Cubulating random groups in the square model

TL;DR

This paper proves that for densities below 3/10, random groups in the square model possess the Haagerup property and are residually finite, by constructing a space with walls leading to a proper action on a CAT(0) cube complex.

Contribution

It introduces a new method to cubulate random groups in the square model using modified hypergraphs and an extended isoperimetric inequality.

Findings

Random groups at density < 3/10 have the Haagerup property.

Such groups are residually finite.

Proper actions on CAT(0) cube complexes are established.

Abstract

Our main result is that for densities $<\frac{3}{10}$ a random group in the square model has the Haagerup property and is residually finite. Moreover, we generalize the Isoperimetric Inequality, to some class of non-planar diagrams and, using this, we introduce a system of modified hypergraphs providing the structure of a space with walls on the Cayley complex of a random group. Then we show that the natural action of a random group on this space with walls is proper, which gives the proper action of a random group on a CAT(0) cube complex.