#### Volume 12, issue 2 (2012)

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Random groups arising as graph products

### Ruth Charney and Michael Farber

Algebraic & Geometric Topology 12 (2012) 979–995
##### 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 $n\to \infty$, random right-angled Artin groups have finite outer automorphism groups, assuming that the probability parameter $p$ is constant and satisfies $0.2929.

##### Keywords
random group, right angled Artin group, hyperbolic group, automorphisms of Artin groups
##### Mathematical Subject Classification 2010
Primary: 20P05
Secondary: 20F36, 57M07