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

Abstract

For random groups in the Gromov density model at d > 3 / 14 d>3/14 , we construct walls in the Cayley complex X 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 d>1/5 and d > 5 / 24 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 d>1/4 .