#### Volume 14, issue 5 (2014)

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

### Richard Gaelan Hanlon and Eduardo Martínez-Pedroza

Algebraic & Geometric Topology 14 (2014) 2783–2808
##### Abstract

If $\mathsc{C}$ is a class of complexes closed under taking full subcomplexes and covers and $\mathsc{G}$ is the class of groups admitting proper and cocompact actions on one-connected complexes in $\mathsc{C}$, then $\mathsc{G}$ 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 $3$, $k\phantom{\rule{0.3em}{0ex}}$–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.

##### Keywords
non-positively curved groups, hyperbolic groups, $\mathrm{CAT}(0)$, diagrammatically reducible, systolic, relatively hyperbolic, towers, van Kampen diagrams, equivariant covers, equivariant towers
Primary: 20F67
Secondary: 57M07