Determining the action dimension of an Artin group by using its complex of abelian subgroups. Issue 4 (8th June 2017)

• Record Type: Journal Article
• Authors: Davis, Michael W.; Huang, Jingyin
• Is Part Of: Bulletin of the London Mathematical Society. Volume 49:Issue 4(2017)
• Page Start: 725
• Page End: 741
• Publication Date: 2017-06-08
• Subjects: 20F36 -- 20F55 -- 20F65 -- 57S30 -- 57Q35 (primary) -- 20J06 -- 32S22 (secondary); Mathematics -- Periodicals; 510
• DOI: 10.1112/blms.12061
• Languages: English
• ISSNs: 0024-6093

Abstract

Abstract: 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 B A . (This is the 'action dimension' of A denoted actdim A .) If H d ( L ; Z / 2 ) ≠ 0, where d = dim L, then actdim A ⩾ 2 d + 2 . Moreover, when the K ( π, 1 ) ‐Conjecture holds for A, the inequality is an equality.