Conical limit sets of hyperbolic subgroups

TL;DR

This paper characterizes conical limit points of hyperbolic subgroups in terms of the Cannon-Thurston map and shows the existence of non-conical limit points when the subgroup is not quasi-convex.

Contribution

It provides a precise description of conical limit points via the Cannon-Thurston map and explores the structure of limit sets for non-quasi-convex subgroups.

Findings

Conical limit points are exactly the points where the Cannon-Thurston map is injective.

When the subgroup is not quasi-convex, non-conical limit points exist.

The study links the geometric properties of subgroups to their limit sets in hyperbolic groups.

Abstract

Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $\Lambda_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $\Lambda_H$ consisting of the points to which the Cannon-Thurston map $\hat i$ injects. Moreover, we show that when $H$ is not quasi-convex in $G$, there exists a non-conical limit point in $\Lambda_H$