Drilling hyperbolic groups

TL;DR

This paper introduces a method called drilling for hyperbolic groups, transforming them into relatively hyperbolic groups with sphere boundaries, which could simplify approaches to the Cannon Conjecture.

Contribution

It defines a new drilling procedure for hyperbolic groups and shows how it can produce relatively hyperbolic groups with sphere boundaries, aiding the study of the Cannon Conjecture.

Findings

Drilling can produce relatively hyperbolic groups with $S^2$ boundary.

Reduces the Cannon Conjecture to a relative version.

Applicable under certain conditions for hyperbolic groups with sphere boundary.

Abstract

Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable.