Group actions on contractible $2$-complexes II

TL;DR

This paper proves that certain groups acting on acyclic 2-complexes must have fixed points, by showing their fundamental groups admit nontrivial unitary representations, thus extending fixed point results for finite group actions.

Contribution

It establishes that for specific groups, the fundamental group of acyclic 2-complexes admits nontrivial unitary representations, completing fixed point theorems for finite group actions on such complexes.

Findings

Groups  PSL_2(q) with q  5 mod 24 or 13 mod 24 have fundamental groups with nontrivial unitary representations.

Every action of a finite group on a finite, contractible 2-complex has a fixed point.

The result extends fixed point theorems to broader classes of group actions on 2-complexes.

Abstract

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, 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.