Subgroups of CAT(0) groups, exotic finiteness properties and non-QI-embeddings into linear groups

Abstract

For every positive integer $n$ we construct an example of a subgroup $L<G$ of a linear $\mathrm{CAT}(0)$ group $G$ such that $L$ is of finiteness type ${\mathcal{F}}_{n-1}$ and not ${\mathcal{F}}_n$, and $L$ does not admit a representation into ${\mathsf{GL}}_d(k)$ which is a quasi-isometric embedding for any local field $k$. We further prove that there is a faithful representation of $L$ into some ${\mathsf{GL}}_{\ell }(\mathbb{C})$ which is not the restriction of any representation of $G$. This generalises a family of fibre products of type ${\mathcal{F}}_2$ not ${\mathcal{F}}_3$ with these properties constructed by the second author.

13 pages