Lifting group actions, equivariant towers and subgroups of non-positively curved groups

Abstract

If $\mathcal{C}$ is a class of complexes closed under taking full subcomplexes and covers and $\mathcal{G}$ is the class of groups admitting proper and cocompact actions on one-connected complexes in $\mathcal{C}$, then $\mathcal{G}$ is closed under taking finitely presented subgroups. As a consequence the following classes of groups are closed under taking finitely presented subgroups: groups acting geometrically on regular $CAT\left(0\right)$ simplicial complexes of dimension $3$, $k$–systolic groups for $k\ge 6$, and groups acting geometrically on $2$–dimensional negatively curved complexes. We also show that there is a finite non-positively curved cubical $3$–complex that is not homotopy equivalent to a finite non-positively curved regular simplicial $3$–complex. We include applications to relatively hyperbolic groups and diagrammatically reducible groups. The main result is obtained by developing a notion of equivariant towers, which is of independent interest.