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

Hanlon, Richard Gaelan¹ ; Martínez-Pedroza, Eduardo²

¹Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s NL A1C 5S7, Canada
²Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s NL A1C 5S7, Canada

Algebraic and Geometric Topology, Tome 14 (2014) no. 5, pp. 2783-2808

Résumé

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.

Détail

DOI : 10.2140/agt.2014.14.2783

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

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