Embeddings of trees of hyperbolic metric spaces and Cannon--Thurston maps

TL;DR

This paper investigates conditions for the existence of Cannon--Thurston maps in trees of hyperbolic spaces and applies these results to graphs of hyperbolic groups, providing new insights into their boundary maps.

Contribution

It establishes new sufficient conditions for the existence of Cannon--Thurston maps in trees of hyperbolic spaces and applies these to graphs of hyperbolic groups, including amalgamated free products.

Findings

Additional conditions for CT map existence are identified.

Examples show CT map failure in general cases.

Results apply to hyperbolic group amalgamations.

Abstract

Given a tree of hyperbolic metric spaces $\pi:X\to T$ a la Bestvina--Feighn (\cite{BF}), and a hyperbolic subspace $Y$ of $X$ with an induced tree of hyperbolic spaces structure over a subtree $S\subset T$, we address the question as to when the Cannon--Thurston (CT) map exists for the inclusion $Y\to X$. In this paper, we find additional sufficient conditions under which the CT map $\partial Y \to \partial X$ exists. However, we show with examples that this may fail to hold in general. These results about trees of spaces are then applied to graphs of hyperbolic groups to prove various existence results for CT maps. A very special instance of these results is the following: \emph{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. Suppose $K_i < G_i$, $i = 1,2$ are hyperbolic subgroups containing $H$ and $K=K_1*_H K_2$. Then $K$ (is hyperbolic and,) the inclusion $K\to G$ admits a CT map if the inclusions $K_i\to G_i$, $i=1,2$ admit CT maps.}