The square model for random groups

TL;DR

This paper introduces the square model for random groups, analyzing their properties at various densities, including triviality, hyperbolicity, Property (T), and freeness, with new isoperimetric inequalities.

Contribution

It defines the square model for random groups and establishes phase transitions and properties at different densities, extending previous models like the triangular model.

Findings

For densities > 1/2, groups are trivial with high probability.

For densities < 1/2, groups are hyperbolic with high probability.

For densities between 1/4 and 1/3, groups lack Property (T).

Abstract

We introduce a new random group model called the square model: we quotient a free group on $n$ generators by a random set of relations, each of which is a reduced word of length four. We prove, as in the Gromov density model, that for densities $> \frac{1}{2}$ a random group in the square model is trivial with overwhelming probability and for densities $<\frac{1}{2}$ a random group is with overwhelming probability hyperbolic. Moreover we show that for densities $\frac{1}{4} < d < \frac{1}{3}$ a random group in the square model does not have Property (T). Inspired by the results for the triangular model we prove that for densities $<\frac{1}{4}$ in the square model, a random group is free with overwhelming probability. We also introduce abstract diagrams with fixed edges and prove a generalization of the isoperimetric inequality.