Algebraic & Geometric Topology

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

Matthew B Day

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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 1n1(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.

Article information

Algebr. Geom. Topol., Volume 12, Number 3 (2012), 1553-1583.

Received: 21 June 2011
Revised: 10 April 2012
Accepted: 25 April 2012
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20] 20E36: Automorphisms of infinite groups [For automorphisms of finite groups, see 20D45] 20F28: Automorphism groups of groups [See also 20E36] 20F69: Asymptotic properties of groups
Secondary: 20F05: Generators, relations, and presentations

right-angled Artin group random graph automorphism group of group


Day, Matthew B. Finiteness of outer automorphism groups of random right-angled {A}rtin groups. Algebr. Geom. Topol. 12 (2012), no. 3, 1553--1583. doi:10.2140/agt.2012.12.1553.

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