Amenable covers of right-angled Artin groups

TL;DR

This paper establishes a precise relationship between the amenable category of right-angled Artin groups and the virtual cohomological dimension of associated Coxeter groups, confirming a conjecture linking these algebraic and topological invariants.

Contribution

It proves that the amenable category of right-angled Artin groups equals the virtual cohomological dimension of their Coxeter counterparts, resolving a conjecture about their topological complexity.

Findings

Amenable category equals the virtual cohomological dimension.

Right-angled Artin groups satisfy the proposed inequality with topological complexity.

The result links algebraic invariants with topological properties of these groups.

Abstract

Let $A_L$ be the right-angled Artin group associated to a finite flag complex $L$. We show that the amenable category of $A_L$ equals the virtual cohomological dimension of the right-angled Coxeter group $W_L$. In particular, right-angled Artin groups satisfy a question of Capovilla--L\"oh--Moraschini proposing an inequality between the amenable category and Farber's topological complexity.