Random group actions on \(\textrm{CAT}(0)\) square complexes

Generalizing ideas in Jahncke (Zufallsgruppen und die eigenschaft FR. Master’s thesis, 2012), we introduce the notion of progression in \(\textrm{CAT}(0)\) $\text{CAT}(0)$ square complexes. Using progression, we are able to build on the proof strategy of Dahmani et al. (Math Ann 349(3): 657–673, 2011) to show any action of a random group with seven or more generators on a \(\textrm{CAT}(0)\) $\text{CAT}(0)$ square complex has a global fixed point.