Boundaries of coned-off hyperbolic spaces

TL;DR

This paper investigates the boundaries of cone-off hyperbolic spaces, showing how their Gromov boundary relates to the original space's boundary and characterizing group elements acting on these cone-offs.

Contribution

It extends previous work by providing a detailed description of the Gromov boundary embedding and characterizing group elements under acylindricity assumptions.

Findings

Gromov boundary of cone-off embeds in original boundary

Under acylindricity, the boundary image is precisely described

Characterization of elliptic and loxodromic elements in group actions

Abstract

Coning off a collection of uniformly quasiconvex subsets of a Gromov hyperbolic space leaves a new space, called the cone-off. Kapovich and Rafi generalized work of Bowditch to show this space is still Gromov hyperbolic. We show that the Gromov boundary of cone-off embeds in the boundary of the original hyperbolic space. (A stronger version of this result was previously obtained by Dowdall and Taylor; see Note in text.) Moreover, under some acylindricity assumptions we give a precise description of the image. As an application, we are able to characterize the elliptic and loxodromic elements of groups acting on certain cone-offs of acylindrical actions.