Random groups arising as graph products

Charney, Ruth; Farber, Michael

doi:10.2140/agt.2012.12.979

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Random groups arising as graph products

Ruth Charney and Michael Farber

Algebraic & Geometric Topology 12 (2012) 979–995

DOI: 10.2140/agt.2012.12.979

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 $0.2929 < p < 1$ .

Keywords

random group, right angled Artin group, hyperbolic group, automorphisms of Artin groups

Mathematical Subject Classification 2010

Primary: 20P05

Secondary: 20F36, 57M07

References

Publication

Received: 1 February 2011

Accepted: 3 December 2011

Published: 4 May 2012

Authors

Ruth Charney
Department of Mathematics Brandeis University Waltham MA 02453 USA

Michael Farber
Warwick Mathematics Institute University of Warwick Coventry CV4 7AL UK

Volume 12, issue 2 (2012)