Subdivision rules for all Gromov hyperbolic groups

TL;DR

This paper demonstrates that all Gromov hyperbolic groups can be represented by finite subdivision rules on the 3-sphere, providing a new way to encode their geometric and quasi-isometric properties.

Contribution

It extends previous work by showing that hyperbolic groups can be described via subdivision rules, including non-cubulated groups, on the 3-sphere.

Findings

Hyperbolic groups can be described by finite subdivision rules.

Provides a boundary-like sequence of cell complexes encoding group properties.

Includes non-cubulated hyperbolic groups in the subdivision framework.

Abstract

This paper shows that every Gromov hyperbolic group can be described by a finite subdivision rule acting on the 3-sphere. This gives a boundary-like sequence of increasingly refined finite cell complexes which carry all quasi-isometry information about the group. This extends a result from Cannon and Swenson in 1998 that hyperbolic groups can be described by a recursive sequence of overlapping coverings by possibly wild sets, and demonstrates the existence of non-cubulated groups that can be represented by subdivision rules.