Bestvina's Normal Form Complex and the Homology of Garside Groups

By Ruth Charney, John Meier, Kim Whittlesey. 2002

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AbstractA Garside group is a group admitting a finite lattice generating set $$\mathcal{D}$$ . Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice $$\mathcal{D}$$ , and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak nonpositive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually Abelian.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/b9321e47d2e20347786617adce599af3bfbd46ce

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Citations in the corpus

Foundations of Garside Theory (2013, unknown): “This program turned out to be rather successful and it led to many developments by a number of diﬀerent authors [8, 11, 12, 16, 18, 17, 48, 47, 59, 111, 118, 127, 128, 158, 157, 165, 181, 192, ...].”
Proof of the $$K(\pi , 1)$$ conjecture for affine Artin groups (2019, background): “te [Bes03] or ane [MS17]. When the noncrossing partition poset [1;w] is a lattice, the dual Artin group is a Garside group, and it admits a standard construction of a classifying space K W ’K(W w;1) [CMW04]. The poset [1;w] is indeed a lattice if W is nite [Bra01, BW02, Bes03, BW08], but this is not”
The dual braid monoid (2001, methodology): “ techniques are modelled on a construction of Bestvina. Following the same approach, Charney, Meier and Whittlesey have extended Bestvina’s construction to the context of an arbitrary Garside monoid ([CMW]). They note that the K(π,1) constructed from the dual monoid has the minimal possible dimensio”
Finite complex reection arrangements are K(; 1) (2015, unknown): “1”
Garside Groups and Yang–Baxter Equation (2009, unknown): “In [8], Dehornoy and Laffont show that whenever M is a Garside monoid, then the resolution defined in [1] is isomorphic to the resolution they define.”
Artin groups of Euclidean type (2013, background): “Proof The initial consequences follow from [17] and [10] with minor modifications to allow for infinite discretely weighted generating sets, and the latter ones are immediate corollaries.”
Non-crossing partition lattices in finite real reflection groups (2007, unknown): “The general construction of K ( A ( W ) , 1)’s is described in [11], using ideas from [4].”
Conjugacy in Garside groups II: structure of the ultra summit set (2006, unknown): “([1] for Artin-Tits groups, [7] for Garside groups.”
Finite complex reflection arrangements are $K(\pi,1)$ (2006, unknown): “Brady and Charney-Meier-Whittlesey, the Garside structure provides a convenient simplicial Eilenberg-McLane K(B(W ), 1) space ([29], [10], [13], [23]).”