Random groups have fixed points on CAT(0) cube complexes

TL;DR

This paper proves that random groups inevitably have fixed points when acting on CAT(0) cube complexes, regardless of whether the action is simplicial, advancing understanding of group actions in geometric group theory.

Contribution

It establishes fixed point properties for random groups acting on CAT(0) cube complexes without the need for simplicial actions, a novel result in the field.

Findings

Random groups have fixed points on CAT(0) cube complexes.

Fixed point property holds without assuming simplicial actions.

Advances understanding of geometric actions of random groups.

Abstract

We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.