On solvable subgroups of automorphism groups of right-angled Artin groups

TL;DR

This paper proves a weak Tits alternative for automorphism groups of right-angled Artin groups, showing they contain either a finite-index nilpotent subgroup or a free subgroup, with criteria based on the defining graph.

Contribution

It establishes a new criterion for the structure of automorphism groups of right-angled Artin groups and explores examples of solvable subgroups, including non-virtually nilpotent cases.

Findings

Automorphism groups contain either a finite-index nilpotent or free subgroup.

A criterion based on the defining graph determines which case occurs.

Examples include solvable subgroups that are not virtually nilpotent.

Abstract

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.