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In this article, we state and prove a general criterion which prevents somegroups from acting properly on finite-dimensional CAT(0) cube complexes. As anapplication, we show that, for every non-trivial finite group $F$, thelamplighter group $F \wr \mathbb{F}_2$ over a free group does not act properlyon a finite-dimensional CAT(0) cube complex (although it acts properly on ainfinite-dimensional CAT(0) cube complex). We also deduce from this generalcriterion that, roughly speaking, given a group $G$ acting on a CAT(0) cubecomplex of finite dimension and an infinite torsion subgroup $L \leq G$, eitherthe 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 $\geq k$.

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