Groups acting on finite dimensional spaces with finite stabilizers

Abstract.

It is shown that every H $\frak {F}$-group G of type $\rm{FP}_\infty$ admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X ^H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a $\Bbb {Z} G$-module which is projective as a $\Bbb {Z} H$-module for all finite $H \le G$ then M has finite projective dimension.