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

Hanlon, Richard Gaelan; Martínez-Pedroza, Eduardo

doi:10.2140/agt.2014.14.2783

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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

DOI: 10.2140/agt.2014.14.2783

Abstract

If $C$ is a class of complexes closed under taking full subcomplexes and covers and $G$ is the class of groups admitting proper and cocompact actions on one-connected complexes in $C$ , then $G$ 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 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

Mathematical Subject Classification 2010

Primary: 20F67

Secondary: 57M07

References

Publication

Received: 9 July 2013

Revised: 5 March 2014

Accepted: 15 March 2014

Published: 6 November 2014

Authors

Richard Gaelan Hanlon
Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s NL A1C 5S7 Canada

Eduardo Martínez-Pedroza
Department of Mathematics and Statistics Memorial University of Newfoundland St. John’s NL A1C 5S7 Canada
http://www.math.mun.ca/~emartinezped/

Volume 14, issue 5 (2014)