The action dimension of right-angled Artin groups

TL;DR

This paper investigates the minimal dimension of contractible manifolds on which right-angled Artin groups act properly, providing calculations that support a conjecture relating $ ext{l}^2$-Betti numbers to action dimension.

Contribution

It computes the action dimension of right-angled Artin groups for many flag complexes, advancing understanding of their geometric properties.

Findings

Calculated action dimensions for numerous right-angled Artin groups

Provided evidence supporting the conjecture relating $ ext{l}^2$-Betti numbers to action dimension

Close to confirming the conjecture that nonzero $ ext{l}^2$-Betti number in degree l implies action dimension ≥ 2l

Abstract

The action dimension of a discrete group $\Gamma$ is the smallest dimension of a contractible manifold which admits a proper action of $\Gamma$. Associated to any flag complex $L$ there is a right-angled Artin group, $A_L$. We compute the action dimension of $A_L$ for many $L$. Our calculations come close to confirming the conjecture that if an $\ell^2$-Betti number of $A_L$ in degree $l$ is nonzero, then the action dimension of $A_L$ is $\ge 2l$.