Subdivision rules for special cubulated groups

TL;DR

This paper introduces explicit subdivision rules for special cubulated groups, enabling the encoding of quasi-isometric properties through tilings on a sphere, and demonstrating their effectiveness in detecting various geometric group properties.

Contribution

The paper provides the first explicit subdivision rules for all special cubulated groups, linking tilings to key geometric and algebraic properties.

Findings

Subdivision rules encode quasi-isometric information

Tilings detect properties like growth and divergence

Examples illustrate the application of subdivision rules

Abstract

We find explicit subdivision rules for all special cubulated groups. A subdivision rule for a group produces a sequence of tilings on a sphere which encode all quasi-isometric information for a group. We show how these tilings detect properties such as growth, ends, divergence, etc. We include figures of several worked out examples.