#### Volume 12, issue 3 (2012)

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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 $Out\left({A}_{\Gamma }\right)$ 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 ${n}^{-1}\left(log\left(n\right)+log\left(log\left(n\right)\right)\right)$ and $1-{n}^{-1}\left(log\left(n\right)+log\left(log\left(n\right)\right)\right)$ bound the range of edge probability functions for which $Out\left({A}_{\Gamma }\right)$ is finite: if the probability of an edge in $\Gamma$ is strictly between these functions as $n$ grows, then asymptotically $Out\left({A}_{\Gamma }\right)$ is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically $Out\left({A}_{\Gamma }\right)$ 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