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 $CAT(0)$ 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 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.

v.4. Version accepted for publication in AGT

References in corpus (2)

• Relative Quasiconvexity using Fine Hyperbolic Graphs
• Coherence and Negative Sectional Curvature in Complexes of Groups

Cited by in corpus (3)

• Large-type Artin groups are systolic
• Subgroups, hyperbolicity and cohomological dimension for totally disconnected locally compact groups
• Closed subsets of a CAT(0) 2-complex are intrinsically CAT(0)