Landing rays and ray Cannon-Thurston maps

TL;DR

This work extends the notion of landing rays from complex dynamics to hyperbolic subgroups of hyperbolic groups, and furnishes sufficient criteria for when the embedding of a subgroup $G_1$ into a hyperbolic group $G$ yields or precludes a Cannon-Thurston map. Central to the approach is a strong JKLO-type criterion and a lamination framework that lets one detect nonexistence of CT maps even when every ray in $G_1$ lands (i.e., a ray-CT map exists). The authors construct several classes of examples—normal subgroups, commensurated subgroups, and endomorphism-driven extensions—where all rays land yet no global CT map exists, and they identify precisely the discontinuity set of ray-CT maps. The results illuminate the subtle boundary behavior of subgroup inclusions in hyperbolic groups and connect coarse geometric phenomena (distortion, lamination structure) with asymptotic geometry, offering a versatile toolkit for generating and understanding non-CT pairs with rich boundary dynamics.

Abstract

For a hyperbolic subgroup H of a hyperbolic group G, we describe sufficient criteria to guarantee the following. 1) Geodesic rays in H starting at the identity land at a unique point of the boundary of G. 2)The inclusion of H into G does not extend continuously to the boundary. As a consequence we obtain sufficient conditions that provide a mechanism to guarantee the non-existence of Cannon-Thurston maps. One such criterion we use extensively is an adaptation of a property proven by Jeon, Kapovich, Leininger and Ohshika. As a consequence we describe a number of classes of examples demonstrating the non-existence of Cannon-Thurston maps. We recover, in the process, a simple counter-example lying at the heart of Baker and Riley's examples.

Theorems & Definitions (115)

• Example 1.2: Basic Example:
• Theorem 1.3
• Lemma 1.4
• Lemma 1.6
• Proposition 1.8
• Example 1.9
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Definition 2.4