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

TL;DR

This paper proves that automorphism and outer automorphism groups of right-angled Artin groups are generally not relatively hyperbolic, except in specific simple cases, clarifying their geometric group theory properties.

Contribution

It establishes the non-relative hyperbolicity of automorphism groups of right-angled Artin groups with complex defining graphs, extending understanding of their geometric structure.

Findings

Automorphism groups are not relatively hyperbolic for graphs with at least 3 vertices.

Outer automorphism groups are not relatively hyperbolic unless virtually isomorphic to simple cases.

Identifies specific conditions under which these groups are or 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.