Stallings folds for CAT(0) cube complexes and quasiconvex subgroups

TL;DR

This paper introduces a higher-dimensional Stallings folding process for group actions on CAT(0) cube complexes, providing a new characterization of quasiconvex subgroups in hyperbolic groups based on hyperplane stabilizer properties.

Contribution

It develops a novel higher-dimensional folding technique and applies it to characterize quasiconvex subgroups via hyperplane stabilizer finiteness properties.

Findings

Higher-dimensional Stallings folding sequence developed

Characterization of quasiconvex subgroups via hyperplane stabilizers

Provides new tools for understanding group actions on CAT(0) cube complexes

Abstract

We describe a higher dimensional analogue of the Stallings folding sequence for group actions on CAT(0) cube complexes. We use it to give a characterization of quasiconvex subgroups of hyperbolic groups which act properly co-compactly on CAT(0) cube complexes via finiteness properties of their hyperplane stabilizers.