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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

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 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.

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
Received: 9 July 2013
Revised: 5 March 2014
Accepted: 15 March 2014
Published: 6 November 2014
Richard Gaelan Hanlon
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s NL A1C 5S7
Eduardo Martínez-Pedroza
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s NL A1C 5S7