A note on torsion subgroups of groups acting on finite-dimensional CAT(0) cube complexes

Abstract

In this article, we state and prove a general criterion which prevents some groups from acting properly on finite-dimensional CAT(0) cube complexes. As an application, we show that, for every non-trivial finite group $F$, the lamplighter group $F\wr {\mathbb{F}}_2$ over a free group does not act properly on a finite-dimensional CAT(0) cube complex (although it acts properly on a infinite-dimensional CAT(0) cube complex). We also deduce from this general criterion that, roughly speaking, given a group $G$ acting on a CAT(0) cube complex of finite dimension and an infinite torsion subgroup $L\le G$, either the normaliser $N_G(L)$ is close to be free abelian or, for every $k\ge 1$, $N_G(L)$ contains a non-abelian free subgroup commuting with a subgroup of $L$ of size $\ge k$.

arXiv comment15 pages, 3 figures. Comments are welcome. arXiv admin note: text overlap with arXiv:1902.04883