Non-relative hyperbolicity of automorphism and outer automorphism groups of right-angled Artin groups

Abstract: Behrstock, Druţu, and Mosher showed in their paper that the general linear group $\mathrm{GL}_n(\mathbb{Z})$ with $n\geq 3$, the automorphism group and the outer automorphism group of a free group of finite rank at least 3 are not relatively hyperbolic. In this talk, we show a generalized version of this theorem using a criterion of non-relative hyperbolicity made by Anderson, Aramayona, and Shackleton. Our result is as follows: The automorphism groups of right-angled Artin groups whose defining graphs have at least 3 vertices are not relatively hyperbolic. We also show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, they are virtually isomorphic to one of the following: a finite group, an infinite cyclic group, or $\mathrm{GL}_2(\mathbb{Z})$. (Joint with Sangrok Oh and Philippe Tranchida)

group theory

Audience: researchers in the topic

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