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

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.