CANNON-THURSTON MAPS FOR TREES OF HYPERBOLIC METRIC SPACES

MAHAN MITRA

Abstract. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T. Let (X_{v}, d_{v}) denote the hyperbolic metric space corresponding to v. Then i : X_{v} → X extends continuously to a map î : X̂_{v} → X̂. This generalizes a Theorem of Cannon and Thurston. The techniques are used to give a new proof of a result of Minsky: Thurston’s ending lamination conjecture for certain Kleinian groups. Applications to graphs of hyperbolic groups and local connectivity of limit sets of Kleinian groups are also given.

1. Introduction

Let G be a hyperbolic group in the sense of Gromov [13]. Let H be a hyperbolic subgroup of G. We choose a finite symmetric generating set for H and extend it to a finite symmetric generating set for G. Let Γ_{H} and Γ_{G} denote the Cayley graphs of H, G respectively with respect to these generating sets. By adjoining the Gromov boundaries ∂Γ_{H} and ∂Γ_{G} to Γ_{H} and Γ_{G}, one obtains their compactifications Γ̂_{H} and Γ̂_{G} respectively.

We’d like to understand the extrinsic geometry of H in G. Since the objects of study here come under the purview of coarse geometry, asymptotic or ‘large-scale’ information is of crucial importance. That is to say, one would like to know what happens ‘at infinity’. We put this in the more general context of a hyperbolic group H acting freely and properly discontinuously on a proper hyperbolic metric space X. Then there is a natural map i : Γ_{H} → X, sending the vertex set of Γ_{H} to the orbit of a point under H, and connecting images of adjacent vertices in Γ_{H} by geodesics in X. Let X̂ denote the Gromov compactification of X.

A natural question seems to be the following:

Research partly supported by Alfred P. Sloan Doctoral Dissertation Fellowship, Grant No. DD 595.

AMS Subject Classification: 20F32, 57M50.

Question: Does the continuous proper map i : Γ_{H} → X extend to a continuous map î: Γ̂_{H} → X̂ ?

Questions along this line have been raised by Bonahon [5]. Related questions in the context of Kleinian groups have been studied by Bonahon [6], Floyd [10] and Minsky [19]. In, [7], [10] or [19], explicit metrics were used. So though some of their results can be thought of as ‘coarse’, the techniques of proof are not. In [20], coarse techniques were used to answer the above question affirmatively for X = Γ_{G}, where G is a hyperbolic group and H a normal subgroup of G. This in turn was a generalization of a theorem of Cannon and Thurston [7]. In this paper, we extend the techniques of [20] to cover examples arising from trees of hyperbolic metric spaces satisfying an extra technical condition introduced by Bestvina and Feighn in [3]: the quasi-isometrically embedded condition. [See Section 3 of this paper or [3] for definitions.]

Definition: Let X and Y be hyperbolic metric spaces and i : Y → X be an embedding. A Cannon-Thurston map î from Ŷ to X̂ is a continuous extension of i. Such a continuous extension will occassionally be called a Cannon-Thurston map for the pair (Y, X). If Y = Γ_{H} and X = Γ_{G} for a hyperbolic subgroup H of a hyperbolic group G, a Cannon-Thurston map for (Γ_{H}, Γ_{G}) will occassionally be referred to as a Cannon-Thurston map for (H, G).

It is easy to see that such a continuous extension, if it exists, is unique.

The main theorem of this paper is :

Theorem 3.8: Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the quasi-isometrically embedded condition. Let v be a vertex of T. Let (X_{v}, d_{v}) denote the hyperbolic metric space corresponding to v. Then i : X_{v} → X extends continuously to a map î: X̂_{v} → X̂.

A direct consequence of Theorem 3.8 above is the following:

Corollary 3.9: Let G be a hyperbolic group acting cocompactly on a simplicial tree T such that all vertex and edge stabilizers are hyperbolic. Also suppose that every inclusion of an edge stabilizer in a vertex stabilizer is a quasi-isometric embedding. Let H be the stabilizer of a vertex or edge of T. Then there exists a Cannon-Thurston map from Γ̂_{H} to Γ̂_{G}.

In [3], Bestvina and Feighn give sufficient conditions for a graph of hyperbolic groups to be hyperbolic. Vertex and edge subgroups are thus natural examples of hyperbolic subgroups of hyperbolic groups.

Essentially all previously known examples of non-quasiconvex hyperbolic subgroups of hyperbolic groups arise this way. Theorem 3.8 shows that these have Cannon-Thurston maps.

Another consequence of Theorem 3.8 above is:

Theorem 4.7: Let Γ be a geometrically tame Kleinian group, such that ℍ^{3}/Γ = M has injectivity radius uniformly bounded below by some ϵ > 0. Then there exists a continuous map from the Gromov boundary of Γ (regarded as an abstract group) to the limit set of Γ in 𝕊_{∞}^{2}.

Theorem 4.7 was independently proven by Klarreich [14], using different techniques.

After some further work and using a theorem of Minsky [18], we are able to give a different proof of another result of Minsky [19] : Thurston’s Ending Lamination Conjecture for geometrically tame manifolds with freely indecomposable fundamental group and a uniform lower bound on injectivity radius.

Theorem 4.13 [19]: Let N_{1} and N_{2} be homeomorphic hyperbolic 3manifolds with freely indecomposable fundamental group. Suppose there exists a uniform lower bound ϵ > 0 on the injectivity radii of N_{1} and N_{2}. If the end invariants of corresponding ends of N_{1} and N_{2} are equal, then N_{1} and N_{2} are isometric.

2. Preliminaries

We start off with some preliminaries about hyperbolic metric spaces in the sense of Gromov [13]. For details, see [8], [11]. Let (X, d) be a hyperbolic metric space. The Gromov boundary X, denoted by ∂X, is the collection of equivalence classes of geodesic rays r : [0, ∞) → Γ with r(0) = x_{0} for some fixed x_{0} ∈ X, where rays r_{1} and r_{2} are equivalent if sup{d(r_{1}(t), r_{2}(t))} < ∞. Let X̂=X ∪ ∂X denote the natural compactification of X topologized the usual way(cf.[11] pg. 124).

The Gromov inner product of elements a and b relative to c is defined by

(a, b)_{c}=1/2[d(a, c) + d(b, c) − d(a, b)].

Definitions: A subset X of Γ is said to be k-quasiconvex if any geodesic joining a, b ∈ X lies in a k-neighborhood of X. A subset X is quasiconvex if it is k-quasiconvex for some k. A map f from one metric space (Y, d_{Y}) into another metric space (Z, d_{Z}) is said to be a (K, ϵ)-quasi-isometric embedding if

\frac{1}{K} (d_{Y} (y_{1}, y_{2})) − ϵ ≤ d_{Z}(f(y_{1}), f(y_{2})) ≤ Kd_{Y} (y_{1}, y_{2}) + ϵ If f is a quasi-isometric embedding, and every point of Z lies at a uniformly bounded distance from some f(y) then f is said to be a quasiisometry. A (K, ϵ)-quasi-isometric embedding that is a quasi-isometry will be called a (K, ϵ)-quasi-isometry.

A (K, ϵ)-quasigeodesic is a (K, ϵ)-quasi-isometric embedding of a closed interval in ℝ. A (K, 0)-quasigeodesic will also be called a Kquasigeodesic.

Let (X, d_{X}) be a hyperbolic metric space and Y be a subspace that is hyperbolic with the inherited path metric d_{Y}. By adjoining the Gromov boundaries ∂X and ∂Y to X and Y, one obtains their compactifications X̂ and Ŷ respectively.

Let i : Y → X denote inclusion.

Definition: Let X and Y be hyperbolic metric spaces and i : Y → X be an embedding. A Cannon-Thurston map î from Ŷ to X̂ is a continuous extension of i.

The following lemma says that a Cannon-Thurston map exists if for all M > 0 and y∈Y, there exists N > 0 such that if λ lies outside an N ball around y in Y then any geodesic in X joining the end-points of λ lies outside the M ball around i(y) in X. For convenience of use later on, we state this somewhat differently. The proof is similar to Lemma 2.1 of [20]

Lemma 2.1. A Cannon-Thurston map from Ŷ to X̂ exists if the following condition is satisfied:

There exists a non-negative function M(N), such that M(N) → ∞ as N → ∞ and for all geodesic segments λ lying outside an N-ball around y_{0} ∈ Y any geodesic segment in Γ_{G} joining the end-points of i(λ) lies outside the M(N)-ball around i(y_{0}) ∈ X.

Proof: Suppose i : Y → X does not extend continuously . Since i is proper, there exist sequences x_{m}, y_{m} ∈ Y and p ∈ ∂Y, such that x_{m} → p and y_{m} → p in Ŷ, but i(x_{m}) → u and i(y_{m}) → v in X̂, where u, v ∈ ∂X and u ≠ v.

Since x_{m} → p and y_{m} → p, any geodesic in Y joining x_{m} and y_{m} lies outside an N_{m}-ball y_{0} ∈ Y, where N_{m} → ∞ as m → ∞. Any bi-infinite geodesic in X joining u, v ∈ ∂X has to pass through some M-ball around i(y_{0}) in X as u ≠ v. There exist constants c and L such that for all m > L any geodesic joining i(x_{m}) and i(y_{m}) in X passes through an (M + c)-neighborhood of i(y_{0}). Since (M + c) is a constant not depending on the index m this proves the lemma. □

3. Trees of Hyperbolic Metric Spaces

We start with a notion closely related to one introduced in [3].

Definition: A tree (T) of hyperbolic metric spaces satisfying the q(uasi) i(sometrically) embedded condition is a metric space (X, d) admitting a map P : X → T onto a simplicial tree T, such that there exist δ,ϵ and K > 0 satisfying the following:

1. For all vertices v ∈ T, X_{v} = P^{−1}(v) ⊂ X with the induced path metric d_{v} is a δ-hyperbolic metric space. Further, the inclusions i_{v} : X_{v} → X are uniformly proper, i.e. for all M > 0, v ∈ T and x, y ∈ X_{v}, there exists N > 0 such that d(i_{v}(x), i_{v}(y)) ≤ M implies d_{v}(x, y) ≤ N.

2. Let e be an edge of T with initial and final vertices v_{1} and v_{2} respectively. Let X_{e} be the pre-image under P of the mid-point of e. Then X_{e} with the induced path metric is δ-hyperbolic.

3. There exist maps f_{e} : X_{e}×[0, 1] → X, such that f_{e}|_{X_{e}×(0,1)} is an isometry onto the pre-image of the interior of e equipped with the path metric.

4. f_{e}|_{X_{e}×{0}} and f_{e}|_{X_{e}×{1}} are (K, ϵ)-quasi-isometric embeddings into X_{v_{1}} and X_{v_{2}} respectively. f_{e}|_{X_{e}×{0}} and f_{e}|_{X_{e}×{1}} will occassionally be referred to as f_{v_{1}} and f_{v_{2}} respectively.

d_{v} and d_{e} will denote path metrics on X_{v} and X_{e} respectively. i_{v}, i_{e} will denote inclusion of X_{v}, X_{e} respectively into X.

The main theorem of this paper can now be stated:

Theorem: 3.8 Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the qi-embedded condition. Let v be a vertex of T. Then i_{v} : X_{v} → X extends continuously to î_{v} : X̂_{v} → X̂.

Some aspects of the proof of the main theorem of this section are similar to the proof of the main theorem of [20]. Given a geodesic segment λ ⊂ X_{v}, we construct a quasi-convex set B_{λ} ⊂ X. It follows from the construction that if λ lies outside a large ball around y_{0} ∈ X_{v}, B_{λ} lies outside a large ball around i_{v}(y_{0}) ∈ X, i.e. for all M ≥ 0 there exists N ≥ 0 such that if λ lies oustside the N-ball around y_{0} ∈ X_{v}, B_{λ} lies outside the M-ball around i_{v}(y_{0}) ∈ X. Combining this with Lemma 2.1 above, the proof of Theorem 3.8 is completed.

For convenience of exposition, T shall be assumed to be rooted, i.e. equipped with a base vertex v_{0}. Let v ≠ v_{0} be a vertex of T. Let v_{−} be the penultimate vertex on the geodesic edge path from v_{0} to v. Let e denote the directed edge from v_{−} to v. Define ϕ_{v} : f_{e_{−}}(X_{e_{−}}×{0}) → f_{e_{−}}(X_{e_{−}}×{1}) by

ϕ_{v}(f_{e_{−}}(x×{0}) = f_{e_{−}}(x×{1}) for x ∈ X_{e_{−}}.

Let μ be a geodesic in (X_{v_{−}}), joining i_{v_{−}}^{−1}(a), i_{v_{−}}^{−1}(b) ∈ i_{v_{−}}^{−1}·f_{e_{−}}(X_{e_{−}}×{0}). Φ_{v}(μ) will denote a geodesic in X_{v} joining ϕ_{v}(a) and ϕ_{v}(b). Let X_{v_{0}} = Y.

For convenience of exposition, we shall modify X, X_{v}, X_{e} by quasiisometric perturbations. Given a geodesically complete metric space (Z, d) of bounded geometry, choose a maximal disjoint collection {N_{1}(z_{α})} of disjoint 1-balls. Then by maximality, for all z ∈ Z there exist z_{α} in the collection such that d(z, z_{α}) < 2. Construct a graph Z_{1} with vertex set {z_{α}} and edge set consisting of distinct vertices z_{α}, z_{β} such that d(z_{α}, z_{β}) ≤ 4. Then Z_{1} equipped with the pathmetric is quasi-isometric to (Z, d). All metric spaces in this section will henceforth be assumed to be graphs of edge length 1 and maps between them will be assumed to be cellular.

We start with a general lemma about hyperbolic metric spaces. This follows easily from the fact that local quasigeodesics in a hyperbolic metric space are quasigeodesics [11]. If x, y are points in a hyperbolic metric space, [x, y] will denote a geodesic joining them.

Lemma 3.1. Given δ > 0, there exist D, C_{1} such that if a, b, c, d are vertices of a δ-hyperbolic metric space (Z, d), with d(a, [b, c]) = d(a, b), d(d, [b, c]) = d(c, d) and d(b, c) ≥ D then [a, b] ∪ [b, c] ∪ [c, d] lies in a C_{1}-neighborhood of any geodesic joining a, d.

Given a geodesic segment λ ⊂ Y, we now construct a quasi-convex set B_{λ} ⊂ X.

Construction of B_{λ}

Choose C_{2} ≥ 0 such that for all e ∈ T, f_{e}(X_{e}×{0}) and f_{e}(X_{e}×{1}) are C_{2}-quasiconvex in the appropriate vertex groups. Let C=C_{1}+C_{2}, where C_{1} is as in Lemma 3.1.

For Z ⊂ X_{v}, let N_{C}(Z) denote the C-neighborhood of Z, that is the set of points at distance less than or equal to C from Z.

\underline{Step 1 :} Let μ ⊂ X_{v} be a geodesic segment in (X_{v}, d_{v}). Then P (μ) = v. For each edge e incident on v, but not lying on the geodesic (in T) from v_{0} to v, choose p_{e}, q_{e} ∈ N_{C}(μ)∩f_{v}(X_{e}) such that d_{v}(p_{e}, q_{e}) is maximal. Let v_{1}, ⋯, v_{n} be terminal vertices of edges e for which d_{v}(p_{e}, q_{e}) > D, where D is as in Lemma 3.1 above. Observe that there are only finitely many v_{i}’s as μ is finite. Define

B^{1}(μ) = i_{v}(μ)∪⋃_{k=1···n}Φ_{v_{i}}(μ)

Note that P (B^{1}(μ)) ⊂ T is a finite tree.

\underline{Step 2 :} Step 1 above constructs B^{1}(λ) in particular. We proceed inductively. Suppose that B^{m}(λ) has been constructed such that the convex hull of P (B^{m}(λ)) ⊂ T is a finite tree. Let {w_{1}, ⋯, w_{n}} = P (B^{m}(λ))\P (B^{m−1}(λ)). (Note that n may depend on m, but we avoid repeated indices for notational convenience.) Assume further that P^{−1}(v_{k})∩B^{m}(λ) is a path of the form i_{v_{k}}(λ_{k}), where λ_{k} is a geodesic in (X_{v_{k}}, d_{v_{k}}). Define

B^{m+1}(λ) = B^{m}(λ)∪⋃_{k=1···n}(B^{1}(λ_{k}))

where B^{1}(λ_{k}) is defined in Step 1 above.

Since each λ_{k} is a finite geodesic segment in Γ_{H}, the convex hull of P (B^{m+1}λ) is a finite subtree of T. Further, P^{−1}(v)∩B^{m+1}(λ) is of the form i_{v}(λ_{v}) for all v ∈ P (B^{m+1}(λ)). This enables us to continue inductively. Define

B_{λ} = ∪_{m≥0}B_{λ}^{m}.

Note finally that the convex hull of P (B_{λ}) in T is a locally finite tree T_{1}.

Quasiconvexity of B_{λ}

We shall now show that there exists C′ ≥ 0 such that for every geodesic segment λ ⊂ Y, B_{λ}⊂X is C′-quasiconvex. To do this we construct a retraction Π_{λ} from (the vertex set of) X onto B_{λ} and show that there exists C_{0} ≥ 0 such that d_{X}(Π_{λ}(x), Π_{λ}(y)) ≤ C_{0}d_{X}(x, y). Let π_{v} : X_{v} → λ_{v} be a nearest point projection of X_{v} onto λ_{v}. Π_{λ} is defined on ⋃_{v∈T_{1}}X_{v} by

Π_{λ}(x) = i_{v}·π_{v}(x) for x ∈ X_{v}.

If x ∈ P^{−1}(T \ T_{1}) choose x_{1} ∈ P^{−1}(T_{1}) such that d(x, x_{1}) = d(x, P^{−1}(T_{1})) and define Π′_{λ}(x) = x_{1}. Next define Π_{λ}(x) = Π_{λ} · Π′_{λ}(x).

The following Lemma says nearest point projections in a δ-hyperbolic metric space do not increase distances much.

Lemma 3.2. Let (Y, d) be a δ-hyperbolic metric space and let μ ⊂ Y be a geodesic segment. Let π : Y → μ map y ∈ Y to a point on μ nearest to y. Then d(π(x), π(y)) ≤ C_{3}d(x, y) for all x, y ∈ Y where C_{3} depends only on δ.

Proof: Let [a, b] denote a geodesic edge-path joining vertices a, b. Recall that the Gromov inner product (a, b)_{c}=1/2[d(a, c)+d(b, c)−d(a, b)]. It suffices by repeated use of the triangle inequality to prove the Lemma when d(x, y) = 1. Let u, v, w be points on [x, π(x)], [π(x), π(y)] and [π(y), x] respectively such that d(u, π(x)) = d(v, π(x)), d(v, π(y)) = d(w, π(y)) and d(w, x) = d(u, x). Then (x, π(y))_{π(x)} = d(u, π(x)). Also, since Y is δ-hyperbolic, the diameter of the inscribed triangle with vertices u, v, w is less than or equal to 2δ (See [1]).

d(u, x) + d(u, v) ≥ d(x, π(x)) = d(u, x) + d(u, π(x))
⇒ d(u, π(x)) ≤ d(u, v) ≤ 2δ
⇒ (x, π(y))_{π(x)} ≤ 2δ

Similarly, (y, π(x))_{π(y)} ≤ 2δ.

i.e. d(x, π(x)) + d(π(x), π(y)) − d(x, π(y)) ≤ 4δ

and d(y, π(y)) + d(π(x), π(y)) − d(y, π(x)) ≤ 4δ

Therefore,

2d(π(x), π(y))
≤ 8δ + d(x, π(y)) − d(y, π(y)) + d(y, π(x)) − d(x, π(x))
≤ 8δ + d(x, y) + d(x, y)
≤ 8δ + 2

Hence d(π(x), π(y)) ≤ 4δ + 1. Choosing C_{3} = 4δ + 1, we are through. □

Lemma 3.3. Let (Y, d) be a δ-hyperbolic metric space. Let μ be a geodesic segment in Y with end-points a, b and let x be any vertex in Y. Let y be a vertex on μ such that d(x, y) ≤ d(x, z) for any z ∈ μ. Then a geodesic path from x to y followed by a geodesic path from y to z is a k-quasigeodesic for some k dependent only on δ.

Proof: As in Lemma 3.2 let u, v, w be points on edges [x, y], [y, z] and [z, x] respectively such that d(u, y) = d(v, y), d(v, z) = d(w, z) and d(w, x) = d(u, x). Then d(u, y) = (z, x)_{y} ≤ 2δ and the inscribed triangle with vertices u, v, w has diameter less than or equal to 2δ (See [1]). [x, y] ∪ [y, z] is a union of 2 geodesic paths lying in a 4δ neighborhood of a geodesic [x, z]. Hence a geodesic path from x to y followed by a geodesic path from y to z is a k−quasigeodesic for some k dependent only on δ. □

Lemma 3.4. Suppose (Y, d) is a δ-hyperbolic metric space. If μ is a (k_{0}, ϵ_{0})-quasigeodesic in Y and p, q, r are 3 points in order on μ then (p, r)_{q} ≤ k_{1} for some k_{1} dependent on k_{0}, ϵ_{0} and δ only.

Proof: [a, b] will denote a geodesic path joining a, b. Since p, q, r are 3 points in order on μ, [p, q] followed by [q, r] is a (k_{0}, ϵ_{0})-quasigeodesic in the δ-hyperbolic metric space Y. Hence there exists a k_{1} dependent

Figure 1

on k_{0}, ϵ_{0} and δ alone such that d(q, [p, r]) ≤ k_{1}. Let s be a point on [p, r] such that d(q, s) = d(q, [p, r]) ≤ k_{1}. Then

(p, r)_{q} = 1/2(d(p, q) + d(r, q) − d(p, r))
= 1/2(d(p, q) + d(r, q) − d(p, s) − d(r, s))
≤ d(q, s) ≤ k_{1}.□

Lemma 3.5. Suppose (Y, d) is δ-hyperbolic. Let μ_{1} be some geodesic segment in Y joining a, b and let p be any vertex of Y. Also let q be a vertex on μ_{1} such that d(p, q) ≤ d(p, x) for x ∈ μ_{1}. Let ϕ be a (K, ϵ)

- quasiisometry from Y to itself. Let μ_{2} be a geodesic segment in Y joining ϕ(a) to ϕ(b) for some g ∈ S. Let r be a point on μ_{2} such that d(ϕ(p), r) ≤ d(ϕ(p), x) for x ∈ μ_{2}. Then d(r, ϕ(q)) ≤ C_{4} for some constant C_{4} dependent only on K, ϵ and δ.

Proof: Since ϕ(μ_{1}) is a (K, ϵ)− quasigeodesic joining ϕ(a) to ϕ(b), it lies in a K′-neighborhood of μ_{2} where K′ depends only on K, ϵ, δ. Let u be a vertex in ϕ(μ_{1}) lying at a distance at most K′ from r. Without loss of generality suppose that u lies on ϕ([q, b]), where [q, b] denotes the geodesic subsegment of μ_{1} joining q, b. [See Figure 1.]

Let [p, q] denote a geodesic joining p, q. From Lemma 3.3 [p, q] ∪ [q, b] is a k-quasigeodesic, where k depends on δ alone. Therefore ϕ([p, q]) ∪ ϕ([q, b]) is a (K_{0}, ϵ_{0})-quasigeodesic, where K_{0}, ϵ_{0} depend on K, k, ϵ. Hence, by Lemma 3.4 (ϕ(p), u)_{ϕ(q)} ≤ K_{1}, where K_{1} depends on K, k, ϵ and δ alone. Therefore,

(ϕ(p), r)_{ϕ(q)}
= 1/2[d(ϕ(p), ϕ(q)) + d(r, ϕ(q)) − d(r, ϕ(p))]
≤ 1/2[d(ϕ(p), ϕ(q)) + d(u, ϕ(q)) + d(r, u)
−d(u, ϕ(p)) + d(r, u)]
= (ϕ(p), u)_{ϕ(q)} + d(r, u)
≤ K_{1} + K′

There exists s ∈ μ_{2} such that d(s, ϕ(q)) ≤ K′

(ϕ(p), r)_{s} = 1/2[d(ϕ(p), s) + d(r, s) − d(r, ϕ(p))]
≤ 1/2[d(ϕ(p), ϕ(q)) + d(r, ϕ(q)) − d(r, ϕ(p))] + K′
= (ϕ(p), r)_{ϕ(q)} + K′
≤ K_{1} + K′ + K′
= K_{1} + 2K′

Also, as in the proof of Lemma 3.2 (ϕ(p), s)_{r} ≤ 2δ

d(r, s) = (ϕ(p), s)_{r}+(ϕ(p), r)_{s}
≤ K_{1}+2K′+2δ
d(r, ϕ(q)) ≤ K_{1}+2K′+2δ+d(s, ϕ(q))
≤ K_{1}+2K′+2δ+K′

Let C_{4}=K_{1}+3K′+2δ. Then d(r, ϕ(q)) ≤ C_{4} and C_{4} is independent of a, b, p. □

d_{T} will denote the metric on T. We are now in a position to prove:

Theorem 3.6. There exists C_{0} ≥ 0 such that d(Π_{λ}(x), Π_{λ}(y)) ≤ C_{0}d_{G}(x, y) for x, y vertices of Γ_{G}.

Proof: It suffices to prove the theorem when d_{G}(x, y) = 1.

Case (a): x, y ∈ P^{−1}(v) for some v ∈ T_{1}. From Lemma 3.2, there exists C_{3} such that d_{v}(π_{v}·i_{v}^{−1}(x), π_{v}·i_{v}^{−1}(y)) ≤ C_{3}. Since embeddings of X_{v} in X are cellular, d(Π_{λ}(x), Π_{λ}(y)) ≤ C_{3}.

Case (b): x ∈ P^{−1}(w) and y ∈ P^{−1}(v) for some v, w ∈ T_{1}.

Since d_{G}(x, y) = 1, v, w are adjacent in T_{1}. Assume, without loss of generality, w = v_{−}.

Recall that

B_{λ} ∩ P^{−1}(v) = i_{v}(λ_{v})
B_{λ} ∩ P^{−1}(w) = i_{w}(λ_{w})

Also, λ_{v} = Φ_{v}(μ_{w}), for some geodesic μ_{w} contained in X_{w}, such that end-points of μ_{w} lie in a C-neighborhood of λ_{w}.

Let z ∈ (X_{w}) denote a nearest point projection of i_{w}^{−1}(x) onto μ_{w}. Then, by Lemma 3.5,

d(i_{w}(z), Π_{λ}·ϕ_{v}(x)) ≤ d(i_{w}(z), ϕ_{v}·i_{w}(z))+d(ϕ_{v}·i_{w}(z), Π_{λ}·ϕ_{v}(x)) ≤ 1+C_{4}.

Since, d(x, y) = 1 = d(x, ϕ_{v}(x)) and i_{v}’s are uniformly proper embeddings, there exists C_{5} > 0 such that d_{v}(ϕ_{v}(x), y) ≤ C_{5} and d(Π_{λ}(ϕ_{v}(x)), Π_{λ}(y)) ≤ C_{3}C_{5}.

Since the end-points of μ_{w} lie in a C-neighborhood of λ_{w}, there exists C_{6}, depending on δ and C such that d(z, Π_{λ}(x)) ≤ C_{6}.

Finally, by the triangle inequality,

d(Π_{λ}(x), Π_{λ}(y)) ≤ C_{6} + 1 + C_{4} + C_{3}C_{5} = C_{7}(say)

Case (c): P ([x, y]) is not contained in T_{1}. Since d(x, y) = 1 P (x) and P (y) belong to the closure T_{2} of the same

component of T \T_{1}. Then P ·Π′_{λ}(x) = P ·Π′_{λ}(y) = v for some v ∈ T. Also d(Π_{λ}(x), Π_{λ}(y)) = d(Π_{λ}·Π′_{λ}(x), Π_{λ}·Π′_{λ}(y)) Let x_{1} = Π′_{λ}(x) and y_{1} = Π′_{λ}(y). Let D and C_{1} be as in Lemma 3.1. If d(Π_{λ}(x_{1}), Π_{λ}(y_{1})) ≥ D, let

i_{v}^{−1}(x_{1}) = u_{1},
i_{v}^{−1}(Π_{λ}(x_{1})) = u_{2},
i_{v}^{−1}(y_{1}) = v_{1},
i_{v}^{−1}(Π_{λ}(y_{1})) = v_{2}.

Then by Lemma 3.1 [u_{1}, u_{2}]∪[u_{2}, v_{2}]∪[v_{2}, v_{1}] is a quasigeodesic lying in a C_{1}-neighborhood of [u_{1}, v_{1}].

Also, x_{1}, y_{1} ∈ i_{v}(X_{v}). Since the image of an edge space in a vertex space is C_{2}-quasiconvex, there exist e ∈ T and x_{2}, y_{2}∈f_{e}(X_{e}×{0}) such that d(x_{2}, Π_{λ}(x_{1})) ≤ C_{1} + C_{2} = C and d(y_{2}, Π_{λ}(y_{1})) ≤ C_{1} + C_{2} = C.

By construction d(Π_{λ}(x_{2}), Π_{λ}(y_{2})) ≤ D. (Else the edge P ([x, y]) of T would be in T_{1}). Therefore,

d(Π_{λ}(x), Π_{λ}(y)) = d(Π_{λ}(x_{1}), Π_{λ}(y_{1}))
≤ 2C + D + 2C
= 4C + D

Choosing C_{0} = max {C_{3}, C_{7}, 4C + D}, we are through. □

To complete the proof of our main Theorem, we need a final Lemma.

Lemma 3.7. There exists A > 0, such that if a ∈ P^{−1}(v)∩B_{λ} for some v ∈ T_{1} then there exists b ∈ i(λ) with d(a, b) ≤ Ad_{T} (P a, P b).

Proof: Let μ be a geodesic path from v_{0} to v in T. Order the vertices on μ so that we have a finite sequence v_{0} = y_{0}, y_{1}, ⋯, y_{n} = v such that d_{T} (y_{i}, y_{i+1}) = 1. and d_{T} (v_{0}, v) = n. Recall further, P (B_{λ}) = T_{1}. Hence y_{i} ∈ T_{1}.

Recall that B_{λ} is of the form ⋃_{v∈T_{1}}i_{v}(λ_{v}).

It suffices to prove that there exists A > 0 independent of v such that if p ∈ i_{y_{j}}(λ_{y_{j}}), there exists q ∈ i_{y_{j−1}}(λ_{y_{j−1}}) with d(p, q) ≤ A.

By construction, λ_{y_{j}} = Φ_{y_{j}}(μ) for some geodesic μ in X_{y_{j−1}} such that end-points of μ lie in a C-neighborhood of λ_{y_{j−1}}. Since ϕ_{y_{j}} is a quasi-isometry, there exists C_{1} such that p lies in a C_{1} neighborhood of ϕ_{y_{j}}(q_{0}) for some q_{0}∈μ. Therefore, d(q_{0}, p) ≤ 1 + C.

Also, since end-points of μ lie in a C-neighborhood of λ_{y_{j−1}}, there exists q ∈ i_{y_{j}}(λ_{y_{j−1}}) with d(q_{0}, q) ≤ C_{2} where C_{2} depends only on δ and C. Choosing A = 1 + C + C_{2}, we are through. □

The main theorem of this paper follows:

Theorem 3.8. Let (X,d) be a tree (T) of hyperbolic metric spaces satisfying the qi-embedded condition. Let v be a vertex of T. Then i_{v} : X_{v} → X extends continuously to î_{v} : X̂_{v} → X̂.

Proof: Without loss of generality, let v_{0} = v be the base vertex of T. To prove the existence of a Cannon-Thurston map, it suffices to show (from Lemma 2.1) that for all M ≥ 0 and x_{0} ∈ X_{v} there exists N ≥ 0 such that if a geodesic segment λ lies oustside the N-ball around x_{0} ∈ X_{v}, B_{λ} lies outside the M-ball around i_{v}(x_{0}) ∈ X.

To prove this, we show that if λ lies outside the N-ball around x_{0} ∈ X_{v}, B_{λ} lies outside a certain M(N)-ball around i_{v}(x_{0}) ∈ X, where M(N) is a proper function from ℕ into itself.

Since X_{v} is properly embedded in X there exists f(N) such that i_{v}(λ) lies outside the f(N)−ball around x_{0} in X and f(N) → ∞ as N → ∞.

Let p be any point on B_{λ}. There exists y ∈ i_{v}(λ) such that d(y, p) ≤ Ad_{T} (P y, P p) by Lemma 3.7. Therefore,

d(x_{0}, p) ≥ d(x_{0}, y) − Ad_{T} (P y, P p)
≥ f(N) − Ad_{T} (P (x_{0}), P p)

By our choice of metric on X,

d(x_{0}, p) ≥ d_{T} (P (x_{0}), P p)

Hence

d(x_{0}, p) ≥ max(f(N) − Ad_{T} (P (x_{0}), P p), d_{T} (P (x_{0}), P p))
≥ \frac{f(N)}{A + 1}

¿From Theorem 3.6 there exists C′ independent of λ such that B_{λ} is a C′-quasiconvex set containing i_{v}(λ). Therefore any geodesic joining the end-points of i_{v}(λ) lies in a C′-neighborhood of B_{λ}.

Hence any geodesic joining end-points of i_{v}(λ) lies outside a ball of radius M(N) where

M(N) = \frac{f(N)}{A+1} − C′

Since f(N) → ∞ as N → ∞ so does M(N). □

The following is a direct consequence of Theorem 3.8 above.

Corollary 3.9. Let G be a hyperbolic group acting cocompactly on a simplicial tree T such that all vertex and edge stabilizers are hyperbolic. Also suppose that every inclusion of an edge stabilizer in a vertex stabilizer is a quasi-isometric embedding. Let H be the stabilizer of a vertex or edge of T. Then there exists a Cannon-Thurston map for (H, G).

4. Geometrically Tame Kleinian Groups

In this section we apply Theorem 3.8 to geometrically tame Kleinian groups.

The convex core of a hyperbolic 3-manifold N (without cusps) is the smallest convex submanifold C(N) ⊂ N for which inclusion is a homotopy equivalence. If C(N) has finite volume, N is said to be geometrically finite. There exists a compact 3-dimensional submanifold M ⊂ N, the Scott core [21] whose inclusion is a homotopy equivalence. The ends of N are in one-to-one correspondence with the components of N − M or, equivalently, the components of ∂M. We say that an end of N is geometrically finite if it has a neighborhood missing C(N). An end of N is simply degenerate if it has a neighborhood homeomorphic to S×ℝ, where S is the corresponding component of ∂M, and if there is a sequence of pleated surfaces homotopic in this neighborhood to the inclusion of S, and exiting every compact set. N is called geometrically tame if all of its ends are either geometrically finite or simply degenerate. In particular, N is homeomorphic to the interior of M. For a more detailed discussion of pleated surfaces and geometrically tame ends, see [23] or [18].

Let inj_{N}(x) denote the injectivity radius at x ∈ N. For the purposes of this section, we shall assume that there exists ϵ_{0} > 0 such that inj_{N}(x) > ϵ_{0} for all x ∈ N. In order to apply Theorem 3.8 we need some preliminary Lemmas.

Let E be a simply degenerate end of N. Then E is homeomorphic to S×[0, ∞) for some closed surface S of genus greater than one.

Lemma 4.1. [23] There exists D_{1} > 0 such that for all x ∈ N, there exists a pleated surface g : (S, σ) → N with g(S)∩B_{D_{1}}(x) ≠ ∅.

The following Lemma follows easily from the fact that inj_{N}(x) > ϵ_{0}:

Lemma 4.2. [6],[23] There exists D_{2} > 0 such that if g : (S, σ) → N is a pleated surface, then dia(g(S)) < D_{2}.

The following Lemma due to Minsky [18] follows from compactness of pleated surfaces.

Lemma 4.3. [18] Fix S and ϵ > 0. Given a > 0 there exists b > 0 such that if g : (S, σ) → N and h : (S, ρ) → N are homotopic pleated surfaces which are isomorphisms on π_{1} and inj_{N}(x) > ϵ for all x ∈ N, then

d_{N}(g(S), h(S)) ≤ a ⇒ d_{T eich}(σ, ρ) ≤ b,

where d_{T eich} denotes Teichmuller distance.

Lemma 4.4. There exist K, ϵ and a homeomorphism h from E to the universal curve over a Lipschitz path in Teichmuller space, such that h is a (K, ϵ)-quasi-isometry.

Proof: We can assume that S×{0} is mapped to a pleated surface S_{0} ⊂ N under the homeomorphism from S×[0, ∞) to E. We shall construct inductively a sequence of ‘equispaced’ pleated surfaces S_{i} ⊂ E exiting the end. Assume that S_{0}, ⋯, S_{n} have been constructed such that:

1. If E_{i} be the non-compact component of E\S_{i}, then S_{i+1} ⊂ E_{i}.

2. Hausdorff distance between S_{i} and S_{i+1} is bounded above by 3(D_{1} + D_{2}).

3. d_{N}(S_{i}, S_{i+1}) ≥ D_{1} + D_{2}.

4. From Lemma 4.3 and condition (2) above there exists D_{3} depending on D_{1}, D_{2} and S such that d_{T eich}(S_{i}, S_{i+1}) ≤ D_{3}

Next choose x ∈ E_{n}, such that d_{N}(x, S_{n}) = 2(D_{1} + D_{2}). Then by Lemma 4.1, there exists a pleated surface g : (S, τ) → N such that d_{N}(x, g(S)) ≤ D_{1}. Let S_{n+1} = g(S). Then by the triangle inequality and Lemma 4.2, if p ∈ S_{n} and q ∈ S_{n+1},

D_{1} + D_{2} ≤ d_{N}(p, q) ≤ 3(D_{1} + D_{2}).

This allows us to continue inductively. The Lemma follows. □

Observe that Ẽ is quasi-isometric to a tree (in fact a ray) of hyperbolic metric spaces by setting T = [0, ∞), with vertex set {n : n ∈ ℕ∪{0}}, edge set {[n − 1, n] : n ∈ ℕ}, X_{n} = S̃_{n} = X_{[n−1,n]}. Further, by Lemma 4.3 this tree of hyperbolic metric spaces satisfies the quasi-isometrically embedded condition. We shall now describe { \atop C(N)} as a tree of hyperbolic metric spaces. Assume M⊂C(N) and ∂M = {F_{1}, ⋯, F_{n}} where F_{i} are pleated surfaces in N cutting off ends E_{i}.

Lemma 4.5. [3] π_{1}(N) is hyperbolic in the sense of Gromov. Also, if i : E → N, denotes inclusion, then i_{∗}π_{1}(E) is a quasiconvex subgroup of π_{1}(N).

Remark: In fact there exists a geometrically finite hyperbolic manifold homeomorphic to N. This is part of Thurston’s monster theorem. See [16] for a different proof of the fact. Also, the limit set of a geometrically finite manifold is locally connected [2]. This shall be of use later.

Note that M̃ ⊂ Ñ is quasi-isometric to the Cayley graph of π_{1}(N). Hence, M̃ is a hyperbolic metric space. Let F̃_{i} ⊂ Ñ represent a lift of F_{i} to Ñ. Then, by Lemma 4.5 above, F̃_{i} is a word-hyperbolic metric space. If Ẽ_{i} is a lift of E_{i} containing F̃_{i} then from our previous discussion, Ẽ_{i} is a ray of hyperbolic metric spaces. Since there are only finitely many ends E_{i}, we have thus shown:

Lemma 4.6. The hyperbolic metric space { \atop C(N)} is quasi-isometric to a tree (T) of hyperbolic metric spaces satisfying the qi-embedded condition. Further, we can choose a base vertex v_{0} of T such that X_{v_{0}} is homeomorphic to M̃.

Applying Theorem 3.8, we get

Theorem 4.7. Let Γ be a geometrically tame Kleinian group, such that ℍ^{3}/Γ = M has injectivity radius uniformly bounded below by some ϵ > 0. Then there exists a continuous map from the Gromov boundary of Γ (regarded as an abstract group) to the limit set of Γ in 𝕊_{∞}^{2}.

The above theorem has been independently proven by Klarreich [14] using different techniques.

Lemma 4.8. Let N be a geometrically tame 3-manifold with inj_{N}(x) > ϵ_{0} > 0 for all x ∈ N. Then the Gromov boundary of π_{1}(N) is locally connected.

Proof: This follows from the fact that there exists a geometrically finite manifold M = ℍ^{3}/Γ homeomorphic to N [16] and that for such an M, the limit set of Γ is locally connected [2]. □

Since a continuous image of a compact locally connected set is locally connected, Lemma 4.8 and Theorem 4.7 give:

Corollary 4.9. Let N = ℍ^{3}/Γ be a geometrically tame 3-manifold with inj_{N}(x) > ϵ_{0} for all x ∈ N. Then the limit set of Γ is locally connected.

Lemma 4.4 shows that there exists a quasi-isometry from a lift Ẽ of an end to the universal cover of a universal curve over a Lipschitz path σ in T eich(S). We show further that σ is a Teichmuller quasigeodesic.

It is well known that geodesics in hyperbolic metric spaces diverge exponentially. The following proposition ‘quasi-fies’ this statement:

Proposition 4.10. Given δ, A_{0} ≥ 0 there exist β>1, B > 0, K ≥ 1 and ϵ ≥ 0 such that if [x, y], [y, z] and [z, w] are geodesics in a δhyperbolic metric space (X, d) with (x, z)_{y} ≤ A_{0}, (y, w)_{z} ≤ A_{0} and d(y, z) ≥ B then any path joining x to w and lying outside a Dneighborhood of [y, z] has length greater than or equal to β^{D}d(y, z), where D = min{(d(x, [y, z]) − 1), (d(w, [y, z]) − 1)}.

Lemma 4.11. σ is a Teichmuller quasigeodesic.

Proof: Let S_{0} = ∂E be a pleated surface containing a closed geodesic l of N. This can always be arranged by taking a simple closed geodesic sufficiently far out in E and mapping in a pleated surface containing it [23]. Construct a sequence of equispaced pleated surfaces as in Lemma 4.4. Ẽ is quasi-isometric to a ray of hyperbolic metric spaces X, with vertex set {n : n ∈ ℕ∪{0}}, edge set {[n − 1, n] : n ∈ ℕ}, X_{n} = S̃_{n} = X_{[n−1,n]}.

Fix x_{0} in S_{0}. Inductively, define x_{n} to be the image of x_{n−1} under the Teichmuller map from S_{n−1} to S_{n}. Let r denote a quasi-isometric embedding of [0, ∞) sending [n, n + 1] to the shortest geodesic from x_{n} to x_{n+1}. Then r is a quasigeodesic in E. Let [a, b] be a lift of l to Ẽ. Let λ ⊂ X be the image of [a, b] under a quasi-isometric homeomorphism h between Ẽ and X, sending S̃_{n} to X_{n}. Construct B_{λ} ⊂ X, as in the previous section. Lifts of r through a, b diverge exponentially.

¿From Theorem 3.6, B_{λ} is quasiconvex and hence a hyperbolic metric space with the inherited metric. Let r_{1}, r_{2} be the images of these lifts through the end-points of λ. Then r_{1}, r_{2} are (K, ϵ)-quasigeodesics diverging exponentially in X. Assume, after reparametrization if necessary, r_{1}(n), r_{2}(n) ∈ X_{n}. Let d_{n} denote the path metric on X_{n}. Then by Proposition 4.10, there exist C_{1} > 1, n ≥ 1 such that d_{N+n}(r_{1}(N + n), r_{2}(N + n)) ≥ C_{1}d_{N}(r_{1}(N), r_{2}(N)) for all N ≥ 0. Hence there exists C_{2} such that d_{T eich}(S_{N}, S_{N+n}) ≥ C_{2} for all N ≥ 0. Since σ was shown to be Lipschitz in Lemma 4.4, this proves that σ is a Teichmuller quasigeodesic. □

So far arguments have been coarse. At this stage, we need to quote a part of the main theorem of [18].

Theorem 4.12. [18] If N is a geometrically tame hyperbolic 3-manifold with indecomposable fundamental group, such that there exists ϵ_{0} > 0 with inj_{N}(x) > ϵ_{0} for all x ∈ N, then each simply degenerate end E of N gives rise to a unique Teichmuller ray r, such that every pleated surface in E lies at a uniformly bounded distance from r. Further, r depends only on the corresponding ending lamination.

That r depends only on the corresponding ending lamination was proven by Masur [15].

Combining Lemma 4.11 and Theorem 4.12 we have a new proof of the main theorem of [19] : the ending lamination theorem for 3-manifolds with freely indecomposable fundamental group and a uniform lower bound on injectivity radius.

Theorem 4.13. Let N_{1} and N_{2} be homeomorphic hyperbolic 3-manifolds with freely indecomposable fundamental group. Suppose there exists a uniform lower bound ϵ > 0 on the injectivity radii of N_{1} and N_{2}. If the end invariants of corresponding ends of N_{1} and N_{2} are equal, then N_{1} and N_{2} are isometric.

Proof: From Lemma 4.11, corresponding simply degenerate ends E_{i1}, E_{i2} of N_{1} and N_{2} are homeomorphic via quasi-isometries to universal curves over Teichmuller quasi-geodesics l_{i1} and l_{i2} lying in bounded neighbourhoods of Teichmuller geodesics l_{i}. Hence corresponding ends are homeomorphic via quasi-isometries to each other. Therefore N_{1}, N_{2} are homeomorphic by a quasi-isometry. Finally, by [22] N_{1} and N_{2} are isometric.

5. Examples

Let H be a hyperbolic subgroup of a hyperbolic group G.

Definition : [12] [9] If i : Γ_{H} → Γ_{G} be an embedding of the Cayley graph of H into that of G, then the distortion function is given by

disto(R) = R^{−1}Diam_{Γ_{0}}(Γ_{0}∩B(R)),

where B(R) is the ball of radius R around 1 ∈ Γ_{G}.

All previously known examples of non-quasiconvex hyperbolic subgroups of hyperbolic groups exhibit exponential distortion. We construct in this section some examples exhibiting greater distortion. Some of these will be shown to have Cannon-Thurston maps. For the rest, existence of Cannon-Thurston maps is not yet known. Further, we shall describe certain examples of free subgroups of P SL_{2}ℂ and show that they exhibit arbitrarily large distortion. The existence of Cannon-Thurston maps for some of these is not yet known.

Our starting point for constructing distorted subgroups of hyperbolic groups is the following Lemma of Bestvina, Feighn and Handel [4]: Lemma 5.1. [4] There exists a hyperbolic group G such that 1 → F → G → F → 1 is exact, where F is free of rank 3.

Let F_{1} ⊂ G denote the normal subgroup. Let F_{2} ⊂ G denote a section of the quotient group. Let G_{1}, ⋯, G_{n} be n distinct copies of G. Let F_{i1} and F_{i2} denote copies of F_{1} and F_{2} respectively in G_{i}. Let

G = G_{1}∗_{H_{1}}G_{2} ∗ ⋯∗_{H_{n−1}}G_{n}

where each H_{i} is a free group of rank 3, the image of H_{i} in G_{i} is F_{i2} and the image of H_{i} in G_{i+1} is F_{(i+1)1}. Then G is hyperbolic. This follows inductively from the fact that the image of H_{i} in G_{i} is quasiconvex in G_{1}∗_{H_{1}}G_{2} ∗ ⋯∗_{H_{i−1}}G_{i} and the main combination theorem of [3].

Let H = F_{11} ⊂ G. Then the distortion of H is superexponential for n > 1. In fact, it can be readily checked that the distortion function is an iterated exponential of height n.

Note further that G_{1}∗_{H_{1}}G_{2} can be regarded as a graph of groups with one vertex and three edges, where the vertex group is G_{1} and edge groups are isomorphic to F. Then from Corollary 3.9, the pair (G_{1}, G_{1}∗_{H_{1}}G_{2}) has a Cannon-Thurston map. Proceeding inductively and observing that a composition of Cannon-Thurston maps is a Cannon-Thurston map, we see that (H, G) has a Cannon-Thurston map.

The next class of examples are not known to have Cannon-Thurston maps:

Our starting point is again Lemma 5.1. Let a_{1}, a_{2}, a_{3} be generators of F_{1} and b_{1}, b_{2}, b_{3} be generators of F_{2}. Then

G = {a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} : b_{i}^{−1}a_{j}b_{i} = w_{ij}}

where w_{ij} are words in a_{i}’s. We add a letter c conjugating a_{i}’s to ‘sufficiently random’ words in b_{j}’s to get G_{1}. Thus,

G_{1} = {a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, c : b_{i}^{−1}a_{j}b_{i} = w_{ij}, c^{−1}a_{i}c = v_{i}}, where v_{i}’s are words in b_{j}’s satisfying a small-cancellation type condition to ensure that G_{1} is hyperbolic. See [13], pg. 151 for details on addition of ‘random’ relations.

It can be checked that these examples have distortion function greater than any iterated exponential.

The above set of examples were motivated largely by examples of distorted cyclic subgroups in [12], pg. 67.

So far, there is no satisfactory way of manufacturing examples of hyperbolic subgroups of hyperbolic groups exhibiting arbitrarily high distortion. It is easy to see that a subgroup of sub-exponential distortion is quasiconvex [12]. Not much else is known. For instance, one does not know if A^{n^{2}} can appear as a distortion function.

The situation is far more satisfactory in the case of Kleinian groups. The following class of examples appears in work of Minsky [17]:

Let S be a hyperbolic punctured torus so that the two shortest geodesics a and b are orthogonal and of equal length. Let S_{0} denote S minus a neighborhood of the cusp. Let N_{δ}(a) and N_{δ}(b) be regular collar neighborhoods of a and b in S_{0}. For n ∈ ℕ, define γ_{n} = a if n is even and equal to b if n is odd. Let T_{n} be the open solid torus neighborhood of γ_{n}×\{n + \frac{1}{2}\} in S_{0}×[0, ∞) given by

T_{n} = N_{δ}(γ_{n})×(n, n + 1)

and let M_{0} = (S_{0})×[0, ∞)\⋃_{n∈ℕ}T_{n}.

Let a(n) be a sequence of positive integers greater than one. Let γ̂_{n} = γ_{n}×{n} and let μ_{n} be an oriented meridian for ∂T_{n} with a single positive intersection with γ̂_{n}. Let M denote the result of gluing to each ∂T_{n} a solid torus T̂_{n}, such that the curve γ̂_{n}^{a(n)}μ_{n} is glued to a meridian. Let q_{nm} be the mapping class from S_{0} to itself obtained by identifying S_{0} to S_{0}×m, pushing through M to S_{0}×n and back to S_{0}. Then q_{n(n+1)} is given by Φ_{n} = D_{γ_{n}}^{a(n)}, where D_{c}^{k} denotes Dehn twist along c, k times. Matrix representations of Φ_{n} are given by

Φ_{2n} = ( \begin{matrix}1 & a(2n) \\ 0 & 1 \end{matrix} )

and

Φ_{2n+1} = ( \begin{matrix}1 & 0 \\ a(2n + 1) & 1 \end{matrix} )

Recall that the metric on M_{0} is the restriction of the product metric. T̂_{n}’s are given hyperbolic metrics such that their boundaries are uniformly quasi-isometric to ∂T_{n} ⊂ M_{0}. Then from [17], M is quasiisometric to the complement of a rank one cusp in the convex core of a hyperbolic manifold M_{1} = ℍ^{3}/Γ. Let σ_{n} denote the shortest path from S_{0}×1 to S_{0}×n. Let \overline{σ_{n}} denote σ_{n} with reversed orientation. Then τ_{n} = σ_{n}γ_{n}\overline{σ_{n}} is a closed path in M of length 2n + 1. Further τ_{n} is homotopic to a curve ρ_{n} = Φ_{1}⋯Φ_{n}(γ_{n}) on S_{0}. Then

Π_{i=1···n}a(i) ≤ l(ρ_{n}) ≤ Π_{i=1···n}(a(i) + 2)

Hence Π_{i=1···n}a(i) ≤ (2n + 1)disto(2n + 1) ≤ Π_{i=1···n}(a(i) + 2)

Since M is quasi-isometric to the complement of the cusp of a hyperbolic manifold and γ_{n}’s lie in a complement of the cusp, the distortion function of Γ is of the same order as the distortion function above. In particular, functions of arbitraily fast growth may be realised. This answers a question posed by Gromov [12] pg. 66.

Manifolds with unbounded a(n)’s are not known to have Cannon-Thurston maps.

Acknowledgements: The author would like to thank his advisor Andrew Casson for helpful comments and Curt Mcmullen for pointing out the results of [2].

References

[1] J. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short. Notes on word hyperbolic groups. Group Theory from a Geometrical Viewpoint (E. Ghys, A. Haefliger, A. Verjovsky eds.), pages 3–63, 1991.

[2] J. Anderson and B. Maskit. Local connectivity of limit sets of Kleinian groups. preprint.

[3] M. Bestvina and M. Feighn. A Combination theorem for Negatively Curved Groups. J. Diff. Geom., vol 35, pages 85–101, 1992.

[4] M. Bestvina, M. Feighn, and M. Handel. Laminations, trees and irreducible automorphisms of free groups. preprint.

[5] F. Bonahon. Geodesic Currents on Negatively Curved Groups. in Arboreal Group Theory (ed. R.C. Alperin), MSRI Publ., vol. 19, Springer Verlag, 1991, pages 143–168.

[6] F. Bonahon. Bouts de varietes hyperboliques de dimension 3. Ann. Math. vol.124, pages 71–158, 1986.

[7] J. Cannon and W. P. Thurston. Group Invariant Peano Curves. preprint.

[8] M. Coornaert, T. Delzant, and A.Papadopoulos. Geometrie et theorie des groupes. Lecture Notes in Math.,vol.1441,Springer Verlag, 1990.

[9] B. Farb. The extrinsic geometry of subgroups and the generalized word problem. Proc. LMS (3) 68, pages 577–593, 1994.

[10] W. J. Floyd. Group Completions and Limit Sets of Kleinian Groups. Invent. Math. vol.57, pages 205–218, 1980.

[11] E. Ghys and P. de la Harpe(eds.). Sur les groupes hyperboliques d’apres Mikhael Gromov. Progress in Math. vol 83, Birkhauser, Boston Ma., 1990.

[12] M. Gromov. Asymptotic Invariants of Infinite Groups. in Geometric Group Theory,vol.2; Lond. Math. Soc. Lecture Notes 182 (1993), Cambridge University Press.

[13] M. Gromov. Hyperbolic Groups. in Essays in Group Theory, ed. Gersten, MSRI Publ.,vol.8, Springer Verlag,1985, pages 75–263.

[14] E. Klarreich. PhD Thesis, SUNY, Stonybrook. in preparation.

[15] H. Masur. Hausdorff dimension of the set of nonergodic foliations of a quadratic differential. Duke Math. J., 66:387–442, 1992.

[16] C. McMullen. Iteration on teichmüller space. Invent. math., 99:425–454, 1990.

[17] Y. N. Minsky. The Classification of Punctured Torus Groups. preprint.

[18] Y. N. Minsky. Teichmuller Geodesics and Ends of 3-Manifolds. Topology, pages 1–25, 1992.

[19] Y. N. Minsky. On Rigidity, Limit Sets, and End Invariants of Hyperbolic 3-Manifolds. J. A.M.S., vol.7, pages 539–588, 1994.

[20] M. Mitra. Cannon-Thurston Maps for Hyperbolic Group Extensions. preprint.

[21] P. Scott. Compact submanifolds of 3-manifolds. Journal L.M.S., pages 246– 250, 1973.

[22] D. Sullivan. On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions. Riemann Surfaces and Related Topics: Proceedings of the 1978 Stonybrook Conference, Ann. of Math. Studies 97, Princeton, 1981.

[23] W. P. Thurston. The Geometry and Topology of 3-Manifolds. Princeton University Notes.

Address : Department of Mathematics, University of California, Berkeley, CA 94720, USA.

email : mitra@math.berkeley.edu