Big Tiles in Hyperbolic Groups

TL;DR

This paper proves that in any word hyperbolic group, for any finite subset, there exists a tile containing it, advancing understanding of the geometric structure of such groups.

Contribution

It establishes the existence of tiles containing arbitrary finite subsets in hyperbolic groups, a novel result in geometric group theory.

Findings

Existence of tiles containing any finite subset in hyperbolic groups

Advancement in understanding the geometric structure of hyperbolic groups

Provides tools for further exploration of tilings in geometric group theory

Abstract

We prove that if $\Gamma $ is a word hyperbolic group and $K$ is a finite subset of $\Gamma $, then $\Gamma $ admits a tile containing $K$.