Isometries of CAT(0) cube complexes are semi-simple

TL;DR

This paper proves that automorphisms of CAT(0) cube complexes are either fixing a point or preserving an axis, and explores implications for group actions with distorted cyclic subgroups.

Contribution

It establishes a dichotomy for automorphisms of CAT(0) cube complexes and links group distortion properties to actions on wall spaces.

Findings

Automorphisms are either fixed point or axis-preserving.

Groups with distorted cyclic subgroups cannot act properly on wall spaces.

Provides structural insight into automorphisms of CAT(0) cube complexes.

Abstract

We show that an automorphism of an arbitrary CAT(0) cube complex either has a fixed point or preserves some combinatorial axis. It follows that when a group contains a distorted cyclic subgroup, it admits no proper action on a discrete space with walls.