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

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 $\mathrm{\partial }Y\to \mathrm{\partial }X$ 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 $G=G_1{\ast }_H{G}_2$ is hyperbolic. Suppose $K_i<G_i$, $i=1,2$ are hyperbolic subgroups containing $H$ and $K=K_1{\ast }_H{K}_2$. 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.}