Minimal volume entropy of RAAG's

Abstract

Bregman and Clay recently characterized which rightangled Artin groups with geometric dimension 2 have vanishing minimal volume entropy. In this note, we extend this characterization to higher dimensions.

Amenable covers of right‐angled Artin groups

“…We also obtain a complete characterisation of right‐angled Artin groups with (non‐)vanishing minimal volume entropy (Theorem 3.9), resolving the cases that were not covered by recent work in [3, 21].…”

“…The following is a complete characterisation of RAAGs with (non‐)vanishing minimal volume entropy. Most cases of Theorem 3.9 appeared in [21, Theorem 1.1], which however left open if $$\widetilde{H}^d(L;\mathbb {Z})=0$$ implies $$\omega (A_L)=0$$ in the case when $$d=2$$. Our resolution of this case goes back to the obstruction theoretical argument in the proof of Corollary 2.5.…”

“…The generalised LS‐category with respect to families of subgroups with controlled growth is closely related to the (non‐)vanishing of minimal volume entropy. The arguments in this section follow [21] to which we refer for the precise definitions.…”

“…Therefore, the amenable category is a meaningful threshold, especially for aspherical spaces. For instance, there are vanishing results in all degrees larger than the amenable category for the comparison map from bounded cohomology to singular cohomology [20, 22], for $$\ell ^2$$‐Betti numbers [32], and for homology growth [21, 33]. The amenable category was systematically studied as an invariant of 3‐manifolds in [15, 16] and for arbitrary spaces recently in [4, 23].…”

“…Our proofs rely on combining upper and lower bounds (Lemma 2.2) with existing results on generalised LS‐category, classifying spaces for families of subgroups, and homology growth from [4, 21, 23, 28, 30, 33].…”