# Rank-one isometries of CAT(0) cube complexes and their centralisers

## TL;DR

This paper classifies infinite-order elements in groups acting on CAT(0) cube complexes into three categories, linking geometric properties with algebraic structures, and provides conditions under which rank-one isometries occur.

## Contribution

It introduces a classification of infinite-order elements based on their geometric and algebraic properties, offering a new algebraic characterization of rank-one isometries.

## Key findings

- Exactly one of three situations occurs for infinite-order elements.
- Provides conditions under which rank-one isometries do not occur.
- Offers algebraic criteria to identify rank-one isometries.

## Abstract

If $G$ is a group acting geometrically on a CAT(0) cube complex $X$ and if $g \in G$ is an infinite-order element, we show that exactly one of the following situations occurs: (i) $g$ defines a rank-one isometry of $X$; (ii) the stable centraliser $SC_G(g)= \{ h \in G \mid \exists n \geq 1, [h,g^n]=1 \}$ of $g$ is not virtually cyclic; (iii) $\mathrm{Fix}_Y(g^n)$ is finite for every $n \geq 1$ and the sequence $(\mathrm{Fix}_Y(g^n))$ takes infinitely many values, where $Y$ is a cubical component of the Roller boundary of $X$ which contains an endpoint of an axis of $g$. We also show that (iii) cannot occur in several cases, providing a purely algebraic characterisation of rank-one isometries.