Finiteness of outer automorphism groups of random right-angled Artin groups. (English) Zbl 1246.05141

Day, Matthew B.

Algebr. Geom. Topol. 12, No. 3, 1553-1583 (2012).

Summary: We consider the outer automorphism group \(\mathrm{Out}(A_\Gamma )\) of the right-angled Artin group \(A_\Gamma\) of a random graph \(\Gamma\) 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(\(A_\Gamma )\) is finite: if the probability of an edge in \(\Gamma \) is strictly between these functions as \(n\) grows, then asymptotically \(\mathrm{Out}(A_\Gamma )\) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically \(\mathrm{Out}(A_\Gamma )\) is almost surely infinite. This sharpens a result of Ruth Charney and Michael Farber.

MSC:

05C80	Random graphs (graph-theoretic aspects)
20E36	Automorphisms of infinite groups
20F28	Automorphism groups of groups
20F69	Asymptotic properties of groups
20F05	Generators, relations, and presentations of groups

Keywords:

right-angled Artin group; random graph; automorphism group of group

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