Fixed Point Sets in Diagrammatically Reducible Complexes

TL;DR

This paper proves that for groups acting on certain reducible 2-complexes with a fine 1-skeleton, the fixed point set, if non-empty, is always contractible, extending fixed point results in geometric group theory.

Contribution

It establishes that fixed point sets are contractible in diagrammatically reducible complexes with a fine 1-skeleton, a weaker condition than local finiteness.

Findings

Fixed point set is contractible if non-empty.

The result applies to complexes with a fine 1-skeleton, a weaker condition than local finiteness.

Provides a fixed point theorem for diagrammatically reducible complexes.

Abstract

Let $H$ be a group acting on a simply-connected diagrammatically reducible combinatorial 2-complex $X$ with fine 1-skeleton. If the fixed point set $X^ H$ is non-empty, then it is contractible. Having fine 1-skeleton is a weaker version of being locally finite.