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Determining the action dimension of an Artin group by using its complex of abelian subgroups. (English) Zbl 1433.20011
Summary: 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 \(L_\oslash\) 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 \(\operatorname{actdim}A\).) If \(H_d(L;\mathbb{Z}/2)\neq 0\), where \(d=\dim L\), then \(\operatorname{actdim}A\geqslant 2d+2\). Moreover, when the \(K(\pi,1)\)-Conjecture holds for \(A\), the inequality is an equality.
MSC:
20F36 Braid groups; Artin groups
20F55 Reflection and Coxeter groups (group-theoretic aspects)
20F65 Geometric group theory
57S30 Discontinuous groups of transformations
57Q35 Embeddings and immersions in PL-topology
20J06 Cohomology of groups
32S22 Relations with arrangements of hyperplanes
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