Balanced walls for random groups

TL;DR

This paper constructs walls in the Cayley complex of a random group at densities below 5/24, enabling a nontrivial action on a CAT(0) cube complex, extending previous results for densities below 1/5.

Contribution

It introduces a new method for defining walls in the Cayley complex of random groups at higher densities, expanding the range of densities where nontrivial actions are known.

Findings

Walls exist for densities < 5/24

Nontrivial actions on CAT(0) cube complexes are possible in this range

Potential extension to densities < 1/4

Abstract

We study a random group G in the Gromov density model and its Cayley complex X. For density < 5/24 we define walls in X that give rise to a nontrivial action of G on a CAT(0) cube complex. This extends a result of Ollivier and Wise, whose walls could be used only for density < 1/5. The strategy employed might be potentially extended in future to all densities < 1/4.