Group actions on contractible $2$-complexes I

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 $q\equiv 3(\mathrm{mod}8)$ or $G=\mathrm{Sz}(2^n)$. The cases $G={\mathrm{PSL}}_2(q)$ with $q\equiv 5(\mathrm{mod}8)$ are addressed in the second part.

To appear in Inventiones mathematicae