Calculating the virtual cohomological dimension of the automorphism group of a RAAG

TL;DR

This paper presents an algorithm to compute the virtual cohomological dimension of automorphism groups of right-angled Artin groups, extending previous results to broader classes of automorphism groups.

Contribution

It introduces a new algorithm applicable to various automorphism groups of RAAGs, utilizing a novel construction of free abelian subgroups to determine their virtual cohomological dimension.

Findings

Algorithm successfully computes the virtual cohomological dimension.

Construction of free abelian subgroups generalizes previous results.

Applicable to untwisted and basis-conjugating automorphism groups.

Abstract

We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.