Random Group Actions on $\mathrm{CAT}(0)$ Square Complexes

TL;DR

This paper extends the concept of progression to $ ext{CAT}(0)$ square complexes and proves that random groups with at least seven generators have a fixed point when acting on such complexes.

Contribution

It introduces the notion of progression in $ ext{CAT}(0)$ square complexes and adapts existing proof strategies to establish fixed point properties for random groups.

Findings

Random groups with ≥7 generators have fixed points on $ ext{CAT}(0)$ square complexes.

The notion of progression is effective in analyzing group actions on complex geometric structures.

The proof builds on and generalizes previous methods from tree actions to square complexes.

Abstract

We generalize ideas of Jahncke from trees to square complexes. We introduce the notion of progression in $\mathrm{CAT}(0)$ square complexes. Using progression, we are able to build on the proof strategy of Dahmani-Guirardel-Przytycki to show any action of a random group with seven or more generators on a $\mathrm{CAT}(0)$ square complex has a global fixed point.