We are motivated by the question that for which class of rightangled Artin groups (RAAG's), the quasi-isometry classification coincides with commensurability classification. This is previously known for RAAG's with finite outer automorphism groups. In this paper, we identify two classes of RAAG's, where their outer automorphism groups are allowed to contain adjacent transvections and partial conjugations, hence infinite. If G belongs to one of these classes, then any other RAAG G is quasi-isometric to G if and only if G is commensurable to G. We also show that in this case, there exists an algorithm to determine whether two RAAG's are quasi-isometric by looking at their defining graphs. Compared to the finite out case, as well as the previous quasi-isometry rigidity results for symmetric spaces, thick Euclidean buildings and mapping class groups, the main issue we need to deal with here is the reconstruction map may not have nice properties as before, or may not even exist. We introduce a deformation argument, as well as techniques from cubulation to deal with this issue.

Suppose that (W,S) is a Coxeter system with associated Artin group A and with a simplicial complex L as its nerve. We define the notion of a ‘standard abelian subgroup’ in A. The poset of such subgroups in A is parameterized by the poset of simplices in a certain subdivision L⊘ of L. This complex of standard abelian subgroups is used to generalize an earlier result from the case of right‐angled Artin groups to case of general Artin groups, by calculating, in many instances, the smallest dimension of a manifold model for BA. (This is the ‘action dimension’ of A denoted actdimA.) If Hdfalse(L;double-struckZ/2false)≠0, where d=dimL, then actdimA⩾2d+2. Moreover, when the K(π,1)‐Conjecture holds for A, the inequality is an equality.

Abstract. We show that every n-quasiflat in an n-dimensional CAT (0) cube complex is at finite Hausdorff distance from a finite union of n-dimensional orthants. Then we introduce a class of cube complexes, called weakly special cube complexes and show that quasi-isometries between their universal covers preserve top dimensional flats. This is the foundational towards the quasi-isometry classification of right-angled Artin groups with finite outer automorphism group.Some of our arguments also extend to CAT (0) spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top dimensional quasiflat in a Euclidean building is Hausdorff close to finite union of Weyl cones, which was previously established in [KL97b, EF97, Wor06] by different methods.

Measure equivalence classification of transvection-free right-angled Artin groups Tome 9 (2022), p. 1021-1067. © Les auteurs, 2022. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D'ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L'accès aux articles de la revue « Journal de l'École polytechnique -Mathématiques » (http://jep.centre-mersenne.org/), implique l'accord avec les conditions générales d'utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l'édition scientifique ouverte www.centre-mersenne.org

No abstract