Hyperbolicities in CAT(0) cube complexes

TL;DR

This survey explores how group actions on CAT(0) cube complexes can reveal hyperbolic properties of groups, providing criteria, models, and applications to various hyperbolic subgroup structures.

Contribution

It offers a comprehensive overview of criteria and models for detecting hyperbolicity in groups acting on CAT(0) cube complexes, with practical applications.

Findings

Criteria for hyperbolicity based on group actions

A model for universal acylindrical actions of cubulable groups

Applications to Morse, stable, and hyperbolically embedded subgroups

Abstract

This paper is a survey dedicated to the following question: given a group acting on some CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as possible, the different criteria we mention are illustrated by applications. We also propose a model for universal acylindrical actions of cubulable groups and give a few applications to Morse, stable and hyperbolically embedded subgroups.