Lifting Group Actions, Equivariant Towers and Subgroups of Non-positively Curved Groups

TL;DR

This paper proves that certain classes of non-positively curved groups are closed under finitely presented subgroups, introduces equivariant towers, and explores geometric properties of complexes related to group actions.

Contribution

It develops the concept of equivariant towers and applies it to demonstrate subgroup closure properties in non-positively curved groups.

Findings

Groups acting on 3-dimensional CAT(0) complexes are subgroup-closed.

K-systolic groups for k≥6 are subgroup-closed.

Certain negatively curved complexes have subgroup-closure properties.

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 $CAT(0)$ simplicial complexes of dimension $3$, $k$-systolic groups for $k\geq 6$, and groups acting geometrically on $2$-dimensional negatively curved complexes. We also show that there is a finite non-positively curved cubical $3$-complex which is not homotopy equivalent to a finite non-positively curved regular simplicial $3$-complex. We included other applications to relatively hyperbolic groups and diagramatically reducible groups. The main result is obtained by developing a notion of equivariant towers which is of independent interest.