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Finiteness of outer automorphism groups of random right-angled {A}rtin groups

Matthew B Day

Algebraic & Geometric Topology 12 (2012) 1553–1583
Abstract

We consider the outer automorphism group Out(AΓ) of the right-angled Artin group AΓ of a random graph Γ on n vertices in the Erdős–Rényi model. We show that the functions n1(log(n) + log(log(n))) and 1 n1(log(n) + log(log(n))) bound the range of edge probability functions for which Out(AΓ) is finite: if the probability of an edge in Γ is strictly between these functions as n grows, then asymptotically Out(AΓ) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(AΓ) is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.

Keywords
right-angled Artin group, random graph, automorphism group of group
Mathematical Subject Classification 2010
Primary: 05C80, 20E36, 20F28, 20F69
Secondary: 20F05
References
Publication
Received: 21 June 2011
Revised: 10 April 2012
Accepted: 25 April 2012
Published: 15 July 2012
Authors
Matthew B Day
Department of Mathematical Sciences, SCEN 301
University of Arkansas
Fayetteville, AR 72701
USA
http://comp.uark.edu/~matthewd