Random groups at density $d<3/14$ act non-trivially on a CAT(0) cube complex

TL;DR

This paper demonstrates that random groups at density less than 3/14 act non-trivially on CAT(0) cube complexes by constructing walls in their Cayley complexes, extending previous density bounds.

Contribution

It introduces a new wall construction in Cayley complexes that allows non-trivial actions on CAT(0) cube complexes at higher densities than previously known.

Findings

Constructed walls in Cayley complexes for random groups at d<3/14.

Extended non-trivial action results to higher densities.

Overcame key combinatorial challenges from prior work.

Abstract

For random groups in the Gromov density model at $d<3/14$, we construct walls in the Cayley complex $X$ which give rise to a non-trivial action by isometries on a CAT(0) cube complex. This extends results of Ollivier-Wise and Mackay-Przytycki at densities $d<1/5$ and $d<5/24$, respectively. We are able to overcome one of the main combinatorial challenges remaining from the work of Mackay-Przytycki, and we give a construction that plausibly works at any density $d<1/4$.