Group actions on contractible $2$-complexes II

Abstract

In this second part we prove that, if $G$ is one of the groups ${\mathrm{PSL}}_2(q)$ with $q>5$ and $q\equiv 5(\mathrm{mod}24)$ or $q\equiv 13(\mathrm{mod}24)$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite $G$-complex admits a nontrivial representation in a unitary group $\mathrm{U}(m)$. This completes the proof of the following result: every action of a finite group on a finite and contractible $2$-complex has a fixed point.

To appear in Inventiones mathematicae