Landing rays and ray Cannon-Thurston maps

Abstract

In this paper, we describe a procedure to construct pairs of hyperbolic groups $H<G$ with the following properties. 1) Every geodesic ray $\gamma$ in $H$ converges to a point $\xi_{\gamma}\in \partial G$. 2) The inclusion of $H$ into $G$ does not extend continuously to $\partial H \to \partial G$. In other words, a Cannon--Thurston map does not exist for this pair of hyperbolic groups. Jeon, Kapovich, Leininger and Ohshika gave a property of conical limit points in the presence of a Cannon--Thurston map. We convert this into a criterion for the existence of Cannon--Thurston maps and use it to prove the non-existence result in (2). We obtain, in particular, a geometric proof of Baker--Riley's counterexample.