Group actions on contractible $2$-complexes I

TL;DR

This paper proves that every finite group action on a finite, contractible 2-complex has a fixed point, using representation theory of the fundamental group, with detailed cases for specific groups.

Contribution

It introduces a new approach to fixed point theorems for group actions on 2-complexes, covering various classes of groups and developing necessary theoretical tools.

Findings

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

Constructs nontrivial representations of the fundamental group for specific G-complexes.

Addresses cases for groups like PSL_2(2^n), PSL_2(q) with q ≡ 3 mod 8, and Sz(2^n).

Abstract

In this series of two articles, we prove that every action of a finite group $G$ on a finite and contractible $2$-complex has a fixed point. The proof goes by constructing a nontrivial representation of the fundamental group of each of the acyclic $2$-dimensional $G$-complexes constructed by Oliver and Segev. In the first part we develop the necessary theory and cover the cases where $G=\mathrm{PSL}_2(2^n)$, $G=\mathrm{PSL}_2(q)$ with $q\equiv 3\pmod 8$ or $G=\mathrm{Sz}(2^n)$. The cases $G=\mathrm{PSL}_2(q)$ with $q\equiv 5\pmod 8$ are addressed in the second part.