## Cannon-Thurston maps for graphs of groups

Recall from our last post that given an infinite hyperbolic subgroup ${H}$ of a hyperbolic group ${G}$, the extension of the inclusion ${i:H \hookrightarrow G}$ to a continuous ${\hat{i}: \partial H \rightarrow \partial G}$ is called a Cannon-Thurston map. By the continuity of ${i}$, if such a map exists it’s unique.

When ${H}$ is quasi-convex in ${G}$, clearly a Cannon-Thurston map exists: equivalent geodesics in ${H}$ are mapped to equivalent quasi-geodesics in ${G}$. However, the general question of when a Cannon-Thurston map exists if ${H}$ is distorted in ${G}$ is still open. In this post we’ll give three partial results directed at answering this question: the original result of Cannon and Thurston, and two generalizations made by Mitra.

1. Three results

1. Cannon-Thurston:
Given ${G \cong \pi_1(M)}$, where ${M}$ is a hyperbolic closed 3-manifold fibering over a circle, and ${H = \pi_1(S)}$, where ${S}$, a closed 2-manifold, is the fiber surface, there is a Cannon-Thurston map ${\hat{i}: \partial H \rightarrow \partial G}$.

2. Mitra:
Suppose $H$ is an infinite hyperbolic normal subgroup of a hyperbolic group $G$. Then there exists a Cannon-Thurston map ${\hat{i}: \partial H \rightarrow \partial G}$.

3. Mitra:
Let ${\Gamma}$ be a finite graph of groups where all vertex groups ${\Gamma_v}$ and edge groups ${\Gamma_e}$ are hyperbolic, and all injections ${\Gamma_e \rightarrow \Gamma_v}$ (where ${e}$ is incident to ${v}$) are quasi-isometric embeddings. Suppose that ${G = \pi_1(\Gamma)}$ is also hyperbolic. Then for any vertex ${v \in \Gamma}$, there is a Cannon-Thurston map ${\partial \Gamma_v \rightarrow \partial G}$.

2. Key lemma

Both of Mitra’s results above make use of the following key lemma.

Lemma 1. Let ${i:Y \hookrightarrow X}$ be a proper embedding of proper hyperbolic spaces. Suppose, given ${y_0 \in Y}$, that there is a non-negative function ${M(N)}$ such that ${M(N) \rightarrow \infty}$ as ${N \rightarrow \infty}$ and for all geodesic segments ${\lambda \subset Y}$ outside an ${N}$-ball around ${y_0 \in Y}$, any geodesic in ${X}$ joining endpoints of ${i(\lambda)}$ is outside the ${M(N)}$-ball around ${i(y_0)}$ (see the illustration below). Then there’s a Cannon-Thurston map ${\hat{i}: \partial Y \rightarrow \partial X}$.

Proof: If ${i: Y \rightarrow X}$ does not extend continuously, there exist sequences ${(x_m), (y_m)}$ in ${Y}$ such that ${x_m \rightarrow p}$ and ${y_m \rightarrow p}$ for some ${p \in \partial Y}$, but ${i(x_m) \rightarrow u}$ and ${i(y_m) \rightarrow v}$ for ${v \neq u}$ in ${\partial X}$.

Note that ${d_Y(y_0, [x_m,y_m]) \rightarrow \infty}$ as ${m \rightarrow \infty}$. On the other hand, for ${m}$ large enough, ${d_X(i(y_0), [i(x_m),i(y_m)])}$ stays uniformly close to ${d_X(i(y_0),[u,v]) > 0}$ (see the discussion of visibility in the previous post). This contradicts the existence of the function ${M}$. $\Box$

3. Trees of hyperbolic spaces

We’ll sketch part of Mitra’s proof of result number 3 above in the original context in which he established it: that of trees of hyperbolic spaces.

A tree ${T}$ of hyperbolic metric spaces is a metric space ${(X,d)}$ admitting a map ${P: X \rightarrow T}$ onto a simplicial tree ${T}$ such that there exist constants ${\delta, \epsilon, K}$ satisfying

1. For all vertices ${v \in T}$, ${X_v := P^{-1}(v) \subset X}$ is path connected, rectifiable, and ${\delta}$-hyperbolic. We denote the induced path metric by ${d_v}$.
2. All inclusions ${X_v \hookrightarrow X}$ are uniformly proper. That is, there exists a function ${f: \mathbb{N} \rightarrow \mathbb{N}}$ such that for all ${x, y \in X_v}$, if ${d(x,y) \le n}$, then ${d_v(x, y) \le f(n)}$.
3. For all edges ${e \in T}$, ${X_e := P^{-1}(e) \subset X}$ is path connected and rectifiable. The induced path metric is denoted ${d_e}$.
4. For every edge ${e \in T}$ there exists a function ${f_e: X_e \times [0,1] \rightarrow X}$ such that ${f_e|_{X_e \times (0,1)}}$ is an isometry onto ${P^{-1}(\mathrm{int}(e))}$.
5. ${f_e|_{X_e \times \{0\}}}$ and ${f_e|_{X_e \times \{1\}}}$ are ${(K, \epsilon)}$-quasi-isometric embeddings into ${X_{v_1}}$ and ${X_{v_2}}$, where ${e}$ connects ${v_1}$ and ${v_2}$.

[Note: although present in Mitra’s definition, we don’t think condition 2 above is necessary. It is equivalent to

2′. There exists a function ${g: \mathbb{N} \rightarrow \mathbb{N}}$ such that for any ${x, y \in X_v}$, if ${d_v(x,y) \ge n}$, then ${d(x, y) \ge g(n)}$, and $g(n) \rightarrow \infty$ as $n \rightarrow \infty$.

This, however, follows from the structure of the space ${X}$. Let ${D=d_v(x,y)}$. In order to shortcut an ${X_v}$-geodesic from ${x}$ to ${y}$, we can’t use a path that travels to more than ${D/2}$ other vertex spaces ${X_w}$, since the distance between such subspaces is at least 1. The length of a path in ${X}$ between ${x}$ and ${y}$ is bounded (up to a constant) below by $\mathrm{min}_{n < D/2} (2n + D/K^n)$ which goes to infinity like $\log(D)$ as $D$ increases.]

One example of a tree of hyperbolic spaces is the “discrete hyperbolic plane.” It’s constructed by gluing squares of side length 1 in as illustrated in the figure below.

The vertical lines correspond to vertex subspaces ${X_v}$, and each ${X_e}$ is glued to the upper vertex subspace by an isometry and to the lower vertex subspace by a ${(2,0)}$-quasi-isometry stretching the metric by a factor of 2.

Theorem 2. Let ${(X, d, T)}$ be a tree of hyperbolic metric spaces. If ${X}$ is hyperbolic, then there exists a Cannon-Thurston map for all ${X_v \hookrightarrow X}$.

The structure of the proof is as follows. Given a geodesic segment ${\lambda}$ in ${X_v}$, a quasiconvex set ${B_\lambda \subset X}$ is constructed containing ${\lambda}$ and satisfying the condition that if ${\lambda}$ is outside an ${N}$-ball around ${y_0 \in X_v}$, then ${B_\lambda}$ is outside an ${M}$-ball around ${i(y_0)}$. As ${N \rightarrow \infty}$, so does ${M \rightarrow \infty}$. Applying the key lemma above finishes the proof.

This proof has the same structure as the proof of Mitra’s other result on normal hyperbolic subgroups. The difference lies in the construction of ${B_\lambda}$.

4. Constructing ${B_\lambda}$

Below is a morally correct, albeit simplified, description of how one constructs ${B_\lambda}$ for a given ${X_v}$-geodesic ${\lambda}$ in stages by starting with ${\lambda}$ and adding pieces of other vertex spaces one by one.

Consider the images of ${f_e}$ intersecting ${\lambda}$ for every edge ${e \in T}$ incident to ${v}$. If the intersection is large enough (this constant depends on a number of other constants defining ${X}$), add the geodesic connecting the farthest points of the preimage of ${\lambda}$ in the neighboring vertex space to ${B_\lambda}$. Now take the resulting geodesic segments in (some of) the spaces ${X_w}$, where ${w}$ is distance 1 away from ${v}$ in ${T}$, and apply the same procedure, without backtracking to ${v}$. Propagate throughout the tree ${T}$ without backtracking to previously visited vertices. See the illustration below.

The bulk of Mitra’s proof consists of showing that the resulting ${B_\lambda}$ is indeed quasi-convex in ${X}$.

As an exercise, pick a ${\lambda}$ in one of vertex spaces of the discrete hyperbolic plane described above, construct a ${B_\lambda}$, and convince yourself that it’s quasi-convex, and that it’s far from a given basepoint ${y_0}$ when ${\lambda}$ itself is.

5. From trees of spaces to graphs of groups

How does Theorem 2 imply the graph-of-groups formulation? Given a graph of groups $\Gamma$ with $G=\pi_1(\Gamma)$, consider its universal cover $\tilde{\Gamma}$ (with the corresponding vertex and edge group assignments, so that $\pi_1(\tilde{\Gamma}) = G$; see below for an example).

Taking Cayley graphs of vertex groups, and gluing them using (Cayley graphs of edge groups) $\times [0,1]$ and the corresponding quasi-isometric embeddings, we get a tree of hyperbolic spaces satisfying the hypotheses of Mitra’s theorem.