose C, C′ to be adjacent, with \overline{C} ∩ \overline{C}′ = π^{−1}(E_{0}).

Let U be the union of all edges E ⊂ T such that π^{−1}(E) ⊂ ϕ(H^{2}). Note that E_{0} ⊂ U, for otherwise there would be a path from C to C′ piercing X in a single point lying in π^{−1}(E_{0}) − ϕ(H^{2}), contradicting that C, C′ are separated by ϕ(H^{2}).

Let E_{n} be the union of all edge paths in T that start with E_{0}, are contained in U, and consist of at most n + 1 edges. We assume by induction that E_{n} contains an edge path consisting of 2n + 1 edges centered on E_{0}. Let γ be the boundary of a regular neighborhood of E_{n} in R^{2}, and write γ = γ_{1} ∗ γ_{2}, where γ_{1}, γ_{2} meet at a transversal to the midpoint of E_{0} (see Figure 2).

The next step of the induction says that E_{n+1} contains an edge path consisting of 2n + 3 edges centered on E_{0}. If this were not true, then one Figure 2: An example of E_{n} with n = 2. If E_{n+1} does not contain an edge path of length 2n + 3 centered on E_{0}, then one of γ_{1} or γ_{2} misses U, because they both miss any edge E′ which intersects E_{n} at a vertex lying at distance less than n from E_{0}, and at least one of them misses any edge which intersects E_{n} at a vertex lying at distance exactly n from E_{0}.

of the two paths γ_{1}, γ_{2} would be disjoint from U (see Figure 2). Suppose γ_{1} is disjoint from U. For each edge E′ that γ_{1} pierces, there exists a point v ∈ E′ such that π^{−1}(v) ⊄ ϕ(H^{2}), and we may assume that γ_{1} pierces E′ at v. We may then lift γ_{1} to a path in R_{3} that misses ϕ(H^{2}) and connects C to C′, a contradiction.

By induction there is a bi-infinite path L ⊂ ⋃_{i=0}^{∞} E_{i} ⊂ U, and so π^{−1}(L) ⊂ ϕ(H^{2}), finishing the proof of step 1.

Step 2. ϕ(H^{2}) is Hausdorff close to the plane Q = π^{−1}(L).

The (K, C) quasi-isometric embedding ϕ has a coarse inverse ψ: ϕ(H^{2}) → H^{2}. The map ψ is a quasi-isometry with constants depending only on K, C. Applying the Packing Theorem (Proposition 4.3) to the restriction of ψ to Q gives a constant K′ = K′(K, C) so that nbhd_{K′}(ψ(Q)) ⊃ H^{2}. It follows that ϕ(H^{2}) ⊂ nbhd_{KK′+C′}(Q). Step 1 showed that Q ⊂ ϕ(H^{2}), and hence the Hausdorff distance between Q and ϕ(H^{2}) is at most KK′ + C.

Step 3. Q is a hyperbolic plane.

Since Q = π^{−1}(L) for some line L in T, it follows that the plane Q is metrically either a hyperbolic plane or a doubled horoball. By step 2, Q is quasi-isometric to H^{2}. But it is well-known that a doubled horoball is not quasi-isometric to H^{2} (see e.g. [ECH^{+}] Figure 7.8). For example, the hyperbolic plane satisfies the “fellow traveller property”, a quasi-isometry invariant that says: for each K ≥ 0 there exists D ≥ 0 such that two K-quasigeodesics with the same endpoints are D-Hausdorff close. But a doubled horoball B_{1} ∪ B_{2} does not satisfy the fellow traveller property: by choosing two points x, y ∈ B_{1} ∩ B_{2} sufficiently far apart, the two geodesics connecting x, y in B_{1} and in B_{2} can be made to have arbitrarily large Hausdorff distance in B_{1} ∪ B_{2}.

This completes the proof of Proposition 4.1. ⋄

5 Horocycles in X_{n}

The results of Proposition 4.1 motivate the following definitions. We define the upper boundary of X_{n}, denoted ∂^{u}X_{n}, to be the set of hyperbolic planes in X_{n}. Note that two distinct hyperbolic planes in X_{n} are an infinite Hausdorff distance from each other. Combining this fact with Proposition 4.1, it follows that any quasi-isometry f: X_{m} → X_{n} induces a map f^{u}: ∂^{u}X_{m} → ∂^{u}X_{n}, characterized as follows: for each Q ∈ ∂^{u}X_{m} we have d_{H}(f(Q), f^{u}(Q)) < ∞, where the bound depends only on the quasi-isometry constants of f.

In this section we use the map f^{u} to investigate how horocycles behave under a quasi-isometry f: X_{m} → X_{n}. This information is used in turn to further pin down the structure of f.

Quasi-isometries preserve horocycles.

Given a quasi-isometry f: X_{m} → X_{n}, define an induced map θ_{f} from the set of horocycles of X_{m} to the set of horocycles of X_{n}, as follows. Consider first a branching horocycle in X_{m}, one of the form σ = π^{−1}(vertex) for some vertex in T_{n}. There exist hyperbolic planes Q_{1}, Q_{2} ⊂ X_{m} such that σ = ∂(Q_{1}∩Q_{2}). Let Q′_{i} = f^{u}(Q_{i}) for i = 1, 2. Let σ′ = ∂(Q′_{1} ∩ Q′_{2}), a branching horocycle in X_{n}. Then we define θ_{f}(σ) = σ′. For any nonbranching horocycle σ ⊂ X_{n}, define θ_{f}(σ) to be the same as θ_{f}(σ_{1}) where σ_{1} is the closest branching horocycle above σ. Note that if \overline{f}: X_{m} → X_{n} is a coarse inverse for f then (θ_{f})^{−1} = θ\overline{f} when restricted to the set of branching horocycles.

Although θ_{f} is only a set map, the next lemma explains the geometric significance of θ_{f}.

Lemma 5.1 (Horocycles are coarsely preserved). Given K ≥ 1, C ≥ 0, there exists a constant λ ≥ 0 so that if f: X_{m} → X_{n} is a (K, C) quasiisometry, then for each horocycle h ⊂ X_{m} we have

d_{H}(f(h), θ_{f}(h)) ≤ λ

Proof. Clearly we may assume that h is a branching horocycle. The idea of the proof is that h is the boundary of the closeness set of two hyperbolic planes, but closeness sets and their boundaries are coarsely preserved by quasi-isometries.

Let Q_{1}, Q_{2} ⊂ X_{m} be the hyperbolic planes such that h = ∂(Q_{1} ∩ Q_{2}). By Proposition 4.1, we have hyperbolic planes Q′_{i} = f^{u}(Q_{i}), i = 1, 2 in X_{n}, satisfying d_{H}(f(Q_{i}), Q′_{i}) ≤ A for some A = A(K, C). We also have a branching horocycle h′ = θ_{f}(h) = ∂(Q′_{1} ∩ Q′_{2}) in X_{n}.

Choose x ∈ h. We must show that x′ = f(x) is close to h′. The set X_{n} − h′ has n + 1 components. The component intersecting Q′_{1} ∩ Q′_{2} is said to be below h′, and the remaining n components are above h′.

Case 1: x′ is above h′

It follows that h′ separates x′ from Q′_{1} ∩ Q′_{2}, and hence d(x′, h′) ≤ d(x′, Q′_{1} ∩ Q′_{2}) ≤ A.

Case 2: x′ is below h′

Choose a point y ∈ Q_{1} − Q_{2} such that d(y, x) = d(y, Q_{2}) = D = 2K(A + C). The point y′ = f(y) satisfies:

d(y′, Q′_{1}) ≤ A

and

d(y′, Q′_{2}) ≥ \frac{D}{K} − C

If y′ is below h′, then the point on Q′_{1} closest to y′ is also in Q′_{2}, and so d(y′, Q′_{2}) = d(y′, Q′_{1}) ≤ A = D/2K − C < D/K − C, a contradiction. It follows that y′ is above h′, and so h′ separates x′ and y′. Therefore d(x′, h′) ≤ d(x′, y′) ≤ Kd(x, y) + C = 2K^{2}(A + C) + C.

This shows that f(h) lies in a bounded neighborhood of h′. Applying a similar argument to the coarse inverse of f, we conclude that h′ lies in a bounded neighborhood of f(h). The bounds clearly depend only on K, C and on A, which in turn depends only on K, C by Proposition 4.1. ⋄

Consider a (K, C) quasi-isometry f: X_{m} → X_{n}, a hyperbolic plane Q ⊂ X_{m}, and the hyperbolic plane Q′ = f^{u}(Q) ⊂ X_{n}. Let f_{Q}: Q → Q′ be the composition of f with the closest point projection to Q′. Since d_{H}(f(Q), Q′) ≤ A it follows that the closest point projection stretches distances by at most a factor depending on A. Applying the preceding lemma we have:

Corollary 5.2. Given K, C there exist constants K′, C′, A such that for each (K, C)-quasi-isometry f: X_{m} → X_{n} and each Q ∈ ∂^{u}(X_{m}), Q′ = f^{u}(Q) ∈ ∂^{u}(X_{n}), the map f_{Q}: Q → Q′ is a (K′, C′)-quasi-isometry of hyperbolic planes, and for each horocycle σ ⊂ Q there is a horocycle σ′ ⊂ Q′ such that d_{H}(f_{Q}(σ), σ′) ≤ A.

Horocycle preserving quasi-isometries of the upper half-plane

Consider a quasi-isometry f: H^{2} → H^{2} which fixes ∞ in the upper halfspace model. The induced boundary map f_{ℓ}: R → R is quasi-symmetric, but typically this boundary map will not be bilipschitz. The following proposition explains under what circumstances f_{ℓ} is bilipschitz. In the next section we will see that this kind of improvement in the quality of f_{ℓ} imposes strong restrictions on quasi-isometries X_{m} → X_{n}.

Recall that for H^{2} we use the upper half plane model with metric (dx^{2} + dy^{2})/y^{2}.

Proposition 5.3 ((bilipschitz)×(translation)). For each K ≥ 1, C ≥ 0, A ≥ 0 there exists D ≥ 0 satisfying the following. Given a (K, C) quasiisometry f: H^{2} → H^{2} which preserves the point ∞, suppose that for each horocycle σ based at ∞, there is a horocycle σ′ = θ_{f}(σ) based at ∞ such that d_{H}(f(σ), σ′) ≤ A. Then the induced boundary map f_{ℓ}: R → R is bilipschitz. Moreover there exists a real number T such d(f(x, y), (f_{ℓ}(x), ye^{T})) ≤ D for all (x, y) ∈ H^{2}.

In other words, the map f is a bounded distance in the sup norm from a map of the form (bilipschitz)×(translation), where the bilipschitz map acts on the line at infinity and by translation we mean a translation of the “hyperbolic height” parameter log(y).

Remark. The sup norm bound D depends only on K, C, A, but the bilipschitz constant for f_{ℓ} and the translation constant T can depend on f in addition to K, C, A. For example, take f to be any homothety (x, y) → (sx, sy), s > 0. However, the proof of Proposition 5.3 will exhibit the following relation: if [a, b] ⊂ R is the “bilipschitz norm” of f_{ℓ}, i.e. the smallest interval such that a|x − y| ≤ |f_{ℓ}(x) − f_{ℓ}(y)| ≤ b|x − y| for all x, y ∈ R, then the interval of ratios [a/e^{T} , b/e^{T}] has bounded size depending only on K, C, A. In other words, f is a “quasi-homothety”.

Proof of Proposition 5.3. By changing the values of K, C we may assume that K, C are Lipschitz quasi-isometry constants for both f and a coarse inverse of f. Also, by changing the value of A we may assume that there is a bijective “Euclidean height transformation” s ↔ s′ such that for each horocycle σ = {y = s}, setting σ′ = θ_{f}(σ) = {y = s′}, we have d_{H}(f(σ), σ′) ≤ A and similarly for a coarse inverse of f. The new values of K, C, A needed to acheive these effects depend only on the old values.

We shall also need the fact that lim_{s→0} s′ = 0.

Consider a rectangle of the form R = [p, q]×[0, s]. Define the eccentricity of R to be ecc(R) = width(R)/height(R) = |p − q|/s. The next lemma describes how eccentricity is distorted by f. The quasi-image of R under f is defined to be the rectangle R′ = [p′, q′]×[0, s′] where p′ = f_{ℓ}(p), q′ = f_{ℓ}(q).

Lemma 5.4 (Eccentricity is quasi-preserved). There exist constants D ≥ 0 depending only on K, C, A and E ≥ 1 depending on K, A, both independent of f, such that for any rectangle R = [p, q] × [0, s] with quasi-image R′ = [p′, q′] × [0, s], if ecc(R) ≥ D then

ecc(R′) ≤ E · ecc(R)

and if ecc(R′) ≥ D then

\frac{1}{E} ecc(R) ≤ ecc(R′)

Proof. Since f preserves ∞, each vertical geodesic x = a maps to a neighborhood of another vertical geodesic x = a′, where the size of the neighborhood depends only on K and C. It follows that there is a constant B = B(K, C) such that each point on the image of x = a is connected to a point of x = a′ by a horocyclic segment of length at most B.

Let l = |p − q|, let γ = Top(R) = [p, q] × s, and note that Length(γ) = l/s = ecc(R). Similarly, Length(γ′) = l′/s′ = ecc(R′), where l′ = |p′ − q′| and γ′ = Top(R′) = [p′, q′] × s′. We need to compare the lengths of γ and γ′ (see Figure 3).

Consider the path f(γ), which lies below the horocycle y = s′e^{A}, and whose endpoints are connected to the vertical geodesics x = p′ and x = q′ by horocyclic segments of length at most B. Concatenating f(γ) with these two horocyclic segments we obtain a path α. From Lemma 2.1 we have

\frac{1}{K} · Length(γ) − C ≤ Length(f(γ)) ≤ K · Length(γ) + C

and so

\frac{1}{K} · ecc(R) − C ≤ Length(f(γ)) ≤ K · ecc(R) + C.

Figure 3: Connect f(γ) to the vertical geodesics x = p′, x = q′ by short horocyclic segments, then project upward to the horocycle y = s′e^{A}.

We also have Length(α) ≤ Length(f(γ)) + 2B

and by choosing ecc(R) ≥ K(2B + C) we have

Length(α) ≤ 2 · Length(f(γ))

and therefore

\frac{1}{2} Length(α) ≤ K · ecc(R) + C.

By further choosing ecc(R) ≥ C we obtain

Length(α) ≤ 2(K + 1) ecc(R).

By projecting α straight upward to the horocycle y = s′e^{A} we obtain

Length(α) ≥ \frac{l′}{s′e^{A}} = ecc(R′)e^{−A}

and thus

ecc(R′) ≤ 2(K + 1)e^{A} ecc(R).

The same argument using the coarse inverse of f shows that if ecc(R′) has the same lower bounds as ecc(R) then

ecc(R) ≤ 2(K + 1)e^{A} ecc(R′)

proving the lemma.

⋄

As a first application of the lemma we show that f_{ℓ} is lipschitz in the large. Given a horocycle y = s_{0}, choose l_{0} sufficiently large so that l_{0}/s_{0} ≥ D. For any p_{0} < q_{0} ∈ R such that |p_{0} − q_{0}| ≥ l_{0}, the first inequality of Lemma 5.4 applies to the rectangle R = [p_{0}, q_{0}] × [0, s_{0}]. Setting s_{1} = s′_{0}, p_{1} = f_{ℓ}(p_{0}), q_{1} = f_{ℓ}(q_{0}), and l_{1} = |p_{1} − q_{1}| we therefore have

\frac{l_{1}}{l_{0}} ≤ E \frac{s_{1}}{s_{0}}

Fixing the horocycle y = s_{0} and its coarse image y = s_{1}, we have shown that f_{ℓ} is lipschitz in the large: if |p−q| ≥ l_{0} then |f_{ℓ}(p)−f_{ℓ}(q)| ≤ (Es_{1}/s_{0})|p−q|.

To show that f_{ℓ} is lipschitz, consider ξ < η ∈ R with m = |ξ − η| < l_{0}. Choose the points p_{0}, q_{0} so that p_{0} < ξ, η < q_{0} and l_{0} = q_{0} − p_{0}. Choose a horocycle y = r so close to the line at infinity that the rectangle R = [ξ, η] × [0, r] has eccentricity m/r ≥ D, and hence the first inequality of Lemma 5.4 applies. Setting m′ = |f_{ℓ}(ξ) − f_{ℓ}(η)| we therefore have

\frac{m′}{m} ≤ E \frac{r′}{r}

Now we find an upper bound for r′/r. We know that r′ → 0 as r → 0, and so we may choose r so close to zero that l_{1}/r′ ≥ D. The second inequality of Lemma 5.4 therefore applies with p = p_{0}, q = q_{0}, and s = r, and we have

\frac{1}{E} \frac{r′}{r} ≤ \frac{l_{1}}{l_{0}}

and so

\frac{m′}{m} ≤ E^{2} \frac{l_{1}}{l_{0}} ≤ E^{3} \frac{s_{1}}{s_{0}}

proving that f_{ℓ} is lipschitz.

A similar argument using the coarse inverse of f gives

\frac{m}{m′} ≤ E^{3} \frac{s_{0}}{s_{1}}

and so

\frac{1}{E^{3}} s_{1}/s_{0} ≤ \frac{m′}{m} ≤ E^{3}s_{1}/s_{0}

Fixing, say, s_{0} = 1 and letting T = log(s_{1}), we have

\frac{1}{E^{3}} e^{T} ≤ \frac{m′}{m} ≤ E^{3}e^{T}

proving that f_{ℓ} is bilipschitz.

The above inequalities also show that for any horocycle y = s_{0}, setting s_{1} = s′_{0} we have

\frac{1}{E^{6}} e^{T} ≤ \frac{s_{1}}{s_{0}} ≤ E^{6}e^{T}

and so

log(s_{0}) + T − log(E^{6}) ≤ log(s_{1}) ≤ log(s_{0}) + T + log(E^{6}).

For any point P = (x_{0}, y_{0}) in the upper half plane, we know that the point f(P) lies in the A-neighborhood of the horocycle y = y′_{0}. The point f(P) also lies in a certain bounded neighborhood of the geodesic x = f_{ℓ}(x_{0}) where the bound depends only on K, C. It follows that d(f(P), (f_{ℓ}(x_{0}), y′_{0})) is bounded above by a constant depending only on K, C, A. From the above inequality with s_{0} = y_{0}, s_{1} = y′_{0} it also follows that

d((f_{ℓ}(x_{0}), y′_{0}), (f_{ℓ}(x_{0}), y_{0}e^{T})) ≤ | log(E^{6})|

completing the proof of the proposition. ⋄

Remark. It is easy to see that for any bilipschitz homeomorphism ϕ: R → R and any T ∈ R, the map (x, y) → (ϕ(x), ye^{T}) is a quasi-isometry of H^{2}. Proposition 5.3 therefore characterizes quasi-isometries that take horocycles based at ∞ to bounded neighborhoods of horocycles based at ∞.

6 The upper and lower boundaries of X_{n}

We have defined the upper boundary ∂^{u}X_{n} and the map on upper boundaries induced by a quasi-isometry. We now define a metric on ∂^{u}X_{n}, and we study how the induced map acts on that metric.

Height function

Choose once and for all a height function on T_{n}, that is a continuous function h: T_{n} → R which maps each oriented edge E ⊂ T_{n} homeomorphically to an oriented interval in R of length log(n). We obtain by composition a height function X_{n} ⟶^{π} T_{n} ⟶^{h} R also denoted h. The choices in the definition of h may be fixed by requiring that the base point of X_{n} map to the origin of R, and by requiring that for each edge E ⊂ T_{n}, the map h | π^{−1}(E) is an isometry on each vertical geodesic segment. We assume that the base point of X_{n} lies on a branching horocycle, and hence for each branching horocycle σ ⊂ X_{n} we have h(σ) = k log(n) for some integer k called the combinatorial height of σ.

Metric on ∂^{u}X_{n}

Given hyperbolic planes Q_{1}, Q_{2} ∈ ∂^{u}X_{n}, define the distance between them to be d(Q_{1}, Q_{2}) = e^{−h(σ)} = n^{−k}, where the horocycle σ = ∂(Q_{1} ∩ Q_{2}) has combinatorial height k. We leave the reader to check that this defines a locally compact metric on ∂^{u}X_{n}. In fact the metric d is an ultrametric, also called a nonarchimedean metric, in other words a metric satisfying d(x, z) ≤ max{d(x, y), d(y, z)} for any x, y, z ∈ ∂^{u}X_{n}.

Note that ∂^{u}X_{n} is isometric to the n-adic rational numbers Q_{n} with the usual n-adic metric. This is the space of all bi-infinite sequences of the form (a_{i}) where a_{i} ∈ {0, …, n − 1} for i ∈ Z, and a_{i} = 0 for i sufficiently close to −∞. In number theory one thinks of (a_{i}) as the infinite series ∑_{i} a_{i}n^{i} (although this is usually only studied when n is a prime). The distance between two distinct n-adic rational numbers (a_{i}), (b_{i}) ∈ Q_{n} is equal to n^{−I}, where I is the largest integer such that a_{i} = b_{i} for all i ≤ I. The isometry between ∂^{u}X_{n} and Q_{n} can be exhibited by choosing, for each vertex v ∈ T_{n}, a bijective labelling of the set of outgoing edges at v by elements of the set {0, …, n − 1}, thereby assigning a label sequence (a_{i}) to each coherently oriented line in T_{n}. The labelling must be chosen so that some coherent line has a label sequence of all zeroes.

Lower boundary

We define the lower boundary of X_{n}, denoted ∂_{ℓ}X_{n}, to be the common line at infinity of any two hyperbolic planes in X_{n}. That is, given hyperbolic planes Q_{1}, Q_{2} ⊂ X_{n}, the inclusions Q_{1}∩Q_{2} ⊂ Q_{i}, i = 1, 2 induce a bijection ∂_{ℓ}Q_{1} ≈ ∂_{ℓ}Q_{2}. Moreover, for any three hyperbolic planes we obtain a commutative triangle

∂_{ℓ}Q_{1} ≈ ∂_{ℓ}Q_{2} ≈ ∂_{ℓ}Q_{3} ≈ ∂_{ℓ}Q_{1}.

We may therefore define ∂_{ℓ}X_{n} by pairwise identifying ∂_{ℓ}Q for all hyperbolic planes Q ⊂ X_{n}. The choice of a height function on X_{n} determines a specific isometry between Q and the upper half plane, taking the height zero horocycle on Q to the horocycle y = 1. This identification determines a specific Euclidean metric on ∂_{ℓ}Q, and these metrics all agree, thereby defining a Euclidean metric on ∂_{ℓ}X_{n}. A quasi-isometry f: X_{m} → X_{n} induces a well-defined map f_{ℓ}: ∂_{ℓ}X_{m} → ∂_{ℓ}X_{n}, as the reader may easily check.

Theorem 6.1 (Boundary maps are bilipschitz). If f: X_{m} → X_{n} is a quasi-isometry, then the induced maps f_{ℓ}: ∂_{ℓ}X_{m} → ∂_{ℓ}X_{n} and f^{u}: ∂^{u}X_{m} → ∂^{u}X_{n} are bilipschitz.

Proof. We have already proved this for f_{ℓ}. Let K, C be quasi-isometry constants for f. There exists a unique map ρ_{j}: X_{j} → H^{2} satisfying the following properties:

• ρ_{j} takes horocycles to horocycles.

• ρ_{j} is an isometry when restricted to each hyperbolic plane in X_{j}.

• ρ_{j} is normalized to take the base point of X_{j} to the point (x, y) = (0, 1).

For each hyperbolic plane Q ⊂ X_{m} define the map \overline{f}_{Q}: H^{2} → H^{2} to be the composition

H^{2} ⟶^{ρ_{m}^{−1}} Q ⟶^{f} X_{n} ⟶^{ρ_{n}} H^{2}.

Claim 6.2. The map \overline{f}_{Q} is at most a distance R_{1} = R_{1}(K, C) in the sup norm from a map of the form (x, y) → (f_{ℓ}(x), ye^{T_{Q}}), where the translation constant T_{Q} depends ostensibly on Q.

To prove the claim, define f̂_{Q} to be the composition

H^{2} ⟶^{ρ_{m}^{−1}} Q ⟶^{f} X_{n} → Q′ = f^{u}(Q) ⟶^{ρ_{n}} H^{2}

where the map X_{n} → Q′ is the closest point projection. By Proposition 4.1 the maps \overline{f}_{Q} and f̂_{Q} are at bounded distance in the sup norm. By Lemma 5.1 the map f̂_{Q} coarsely preserves horocycles. By Proposition 5.3 the map f̂_{Q} is a bounded distance in the sup norm from a map of the form (x, y) → (f_{ℓ}(x), ye^{T_{Q}}). All the distances in this argument depend only on K, C, proving the claim.

We can remove the dependence of the translation constant on Q, using the fact that any two hyperbolic planes intersect below some horocycle, and on that region of intersection the translation constants must nearly agree. More formally:

Claim 6.3. There exists T ∈ R (depending on f) and R_{2} ≥ 0 (depending only on K, C) such that for each hyperbolic plane Q ⊂ X_{m}, the map \overline{f}_{Q} is at most a distance R_{2} in the sup norm from the map

(x, y) → (f_{ℓ}(x), ye^{T})

To prove this claim, set \overline{f}_{i} = \overline{f}_{Q_{i}} and T_{i} = T_{Q_{i}}, and it suffices to bound |T_{1} − T_{2}| independent of the hyperbolic planes Q_{1}, Q_{2}. Choose a point P ∈ Q_{1} ∩ Q_{2}, and let (x, y) = ρ_{m}(P). By the previous claim, for i = 1, 2 we have

d(\overline{f}_{i}(x, y), (f_{ℓ}(x), ye^{T_{i}})) ≤ R_{1}

However, \overline{f}_{1}(x, y) = \overline{f}_{2}(x, y) and so

|T_{1} − T_{2}| = d((f_{ℓ}(x), ye^{T_{1}}), (f_{ℓ}(x), ye^{T_{2}})) ≤ 2R_{1}

proving the claim.

To prove Theorem 6.1, consider hyperbolic planes Q_{1}, Q_{2} ∈ ∂^{u}(X_{m}) and Q′_{i} = f^{u}(Q_{i}) ∈ ∂^{u}(X_{n}). Let σ = ∂(Q_{1} ∩ Q_{2}) and let σ′ = ∂(Q′_{1} ∩ Q′_{2}). We must show that

log(d(Q′_{1}, Q′_{2})) − log(d(Q_{1}, Q_{2})) = −h(σ′) + h(σ)

is bounded above and below by constants that depend only on f, not on Q_{1} and Q_{2}. For each p ∈ σ, it follows from Lemma 5.1 that |h(σ′) − h(f(p))| is bounded depending only on K, C. Also, the second claim above shows that |h(f(p)) − h(σ) − T | is bounded depending only on K, C. It follows that |h(σ′) − h(σ) − T | is bounded, finishing the proof of the theorem. ⋄

7 Proof of main theorem

Recall the statement:

Theorem 7.1 (Quasi-isometric iff commensurable). For integers m, n ≥ 2, the groups BS(1, m) and BS(1, n) are quasi-isometric if and only if they are commensurable. This happens if and only if there exist integers r, j, k > 0 such that m = r^{j} and n = r^{k}.

Proof. The group BS(1, m) = ⟨a, b | aba^{−1} = b^{m}⟩ has a finite index subgroup isomorphic to BS(1, m^{I}), generated by a^{I}, b. This subgroup is the kernel of the homomorphism BS(1, m) → Z/IZ taking a → 1, b → 0, and so the index equals I. The groups BS(1, r^{j}) and BS(1, r^{k}) are therefore commensurable, and hence quasi-isometric.

If BS(1, m) and BS(1, n) are quasi-isometric, then by the observation of Svarc-Milnor there is a quasi-isometry f: X_{m} → X_{n}. By Theorem 6.1, f induces a map f^{u}: Q_{m} ≈ ∂^{u}X_{m} → ∂^{u}X_{n} ≈ Q_{n} which is a bilipschitz homeomorphism.

The n-adic integers Z_{n} are defined to be the subset of Q_{n} consisting of all (a_{i}) ∈ Q_{n} where a_{i} = 0 for i < 0. Given a ∈ {0, …, n − 1} and I ∈ Z, the set

C(a, I) = {(b_{i}) ∈ Q_{n} | b_{i} = 0 if i < I, b_{I} = a}

is called a clone of Z_{n}. Noting that Z_{m} has diameter 1 in Q_{m}, it follows that the set f^{u}(Z_{m}) has bounded diameter in Q_{n}. But every bounded diameter subset of Q_{n} is contained in some clone C(a, I) of Z_{n}. The clone C(a, I) is isometric to Z_{n} with its metric scaled by n^{−I}. Thus, multiplying the metric on f^{u}(Z_{m}) by n^{I}, we obtain a bilipschitz embedding Z_{m} → Z_{n}.

The proof of Theorem 7.1 is completed by applying the following theorem, whose proof is found in the appendix as Corollary 10.11:

Theorem 7.2 (D. Cooper). For integers m, n ≥ 2, if there is a bilipschitz embedding Z_{m} → Z_{n} then there exist integers r, j, k > 0 such that m = r^{j}, n = r^{k}.

⋄

8 The quasi-isometry group

Given a metric space X, the quasi-isometry group QI(X) is defined as follows. Define an equivalence relation on quasi-isometries where f ≡ g if f and g have bounded distance in the sup norm. Let QI(X) be the set of equivalence classes of quasi-isometries from X to itself. Composition of quasiisometries gives a well-defined group structure on QI(X). A quasi-isometry between two metric spaces induces an isomorphism between quasi-isometry groups. In particular, QI(BS(1, n)) ≈ QI(X_{n}).

Let Bilip(Y) be the group of bilipschitz homeomorphisms of a metric space Y.

Theorem 8.1 (Quasi-isometry group). QI(BS(1, n)) is isomorphic to

Bilip(R) × Bilip(Q_{n}).

Before proving this theorem we need to state a basic principle, dubbed the “rubber band principle” by Rich Schwartz. The trivial but useful idea is that if one takes any size chain of rubber bands connecting two distant points, and if each rubber band is stretched by a factor of at most K, then the whole chain is stretched by at most a factor of K.

Lemma 8.2 (Rubber Band Principle). For all L, M > 0, there exists C > 0 so that the following holds: Suppose X and Y are path metric spaces and f : X → Y is a map. Suppose that there are collections of isometrically embedded subspaces C_{X} of X and C_{Y} of Y satisfying:

• Any two points in X can be connected by an M-quasigeodesic which is a concatenation of a finite number of subpaths, each lying in an element of C_{X}; similarly for Y.

• f induces a one-to-one correspondence between elements of C_{X} and C_{Y}.

• f restricts to a L-quasi-isometry between corresponding elements of C_{X} and C_{Y}.

Then f : X → Y is a C-quasi-isometry.

We leave the proof of the Rubber Band Principle as an easy exercise.

Proof of Theorem 8.1. Choosing a base point of X_{n} induces maps

QI(X_{n}) ↦ Bilip(∂_{ℓ}X_{n}) ≈ Bilip(R)
f ↦ f_{ℓ}

and

QI(X_{n}) ↦ Bilip(∂^{u}X_{n}) ≈ Bilip(Q_{n})
f ↦ f^{u}

These two maps are obviously homomorphisms, and so we obtain a homomorphism QI(X_{n}) → Bilip(∂_{ℓ}X_{n}) × Bilip(∂^{u}X_{n}).

To show that the map is onto, consider α ∈ Bilip(∂_{ℓ}X_{n}), β ∈ Bilip(∂^{u}X_{n}). Choose a constant T ∈ R so that d(β(Q), β(Q′)) ≤ e^{T} d(Q, Q′) for all Q, Q′ ∈ ∂^{u}X_{n}. Given Q ∈ ∂^{u}X_{n}, define the map f_{Q}: Q → Q′ = β(Q) to be the composition

Q ⟶^{ρ_{n}} H^{2} ⟶^{α×τ_{T}} H^{2} ⟶^{ρ_{n}^{−1}} Q′

where (α×τ_{T})(x, y) = (α(x), τ_{T} (y)) = (α(x), ye^{−T}). Note that f_{Q} is a quasiisometry from Q to Q′, with constants that are independent of Q. Given Q_{1}, Q_{2} ∈ ∂^{u}(X_{n}), we shall show that the maps f_{i} = f_{Q_{i}} agree on the overlap Q_{1} ∩ Q_{2}. Let σ = ∂(Q_{1} ∩ Q_{2}) and let σ′ = ∂(Q′_{1} ∩ Q′_{2}). By the choice of T it follows that h(σ′) − h(σ) = − log(d(β(Q′_{1}), β(Q′_{2}))) + log(d(Q_{1}, Q_{2})) ≥ −T, and so h(σ′) ≥ h(σ)−T. It follows that if x ∈ Q_{1} ∩Q_{2} then h(x) ≤ h(σ) and hence h(f_{1}(x)) = h(f_{2}(x)) ≤ h(σ′), so f_{1}(x) and f_{2}(x) both lie in Q′_{1} ∩ Q′_{2}. But the points f_{1}(x), f_{2}(x) are both taken by ρ_{n} to the same point of H^{2}, and ρ_{n} is 1 − 1 on Q′_{1} ∩ Q′_{2}, and so it follows that f_{1}(x) = f_{2}(x).

We may now paste together the maps f_{Q}, as Q ranges over ∂^{u}(X_{n}), to get a map f: X_{n} → X_{n}. This map is a quasi-isometry on each hyperbolic plane Q, with constants independent of Q. By the Rubber Band Principle applied to the collection of isometrically embedded hyperbolic planes in X_{n}, we have that f is a quasi-isometry. It is clear that f_{ℓ} = α and f^{u} = β, proving that QI(BS(1, n)) maps onto Bilip(R) × Bilip(Q_{n}).

To show that the map is injective, given a quasi-isometry f: X_{n} → X_{n}, suppose that f_{ℓ} is the identity on ∂_{ℓ}X_{n} and f^{u} is the identity on ∂^{u}X_{n}. It follows that for each hyperbolic plane Q ⊂ X_{n}, f(Q) is a bounded Hausdorff distance from Q, and so we obtain a horocycle preserving quasi-isometry from H^{2} to itself defined as the composition

H^{2} ⟶^{ρ_{n}^{−1}} Q ⟶^{f} X_{n} → Q ⟶^{ρ_{n}} H^{2}

where X_{n} → Q is the closest point projection. The quasi-isometry constants of this map depend only on those of f. This quasi-isometry of H^{2} is therefore a bounded distance from a quasi-isometry of the form (x, y) → (f_{ℓ}(x), ye^{T}) = (x, ye^{T}) for some constant T = T (Q). As remarked earlier, if [a, b] is the smallest interval containing the bilipschitz constants for f_{ℓ}, then the interval [a/e^{T} , b/e^{T}] is bounded above and below by positive constants depending only on the quasi-isometry constants of f. In the present situation where a = b = 1, it follows that T = T (Q) is bounded independent of Q. The distance between (x, ye^{T}) and (x, y) is therefore bounded, and it follows that f is a bounded distance from the identity map in the sup norm. ⋄

Remark. The quasi-isometry group QI(G) has a natural topological group structure, and it can be proved that the isomorphism

QI(BS(1, n)) ≈ Bilip(R) × Bilip(Q_{n})

is an isomorphism of topological groups.

9 Is BS(1, n) a lattice?

Here is an easy argument to show that BS(1, n) is not a lattice in any 1connected solvable real Lie group S. If it were then it must be cocompact since all lattices in a solvable Lie group are cocompact. Since S is simply connected it is actually contractible (the matrix exponential map is a diffeomorphism), hence Γ\S would be a K(Γ, 1) space. But BS(1, n) has (virtual) cohomological dimension 2, hence S would have to be two-dimensional. But the only connected two-dimensional (real) solvable Lie group admitting a lattice is R^{2}, which gives a contradiction since BS(1, n), n ≥ 2 is not abelian.

We also know that BS(1, n) is not a subgroup of any group G acting properly discontinuously and cocompactly on any space X which is nonpositively curved in the CAT(0) sense. In particular BS(1, n) is not a subgroup of a cocompact lattice in any (real, p-adic, or product of real and p-adic) semisimple Lie group. The reason is that BS(1, n) contains an infinitely generated rank 1 abelian group Q, namely the kernel of the homomorphism BS(1, n) → Z taking a → 1, b → 0. The translation number of any g ∈ G is defined to be τ_{g} = inf_{x∈X} d(x, g(x)). The number τ_{g} is bounded away from zero as g ranges over the infinite order elements of G, and τ_{g^{n}} = nτ_{g}. If Q were a subgroup of G, then Q would have elements with nonzero translation numbers arbitrarily close to zero, a contradiction.

We should note that, at least for prime p, BS(1, p) in some sense exhibits a mixture of real and p-adic behaviour as follows: BS(1, p) is an infinite index subgroup of GL(2, Z[1/p]), which is a discrete subgroup of GL(2, R) × GL(2, Q_{p}) via the representation M ↦ (M, M). The group P GL(2, R) × P GL(2, Q_{p}) acts transitively by isometries on H^{2} × T, where T is the Bruhat-Tits tree for P GL(2, Q_{p}) (see [Se]). Furthermore, P GL(2, Z[1/p]) acts properly discontinuously and isometrically with cofinite volume (but not cocompactly) on H^{2} × T [Se]. By restriction we then obtain an isometric, properly discontinuous action of BS(1, p) on H^{2} × T. This action has infinite covolume (as it must by the above comments).

10 Appendix: Bilipschitz homeomorphisms of Self-Similar Cantor Sets by D. Cooper

Introduction

The study of properties of a metric space which are preserved by bilipschitz homeomorphism occurs in the study of groups via a word metric. It was also studied in [CP] for a certain type of Cantor set embedded in the real line. The Cantor sets concerned are mild generalizations of the original middle-third Cantor set. This Cantor set has the basic property that it is the union of two exact copies of itself each scaled down in size by a factor of 1/3. The generalization allows finitely many linear scale factors. It is easy to see that the Hausdorff dimension of this type of Cantor set depends only on these scale factors. Now a bilipschitz homeomorphism preserves Hausdorff dimension, and so a natural question is what, if any, further invariants other than Hausdorff dimension are there. An almost complete answer was given in [CP], using invariants derived from the Hausdorff measure. This work was generalized to analogous Cantor sets in Euclidean space of dimension n by H. Vuong in his thesis, [Vu1], [Vu2]. In this section, we generalize in a different direction to abstract metric Cantor sets which possess a certain linear self-similarity structure. There is a further generalization to a much wider class of Cantor set, where the self-similarity structure is smooth rather than linear. This will not be dealt with here.

The main result is Theorem (10.6) which states that every bilipschitz homeomorphism between two Cantor sets of the type we consider has constant Radon-Nikodym derivative on some clopen. The reason is, roughly speaking, that a bilipschitz map only permits a limited amount of metric distortion. The self-similar bumpiness of a Cantor set forces non-linearity to come in discrete amounts. Thus a finite number of discrete distortions exhausts the amount of metric distortion possible. Actually, this argument should be made at the level of Hausdorff measure rather than at the level of the metric.

The main ideas are already in [CP], except that there the topological properties of the real line are heavily used. This obstacle was overcome in [Vu1] with the introduction of separation defined below.

Cantor Sets

A map f : X ⟶ Y between metric spaces is called a similarity map if there is a constant K > 0 such that d_{Y} (fx_{1}, fx_{2}) = K d_{X}(x_{1}, x_{2}) for all x_{1}, x_{2} in X. Consider a Cantor set C equipped with a metric d_{C} then a clone structure on C is the following data. Let A_{1}, A_{2}, ⋯, A_{n} be a partition of C into clopens and for each A_{i} let τ_{i} be a similarity map of A_{i} onto C. We will call such a Cantor set a clone Cantor set.

Any composition of the maps τ_{i} is called a clone map, (we include the identity map as a clone map) thus the set of clone maps forms a semi-group. A clone of C is the pre-image of C under a clone map. Thus C and each of the A_{i} are clones. More generally any pre-image of a clone under a clone map is also a clone. For every clone there is a clone map sending the clone onto C. The clone structure is not unique, since one may replace any clone by finitely many clones which partition it. The level of a clone is defined inductively. The clone C is level 0, and a clone has level n if the smallest clone properly containing it has level n − 1.

Two clones are either disjoint or else one is a subset of the other. Thus one may construct a tree where the vertices are clones and there is an edge between a clone of level n and any clone of level n + 1 that it meets (hence contains.)

Lemma 10.1. Every clopen is a finite union of clones.

Given a subset A of a Cantor set C define the separation of A to be the minimum distance between A and its complement,

sep(A) = inf {d_{C}(x, y) : x ∈ A, y ∈ C − A }.

By convention, sep(C) = ∞. Recall that the diameter of a metric space is the supremum of the distance between points in the metric space. Now define the relative separation of A to be rel(A) = sep(A)/diam(A). We will make frequent use of the minimum separation between level-1 clones:

α(C) = min{ sep(A) : A is a level-1 clone of C }.

An ϵ-cover of a metric space, X, is a cover by sets each of which has diameter at most ϵ. Given a cover U = { U_{i} } of X and δ > 0 define

H_{δ}(U) = ∑_{i} [diam(U_{i})]^{δ}.

The Hausdorff-δ outer measure of X is the limit as ϵ → 0 of the inf H_{δ}(U_{ϵ}) where the infimum is taken over all ϵ-covers, U_{ϵ}, of X. We will assume that the Hausdorff measure μ_{C} of C in its Hausdorff dimension is finite and not zero.

A map f : (X, μ_{X}) ⟶ (Y, μ_{Y}) between measure spaces is called measure linear if there is a constant K > 0 such that for every measurable set A in X we have μ_{Y} (fA) = Kμ_{X}(A). In other words, the Radon-Nikodym derivative of f is K everywhere. We also describe this by saying that the map is linear at the level of measure theory. A similarity map between metric spaces is measure linear for Hausdorff measures, thus in particular clone maps are measure linear. Similarly the notion of bilipschitz at the level of measure theory may be defined. Let f be a bilipschitz homeomorphism between two such Cantor sets with bilipschitz constant K. Then f is K^{δ}-bilipschitz for δ-dimensional Hausdorff measure. It follows that the two Cantor sets have the same Hausdorff dimension.

We will use the term approximates to mean that the ratio of two numbers is bounded away from 0 and ∞ by numbers depending only on certain constants.

The following says that the separation of a clone approximates its diameter.

Lemma 10.2. Given a clone Cantor set C there is a constant ξ(C) > 1 such that for every clone A of C with A ≠ C we have

ξ(C)^{−1} ≤ rel(A) ≤ ξ(C).

Proof. Let x, y be points in A and C − A of minimal distance apart and let B be the smallest clone containing containing both of them. Let τ be the clone map taking B onto C, thus rel(τA) = rel(A). After replacing A by τA we may assume that B = C and τ is the identity.

Now x, y are in different level-1 clones by choice of B so d_{C}(x, y) ≥ α(C). Let D be the smallest clone properly containing A. Suppose that diam(D) < α. Then there every point in D − A is closer to A than x is to y, a contradiction. Thus diam(D) ≥ α and there are only finitely many clones this large, so that there is a bound on level(D). By choice of D we have level(A) = level(D) + 1 and so there are only finitely many possibilities for A. Hence ξ(C) is the maximum of a finite set. ⋄

The next result says that the smallest clone containing a given set has approximately the same diameter as that set.

Lemma 10.3. Given a clone Cantor set C there is a constant μ(C) with the following property. Let X be any subset of C of positive diameter and let A be the smallest clone of C containing X. Then

diam(X) ≤ diam(A) ≤ diam(X)μ(C).

Proof. Let τ be the clone map sending A onto C. Then τ(X) is not contained in any level-1 clone of C by definition of A. Thus τ(X) contains points in two different level-1 clones of C, and so diam(τX) ≥ α(C). Hence

\frac{diam(A)}{diam(X)} = \frac{diam(τA)}{diam(τX)} ≤ \frac{diam(C)}{α(C)}.

Taking μ(C) = diam(C)α(C)^{−1} gives the result. ⋄

The clone structure of a clone Cantor set provides a natural way to magnify phenomena, in particular a way to magnify subsets. Given a subset, A, of a clone Cantor set C there is some smallest clone, B, which contains A. Now B is a copy of C but linearly scaled down. The clone structure provides an identification, τ : B ⟶ C, of B with C. Then τA is a magnified copy of A in C, in particular the diameter of τA is at least α. This technique is used to enlarge phenomena which happen on a very small scale.

Lemma 10.4. Given K > 1, ϵ > 0 and clone Cantor sets C, C′ there is a constant M with the following property. Let f be any K-bilipschitz map of C onto a clopen in C′, and suppose that rel(fC) ≥ ϵ. Let A be any clone of C. If P′ is the smallest clone of C′ containing fA and τ′ is the clone map taking P′ onto C′ then there are at most M = M(C, C′, K, ϵ) possibilities for the image τ′(fA) as a subset of C′.

Proof. Since f is K-bilipschitz diam(fA) ≤ K diam(A). Choose x in fA and y in C′ − fA minimizing the distance between x and y. If y is in C′ − fC then sep(fA) ≥ sep(fC) and diam(fA) ≤ diam(fC) so rel(fA) ≥ ϵ. Otherwise if y is in fC then,

sep(fA) = d_{C′}(x, y) ≥ K^{−1}d_{C}(f^{−1}x, f^{−1}y) ≥ K^{−1}sep(A).

Using lemma (10.2) we get that sep(fA) ≥ K^{−1}diam(A)ξ(C)^{−1}. Hence in either case

rel(fA) ≥ min (\frac{K^{−1}diam(A)ξ(C)^{−1}}{K diam(A)} , ϵ) ≡ γ.

Since P′ is the smallest clone containing fA it follows that C′ is the smallest clone containing τ′fA thus diam(τ′(fA)) ≥ α(C′). Since τ′ is a similarity, rel(τ′(fA)) ≥ rel(fA) ≥ γ, thus

sep(τ′(fA)) = rel(τ′(fA))diam(τ′(fA)) ≥ γα(C′) ≡ β.

Consider a clone B of C′ contained in τ′(fA) and not a proper subset of any other clone contained in τ′(fA). We will call such a clone a maximal clone in τ′(fA). Let D be a clone containing B with level(D) = level(B)−1. There is a point y ∈ D which is not in τ′(fA) for otherwise B would not be maximal. Hence sep(τ′(fA)) ≤ diam(D). Combining this with the previous inequality gives diam(D) ≥ β. The argument of the last paragraph of lemma (10.2) again shows that level(D) is bounded and so there only a bounded number of possibilities for B. This bound depends only on β and C′.

Thus there is a bounded number of choices for the collection of maximal clones. By lemma (10.1) the clopen τ′(fA) is a finite union of (maximal) clones, hence there are a bounded number of possibilities for τ′(fA). ⋄

Corollary 10.5. Given K > 1, ϵ > 0 and clone Cantor sets C, C′ there is a finite set S of positive real numbers with the following propery. Suppose that f is any K-bilipschitz map of C onto a clopen in C′, and suppose that rel(fC) ≥ ϵ. If B is any clone of level n containing a clone D of level n + 1 in C then μ_{C′}(fD)/μ_{C′}(fB) is in S.

Proof. Let P′ be the smallest clone of C′ containing fB and τ′ the clone map of P′ onto C′. Since τ′ is a similarity

\frac{μ_{C′}(fD)}{μ_{C′}(fB)} = \frac{μ_{C′}(τ′(fD))}{μ_{C′}(τ′(fB))}.

Then there are finitely many possible images τ′(fB) by lemma (10.4). We will show that there are only finitely many possibilities for the set τ′(fD), then the result follows.

The ratio diam(D)/diam(B) is one of the finitely many numbers diam(A)/diam(C) for A a level-1 clone of C, and is thus bounded away from 0. By lemma (10.3) diam(P′) approximates diam(fB) which approximates diam(B) since f is bilipschitz. Let Q′ be the smallest clone of C′ containing fD. Then diam(Q′) approximates diam(D) and so the ratio diam(Q′)/diam(P′) approximates diam(D)/diam(B) and so is bounded away from 0. Now

\frac{diam(Q′)}{diam(P′)} = \frac{diam(τ′(Q′))}{diam(τ′(P′))}

and τ′(P′) = C′ hence diam(τ′Q′) is bounded away from 0.

There are only finitely many clones τ′Q′ in C′ which are this large. If we map Q′ onto C′ by a clone map ϕ, then ϕτ′(fD) is one of finitely many possibilities by lemma (10.4), and since there are only finitely many possible choices for ϕ there are only finitely many possible sets τ′(fD). ⋄

The following is the basic result in this subject.

Theorem 10.6. Suppose that C, C′ are clone Cantor sets and that f is a bilipschitz map of C onto a clopen subset of C′. Then there is a clopen A in C such that the restriction f|A of f to A is measure linear.

Proof. Given a measurable subset A of C of positive measure define the mass ratio of A to be

MR(A) = \frac{μ_{C′}(fA)}{μ_{C}(A)}.

This is the multiplicative factor by which f changes the mass of A. Since f is bilipschitz the mass ratio function is bounded above, let L be the supremum. We remark that

MR(A ∪ B) ≤ max{MR(A), MR(B)}

and the inequality is strict unless the mass ratios are equal. Thus given ϵ > 0 there is a clone A of C with MR(A) > L − ϵ. We will show that for ϵ small enough, that f|A is measure linear.

Assertion. There is a finite set S of real numbers such that for every pair of clones E ⊂ D of C with level(E) = level(D)+1 then MR(E)/MR(D) is in S.

Assuming this, suppose that f|A is not measure linear. Then there is some clone E in A such that MR(A) < MR(E). Otherwise for every clone E in A we would have MR(E) ≤ MR(A) but this would imply, by the remark, there is always equality and this implies that f|A is measure linear.

Thus if f|A is not measure linear we may choose a clone E in A with MR(A) < MR(E) and with n = level(E) minimal. Note that if D is any clone in A of level less than n then MR(D) = MR(A). Let D be a clone in A containing E with level(D) = n − 1. Then MR(E)/MR(D) is in S. Define λ to be the minimum of the elements of S which are larger than 1. Thus MR(E)/MR(D) ≥ λ. If we choose ϵ such that (L − ϵ)λ > L then

MR(E) > λMR(D) = λMR(A) > λ(L − ϵ) > L

which contradicts the definition of L, proving that f|A is measure linear. To prove the assertion observe that

\frac{MR(E)}{MR(D)} = \frac{μ_{C′}(fE)/μ_{C′}(fD)}{μ_{C}(E)/μ_{C}(D)}.

Since level(E) = level(D) + 1, as discussed in the proof of (10.5), there are only finitely many values for the ratio μ_{C}(E)/μ_{C}(D). By (10.5), applied with ϵ = rel(fC), there are only finitely many values for μ_{C′}(fE)/μ_{C′}(fD), and this proves the assertion. ⋄

One may now define various invariants of the bilipschitz type of clone Cantor sets. Let C be a clone Cantor set with Hausdorff measure μ_{C} in its Hausdorff dimension. We will call two sets of real numbers A and B similar if there are non-zero scalars α, β such that

αA ⊂ B  and  βB ⊂ A.

The clopen invariant of C is the similarity class of the countable set of real numbers {μ_{C}(A) : A a clopen in C }.

Corollary 10.7. If C, C′ are clone Cantor sets and there is a bilipschitz map of C onto a clopen in C′, then C and C′ have the same clopen invariant.

Proof. By the theorem, there is a clopen, hence a clone, A in C and a measure linear map of A onto a clopen in C′. The set A is a clone Cantor set and the clone map of A onto C shows that it has the same clopen invariant as C. Let α be the Radon-Nikodym derivative of the measure linear map of A into C′. The image of a clopen B in A is a clopen in C′ with α times the measure of B. Using that the inverse of a bilipschitz map is bilipschitz, one obtains the reverse relationship. ⋄

There is another invariant which measures the expansions of the similarity self maps of a clone Cantor set. A clone map is an example of such a map. In order to obtain a notion which is an invariant of bilipschitz homeomorphism, one must allow for conjugating a map which is a metric similarity by a bilipschitz map which is measure preserving, and this leads to the following.

Definition. Given K > 1 we will call a map ψ : (X, d_{X}) ⟶ (Y, d_{Y}) between metric spaces K-quasi-conformal if

K^{−1} ≤ \frac{d(ψx, ψy)/d(x, y)}{d(ψx, ψz)/d(x, z)} ≤ K

for all distinct points x, y, z. Thus a similarity is 1-quasi-conformal. A Kbilipschitz map is K^{2}-quasi-conformal. The composition of K and K′ quasiconformal maps is (KK′)-quasi-conformal. Suppose that x, y, w are distinct then

K^{−1} ≤ \frac{d(ψy, ψw)/d(y, w)}{d(ψy, ψx)/d(y, x)} ≤ K

thus

K^{−2} ≤ \frac{d(ψy, ψw)/d(y, w)}{d(ψx, ψz)/d(x, z)} ≤ K^{2}.

Regard x and z as fixed, then we see that ψ is M-bilipschitz with M = K^{2} max(L, 1/L) where L = d(ψx, ψz)/d(x, z).

Lemma 10.8. Suppose that ψ : X ⟶ Y is a K-quasi-conformal bijection. Then ψ is M-bilipschitz with M = K^{4} max(N, 1/N) where N = diam(Y)/diam(X).

Proof. Choose x, y, z, w in X with d_{X}(y, w) = diam(X) and d_{Y} (ψx, ψz) = diam(Y). Then

K^{−2} ≤ \frac{d_{Y} (ψy, ψw)/d_{X}(y, w)}{d_{Y} (ψx, ψz)/d_{X}(x, z)} = \frac{d_{Y} (ψy, ψw)}{d_{X}(y, w)} \frac{d_{X}(x, z)}{diam(Y)}.

Now d_{X}(x, z) ≤ diam(X) thus

(K^{2}N^{−1})^{−1} = K^{−2} \frac{diam(Y)}{diam(X)} ≤ \frac{d_{Y} (ψy, ψw)}{d_{X}(y, w)} ≤ \frac{diam(Y)}{diam(X)} = N ≤ K^{2}N.

It follows that ψ is M-bilipschitz with M = K^{2} max(K^{2}N, K^{2}N^{−1}). ⋄ Definition. A map σ : (X, d_{X}) ⟶ (X, d_{X}) of a metric space (X, d_{X}) to itself has bounded distortion or is BD if there is a constant K > 1 such that for every integer n > 0 the map σ^{n} is K-quasi-conformal. Informally, the iterates of σ are almost similarities. For example the conjugate of a contracting similarity by a bilipschitz map is BD. This definition is not quite equivalent to the one in [CP], but is better for present purposes. Definition. A local similarity of a clone Cantor set C is a continuous map σ : U ⟶ U defined on a non-empty clopen U of C which is a homeomorphism onto V = σU. We require that V is a clopen contained in U, and the following properties:

(1) σ has a unique fixed point x called a similarity point of C.

(2) σ is measure linear with respect to Hausdorff measure. The Radon-Nikodym derivative of σ is called the scale factor of σ.

(3) σ is BD.

If x is a similarity point of C we define the similarity group S_{x}(C) of C at x to be the subgroup of the multiplicative group R^{+} generated by the scale factors of the local similarities of C fixing x.

Corollary 10.9. Suppose that C and C′ are clone Cantor sets and there is a bilipschitz homeomorphism of C onto a clopen subset of C′. If x is a similarity point of C there is a similarity point y in C′ such that S_{x}(C) = S_{y}(C′).

Proof. By theorem (10.6) there is a clopen A of C on which the given bilipschitz homeomorphism

ϕ : C ⟶ C′

is measure linear. The orbit of x under the semi-group of clone maps of C contains a point in every clone and hence contains a point z in A. Clearly S_{x}(C) = S_{z}(C). We claim that the conjugate of a local similarity by a measure-linear bilipschitz homeomorphism is a local similarity. To verify this, let σ : A ⟶ A be a local similarity defined on the clopen A of C and with fixed point z. Let σ′ = ϕ ∘ σ ∘ ϕ^{−1} then σ′ has a unique fixed point ϕ(z). Since ϕ is a homeomorphism, ϕA is a clopen of C and so the domain of σ′ is the clopen ϕA. Since ϕ|A and σ are measure linear, σ′ is measure linear. Since σ is BD then the iterates σ^{n} are all K-quasi-conformal. Also ϕ is bilipschitz and thus M-quasi-conformal for some M. Thus

σ′^{n} = ϕ ∘ σ^{n} ∘ ϕ^{−1}

is (M^{2}K)-quasi-conformal. Hence σ′ is BD. ⋄

An Application to n-adic Cantor sets

Given n > 1 let C_{n}, or just C, be the Cantor set of sequences (x_{0}, x_{1}, ⋯) where x_{i} is an element of {0, ⋯, n − 1}. The metric on C_{n} is defined by d_{n}(x, y) = n^{−p} if p is minimal subject to x_{p} ≠ y_{p}. Thus C has diameter 1. The shift map σ mapping C onto itself is defined by σ(x_{0}, x_{1}, ⋯) = (x_{1}, x_{2}, ⋯). Define clones A_{i} for 0 ≤ i ≤ n − 1 to be the subsets of C with first digit i. Define a clone map τ_{i} = σ|A_{i} which linearly maps A_{i} onto C.

Proposition 10.10. The Hausdorff dimension of the Cantor set C is 1 and it has measure 1 in this dimension.

Proof. There is a map ϕ of C onto the unit interval [0, 1] given by regarding a sequence (x_{0}, x_{1}, ⋯) as an expansion 0.x_{0}x_{1}x_{2} ⋯ in base-n of a real number. Consider a subset A in C with diameter R. Choose p so that n^{−p} ≤ R < n^{1−p}. Then all points in A consist of sequences which have the same digits up to and including x_{p−1} and therefore ϕA has Euclidean diameter at most n^{−p}. Thus

diameter_{C}(A) ≥ Euclidean diameter(ϕA).

Thus the sum of the diameters of the sets in any cover of C is at least 1 and so the Hausdorff 1-measure of C is at least 1. Given ϵ > 0 there is a cover of C by level-k clones each of which has diameter in the d_{C}-metric n^{−k} < ϵ. There are n^{k} such clones, so the sum of their diameters is 1. Therefore the Hausdorff 1-measure is at most 1. ⋄

Since clone maps are measure linear it follows that the measure of a clone of C equals the Euclidean length of its image under ϕ. Thus ϕ is measure preserving. Hence the clopen invariant of C_{n} is the similarity class of [0, 1] ∩ Z[1/n].

Corollary 10.11. There is a bilipschitz homeomorphism of the clone Cantor set C_{n} onto a clopen subset of C_{m} if and only if there are integers i, j > 0 such that n^{i} = m^{j}.

Proof. If n^{i} = m^{j} then it is clear that the Cantor sets are bilipschitz homeomorphic. For the converse, suppose that σ is a local similarity of C_{n} with scale factor s < 1 for Hausdorff measure. We first show that s is a rational power of n.

Let A be a clone in the domain of σ which contains the fixed point of σ. Then some iterate, σ^{m}, of σ maps A into A. We replace σ with σ^{m}|A. The scale factor of σ^{m} is s^{m} and it suffices to prove that s^{m} is a rational power of n. By this means we may suppose that the domain of σ is a clone A.

Since σ is BD there is L > 1 such that for all p > 0 we have that σ^{p} is L-quasi-conformal. Define α ≡ α(C_{n}). The lemma (10.4) provides a constant, M = M(C_{n}, C_{n}, K, ϵ), for the case that C and C′ are both C_{n}, and K = L^{6}α^{−1}, and ϵ = K^{−1}α. Given an integer a > 0, define a new metric, d_{new} on C_{n} which is n^{a} times the n-adic metric thus d_{new} = n^{a}d_{n}. There is a unique clone, D, of diameter 1 in the d_{new}-metric which contains the unique fixed point of σ. We may choose a large enough that D is contained in σ^{2M}A. Then D with the metric d_{new} is a clone Cantor set isometric to C_{n} with the n-adic metric. Also σ|D is a local similarity of D with scale factor s. Also for all p > 0 we have σ^{p} is L-quasi-conformal. We now replace (C_{n}, d_{n}) and σ by (D, d_{new}) and σ|D. By means of this we may assume our original local similarity, σ, admits an extension to a local similarity, \overline{σ}, defined on all of a clone Cantor set, E, of large diameter containing C_{n} as a clone. Furthermore we may assume that the image of \overline{σ}^{2M} contains C_{n}.

Suppose that A_{p} is the smallest clone containing σ^{p}C_{n}. If τ_{p} is the clone map taking A_{p} onto C_{n}, define ψ_{p} ≡ τ_{p}σ^{p} then ψ_{p}C_{n} is not contained in any level-1 clone of C_{n} thus diam(ψ_{p}C_{n}) ≥ α. The clone map τ_{p} has domain the clone A_{p} which it maps bijectively to C_{n}. Define B_{p} = (\overline{σ})^{−p}A_{p} and define a bijection

\overline{ψ}_{p} ≡ τ_{p}\overline{σ}^{p} : B_{p} ⟶ C_{n}.

Since \overline{σ}^{p} is L-quasi-conformal and τ_{p} is linear \overline{ψ}_{p} is also L-quasi-conformal. Thus, in particular, ψ_{p} is L-quasi-conformal. Apply lemma (10.8) to

ψ_{p} : X ≡ C_{n} ⟶ Y ≡ ψ_{p}C_{n}.

Then diam(X) = 1 and diam(Y) = diam(ψ_{p}C_{n}) ≥ α so α ≤ N ≤ 1 thus ψ_{p} is (L^{4}α^{−1})-bilipschitz. Since \overline{ψ}_{p} is L-quasi-conformal and (L^{4}α^{−1})- bilipschitz on C_{n} it follows that \overline{ψ}_{p} is K-bilipschitz with K = L^{6}α^{−1}.

We wish to apply the lemma (10.4) to the maps ψ_{p} : C_{n} ⟶ C_{n} with 1 ≤ p ≤ 2M. These maps are all K-bilipshitz. We claim that sep(ψ_{p}C_{n}) ≥ ϵ. Since p ≤ 2M we have \overline{σ}^{p}E contains C_{n} thus \overline{ψ}_{p}(B_{p}) = C_{n}. Since \overline{ψ}_{p} is Kbilipschitz sep(ψ_{p}C_{n}) is at least K^{−1}d_{E}(C_{n}, B_{p} − C_{n}). Now d_{E}(C_{n}, B_{p} − C_{n}) ≥ α because C_{n} is a clone in E and the separation of C_{n} in E is therefore bigger than the separation of any clone contained in C_{n}. Thus sep(ψ_{p}A) ≥ K^{−1}α = ϵ, which proves the claim.

By lemma (10.4) there are M = M(C_{n}, C_{n}, K, ϵ) possibilities for ψ_{p}C_{n}. So for some 1 ≤ k < j ≤ 2M we have that ψ_{j}C_{n} = ψ_{k}C_{n}. The metric scale factor of every clone map of C_{n} is an integral power of n. Since the Hausdorff dimension of C_{n} is 1 this means that the scale factor for Hausdorff measure of a clone map is also an integral power of n. Thus τ_{j}, τ_{k} are measure linear with scale factors for Hausdorff measure which are powers of n. Thus the scale factor for Hausdorff measure for ψ_{j} and ψ_{k} is n^{a}s^{j} = n^{b}s^{k} and so s is a rational power of n.

By (10.9) we have that s is also the scale factor of a local similarity of C_{m}. The above then implies that s is also a rational power of m. ⋄

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

Lee Mosher: Dept. of Math. and C.S., Rutgers Univ., Newark Newark, NJ 07102 E-mail: mosher@andromeda.rutgers.edu

Daryl Cooper: Dept. of Math., Univ. of California, Santa Barbara Santa Barbara, CA 93106 E-mail: cooper@math.ucsb.edu