Minimal Volume Entropy of RAAG's

Abstract

Bregman and Clay recently characterized which right-angled 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 [HS,BC21].…”

“…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 [Gro82,Iva85], for ℓ 2 -Betti numbers [Sau09], and for homology growth [Sau16,HS]. The amenable category was systematically studied as an invariant of 3-manifolds in [GLGAH13,GLGAH14] and for arbitrary spaces recently in [CLM, LM].…”

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].…”

“…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 𝓁 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].…”