If
is a
class of complexes closed under taking full subcomplexes and covers and
is the
class of groups admitting proper and cocompact actions on one-connected complexes in
, then
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
simplicial complexes
of dimension
,
–systolic groups for
, and groups acting
geometrically on
–dimensional
negatively curved complexes. We also show that there is a finite non-positively curved cubical
–complex
that is not homotopy equivalent to a finite non-positively curved regular simplicial
–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.
