Recall from our last post that given an infinite hyperbolic subgroup of a hyperbolic group , the extension of the inclusion to a continuous is called a Cannon-Thurston map. By the continuity of , if such a map exists it’s unique.
When is quasi-convex in , clearly a Cannon-Thurston map exists: equivalent geodesics in are mapped to equivalent quasi-geodesics in . However, the general question of when a Cannon-Thurston map exists if is distorted in 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
Given , where is a hyperbolic closed 3-manifold fibering over a circle, and , where , a closed 2-manifold, is the fiber surface, there is a Cannon-Thurston map .
Suppose is an infinite hyperbolic normal subgroup of a hyperbolic group . Then there exists a Cannon-Thurston map .
Let be a finite graph of groups where all vertex groups and edge groups are hyperbolic, and all injections (where is incident to ) are quasi-isometric embeddings. Suppose that is also hyperbolic. Then for any vertex , there is a Cannon-Thurston map .
2. Key lemma
Both of Mitra’s results above make use of the following key lemma.
Lemma 1. Let be a proper embedding of proper hyperbolic spaces. Suppose, given , that there is a non-negative function such that as and for all geodesic segments outside an -ball around , any geodesic in joining endpoints of is outside the -ball around (see the illustration below). Then there’s a Cannon-Thurston map .
Proof: If does not extend continuously, there exist sequences in such that and for some , but and for in .
Note that as . On the other hand, for large enough, stays uniformly close to (see the discussion of visibility in the previous post). This contradicts the existence of the function .
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 of hyperbolic metric spaces is a metric space admitting a map onto a simplicial tree such that there exist constants satisfying
- For all vertices , is path connected, rectifiable, and -hyperbolic. We denote the induced path metric by .
- All inclusions are uniformly proper. That is, there exists a function such that for all , if , then .
- For all edges , is path connected and rectifiable. The induced path metric is denoted .
- For every edge there exists a function such that is an isometry onto .
- and are -quasi-isometric embeddings into and , where connects and .
[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 such that for any , if , then , and as .
This, however, follows from the structure of the space . Let . In order to shortcut an -geodesic from to , we can’t use a path that travels to more than other vertex spaces , since the distance between such subspaces is at least 1. The length of a path in between and is bounded (up to a constant) below by which goes to infinity like as 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 , and each is glued to the upper vertex subspace by an isometry and to the lower vertex subspace by a -quasi-isometry stretching the metric by a factor of 2.
Theorem 2. Let be a tree of hyperbolic metric spaces. If is hyperbolic, then there exists a Cannon-Thurston map for all .
The structure of the proof is as follows. Given a geodesic segment in , a quasiconvex set is constructed containing and satisfying the condition that if is outside an -ball around , then is outside an -ball around . As , so does . 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 .
Below is a morally correct, albeit simplified, description of how one constructs for a given -geodesic in stages by starting with and adding pieces of other vertex spaces one by one.
Consider the images of intersecting for every edge incident to . If the intersection is large enough (this constant depends on a number of other constants defining ), add the geodesic connecting the farthest points of the preimage of in the neighboring vertex space to . Now take the resulting geodesic segments in (some of) the spaces , where is distance 1 away from in , and apply the same procedure, without backtracking to . Propagate throughout the tree without backtracking to previously visited vertices. See the illustration below.
The bulk of Mitra’s proof consists of showing that the resulting is indeed quasi-convex in .
As an exercise, pick a in one of vertex spaces of the discrete hyperbolic plane described above, construct a , and convince yourself that it’s quasi-convex, and that it’s far from a given basepoint when 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 with , consider its universal cover (with the corresponding vertex and edge group assignments, so that ; see below for an example).
Taking Cayley graphs of vertex groups, and gluing them using (Cayley graphs of edge groups) and the corresponding quasi-isometric embeddings, we get a tree of hyperbolic spaces satisfying the hypotheses of Mitra’s theorem.