Boundaries of Dehn fillings

TL;DR

This paper explores how Dehn fillings affect the boundaries of relatively hyperbolic groups, showing many fillings preserve boundary properties and linking the Cannon conjecture to a relative version.

Contribution

It introduces the study of boundary behavior under Dehn filling for toral relatively hyperbolic groups and connects the Cannon conjecture to a relative setting.

Findings

Many Dehn fillings produce hyperbolic groups with 2-sphere boundary

Dehn fillings can preserve boundary topologies in relatively hyperbolic groups

The Cannon conjecture implies a relative version of itself

Abstract

We begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular we show many Dehn fillings of a toral relatively hyperbolic group with 2-sphere boundary are hyperbolic with 2-sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture.