Embeddings of trees of hyperbolic spaces and Cannon--Thurston maps
Rakesh Halder, Pranab Sardar
TL;DR
This work develops a geometric framework of trees of hyperbolic spaces to study CT maps for subgroups under amalgamation and graph-of-groups constructions. It proves that if each vertex inclusion $K_i o G_i$ admits a CT map and a projection hypothesis holds across the edge spaces, then the amalgamated inclusion $K=K_1*_H K_2 o G=G_1*_H G_2$ also admits a CT map, with extensions to HNN extensions and subgraphs of groups. The approach relies on flow spaces, Mitra-type retractions, and boundary flow to propagate quasiconvexity and boundary data through the tree, yielding a main CT theorem and associated lamination results. The paper also surveys applications, provides examples showing the necessity of hypotheses in some cases, and discusses nonexistence results to delineate the boundaries of CT-map existence in graph-of-hyperbolic-group contexts.
Abstract
Suppose $G_1$ and $G_2$ are hyperbolic groups with a common quasiconvex subgroup $H$, and the free product with amalgamation $G = G_1 *_H G_2$ is hyperbolic. Let $K_i < G_i$, $i = 1,2$, be quasiconvex subgroups containing $H$, and let $K = K_1 *_H K_2$. It follows from the work of Bestvina and Feighn that $K$ is hyperbolic, and it follows from the work of the second named author with M. Kapovich that the inclusion $K \to G$ extends continuously to the (Gromov) boundary. The present paper arose from the question of whether the same conclusion holds when $K_i < G_i$ are no longer quasiconvex but merely hyperbolic. As an application of the main result of this paper, we answer this question affirmatively, provided each inclusion $K_i \to G_i$ extends continuously to the boundary.