The action dimension of right-angled Artin groups

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 $\ge 2l$.

16 pages

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