Mathematical Research Letters

Volume 23 (2016)

Number 1

On the number of outer automorphisms of the automorphism group of a right-angled Artin group

Pages: 145 – 162



Neil J. Fullarton (Department of Mathematics, Rice University, Houston, Texas, U.S.A.)


We show that for any natural number $N$ there exists a right-angled Artin group $A_{\Gamma}$ for which $\mathrm{Out}(\mathrm{Aut}(A_{\Gamma}))$ has order at least $N$. This is in contrast with the cases where $A_{\Gamma}$ is free or free abelian: for all $n$, Dyer–Formanek and Bridson–Vogtmann showed that $\mathrm{Out}(\mathrm{Aut}(F_n)) = 1$, while Hua–Reiner showed $\lvert \mathrm{Out}(\mathrm{Aut}(\mathbb{Z}^n)) \rvert \leq 4$. We also prove the analogous theorem for $\mathrm{Out}(\mathrm{Out}(A_{\Gamma}))$. These theorems fit into a wider context of algebraic rigidity results in geometric group theory. We establish our results by giving explicit examples; one useful tool is a new class of graphs called austere graphs.

Accepted 16 September 2014

Published 25 May 2016