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Lifting group actions, equivariant towers and subgroups of non-positively curved groups. (English) Zbl 1335.20045
Summary: 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 $$k\geq 6$$, 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.

##### MSC:
 20F67 Hyperbolic groups and nonpositively curved groups 57M07 Topological methods in group theory 20F65 Geometric group theory 20E07 Subgroup theorems; subgroup growth 20F05 Generators, relations, and presentations of groups
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