Automorphism and outer automorphism groups of Right-angled Artin groups are not relatively hyperbolic

Abstract

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least 3 vertices are not relatively hyperbolic. We then show that the outer automorphism groups are not relatively hyperbolic, if they are not virtually isomorphic to a right-angled Artin group whose defining graph is either a single vertex or disconnected.

Figures (2)

• Figure 1: Typical examples of $\Gamma$ with $\mathop{\mathrm{Out}}\nolimits(A_{\Gamma})$ relatively hyperbolic.
• Figure 2: The graph $\Gamma$ where $\mathop{\mathrm{Out}}\nolimits^*(A_\Gamma)$ is isomorphic to $\mathop{\mathrm{Aut}}\nolimits^*(\mathop{\mathrm{\mathbb{F}}}\nolimits_2)$.

Theorems & Definitions (6)

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