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

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 $S{C}_G(g)=\{h\in G\mid \exists n\ge 1,[h,g^n]=1\}$ of $g$ is not virtually cyclic; (iii) ${\mathrm{Fix}}_Y(g^n)$ is finite for every $n\ge 1$ 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.

arXiv comment20 pages, 4 figures. Comments are welcome. arXiv admin note: text overlap with arXiv:1902.04883