Cannon-Thurston maps for trees of hyperbolic metric spaces

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 mapî : X v → X. 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.

Super-exponential distortion of subgroups of CAT(−1) groups

“…1/ groups containing free subgroups with arbitrary iterated exponential distortion, and distortion higher than any iterated exponential. The construction parallels that of Mahan Mitra in [2] but our groups are the fundamental groups of locally CAT. 1/ 2-complexes.…”

“…This makes it easy to see that the exponential distortions compose as required. Also, the example in [2] with distortion higher than any iterated exponential is of the form .F 3 Ì F 3 / Ì ‫;ޚ‬ with the generator of ‫ޚ‬ conjugating the generators of the first F 3 to "sufficiently random" words in the generators of the second F 3 . In contrast, our group can be described explicitly, without recourse to random words, allowing for an explicit check that our group is CAT.…”

“…1/ fashion. One of our motivations for producing these examples was that it was not immediate from the description that the examples in [2] had the appropriate iterated exponential distortions. Mitra's examples are graphs of groups with underlying graph a segment of length n, where the vertex groups are hyperbolic F 3 Ì F 3 groups, and each edge identifies the kernel F 3 in one vertex group with the second F 3 factor in the adjacent vertex group.…”

“…1/ 2-complexes. The building blocks used in [2] are hyperbolic F 3 Ì F 3 groups, which are not known to be CAT.0/. Our building blocks are graphs of groups where the vertex and edge groups are all free groups of equal rank and the underlying graph is a bouquet of a finite number of circles.…”

Primitive stable representations of free Kleinian groups

“…Starting with the pioneering work of Cannon and Thurston [7] for closed 3-manifolds fibring over a circle, Cannon-Thurston maps have been generalised in several ways by Bowditch [3], Klarreich [14], McMullen [22], Minsky [23] and Mj [26,27,28,29,30,31] (see also [25], [36]). Mj proved the existence of Cannon-Thurston maps for Kleinian surface groups in [29], and described the points identified by Cannon-Thurston maps in [30](see [33] for the case of punctured surfaces).…”