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

15 pages