Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups

TL;DR

This paper calculates the virtual cohomological dimension of certain automorphism groups of free groups and right-angled Artin groups, providing bounds and exact values in specific cases, advancing understanding of their algebraic structure.

Contribution

It introduces new bounds and exact calculations for the virtual cohomological dimension of automorphism groups of 2D RAAGs and partially symmetric automorphisms of free groups.

Findings

VCD bounds for automorphism groups of free groups

Exact VCD for RAAGs with tree defining graphs

New techniques for computing cohomological dimensions

Abstract

We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.