# Median sets of isometries in CAT(0) cube complexes and some applications

## TL;DR

This paper introduces median sets for isometries in CAT(0) cube complexes, providing new tools for understanding group actions and deriving several structural and geometric results.

## Contribution

It defines median sets for isometries in CAT(0) cube complexes and applies this concept to prove new theorems about group actions and structures.

## Key findings

- Cubulation of centralisers in CAT(0) cube complexes
- A splitting theorem for group actions
- Dehn twists are elliptic in all actions on CAT(0) cube complexes

## Abstract

In this article, we associate to isometries of CAT(0) cube complexes specific subspaces, referred to as \emph{median sets}, which play a similar role as minimising sets of semisimple isometries in CAT(0) spaces. Various applications are deduced, including a cubulation of centralisers, a splitting theorem, a proof that Dehn twists in mapping class groups must be elliptic for every action on a CAT(0) cube complex, a cubical version of the flat torus theorem, and a structural theorem about polycyclic groups acting on CAT(0) cube complexes.