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

TL;DR

This paper constructs specific subgroups within linear CAT(0) groups that have particular finiteness properties and cannot be quasi-isometrically embedded into linear groups over local fields, revealing new exotic behaviors.

Contribution

It introduces new examples of subgroups with precise finiteness properties that defy quasi-isometric embedding into linear groups over local fields, extending previous constructions.

Findings

Existence of subgroups of type F_{n-1} not F_n within linear CAT(0) groups.

These subgroups cannot be quasi-isometrically embedded into any linear group over local fields.

There are faithful complex linear representations of these subgroups that do not extend to the ambient group.

Abstract

For every positive integer $n$ we construct an example of a subgroup $L< G$ of a linear ${\rm CAT}(0)$ 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.