Random groups arising as graph products

Abstract

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdös and Rényi model of a random graph and find precise threshold functions for hyperbolicity (or relative hyperbolicity). We also study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as n → ∞, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter p is constant and satisfies 0:2929 < p < 1.