Groups acting on finite dimensional spaces with finite stabilizers

Abstract

It is shown that every HF fraktur sign-group G of type FP∞ 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 XH 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 ℤG-module which is projective as a ℤH-module for all finite H ≤ G then M has finite projective dimension.