From cut sets to cube complexes

TL;DR

This paper presents a method to derive cube complex actions from topological spaces with divisions, offering an alternative to Sageev's construction for hyperbolic groups using quasiconvex subgroups.

Contribution

It introduces a new approach to construct cube complex actions from topological spaces with divisions, extending Sageev's method to hyperbolic and relatively hyperbolic groups.

Findings

Provides a construction of cube complex actions from topological spaces with divisions.

Offers an alternative to Sageev's construction for hyperbolic groups.

Applicable to groups with no peripheral splittings.

Abstract

In this paper, we obtain an action on a cube complex from an action on a path-connected topological space with a system of divisions. In the settings of hyperbolic groups or relatively hyperbolic groups with no peripheral splittings, our result provides an alternate route to Sageev's construction of a cube complex action from a collection of (relatively) quasiconvex subgroups of a (relatively) hyperbolic group.