Finiteness of outer automorphism groups of random right-angled Artin groups

Day, Matthew B

doi:10.2140/agt.2012.12.1553

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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

DOI: 10.2140/agt.2012.12.1553

Abstract

We consider the outer automorphism group $Out (A_{Γ})$ of the right-angled Artin group $A_{Γ}$ of a random graph $Γ$ on $n$ vertices in the Erdős–Rényi model. We show that the functions $n^{- 1} (log (n) + log (log (n)))$ and $1 - n^{- 1} (log (n) + log (log (n)))$ bound the range of edge probability functions for which $Out (A_{Γ})$ is finite: if the probability of an edge in $Γ$ is strictly between these functions as $n$ grows, then asymptotically $Out (A_{Γ})$ is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically $Out (A_{Γ})$ 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

Volume 12, issue 3 (2012)