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

## TL;DR

This paper establishes a criterion that restricts certain groups from acting properly on finite-dimensional CAT(0) cube complexes, with implications for groups like lamplighter groups and their torsion subgroups.

## Contribution

It provides a new general criterion to determine when groups cannot act properly on finite-dimensional CAT(0) cube complexes, extending understanding of group actions in geometric group theory.

## Key findings

- Lamplighter groups over free groups do not act properly on finite-dimensional CAT(0) cube complexes.
- The normaliser of an infinite torsion subgroup in such groups is either nearly free abelian or contains non-abelian free subgroups.
- The criterion applies broadly to analyze group actions 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 $L \leq G$, either the normaliser $N_G(L)$ is close to be free abelian or, for every $k \geq 1$, $N_G(L)$ contains a non-abelian free subgroup commuting with a subgroup of $L$ of size $\geq k$.