Homological stability for Artin monoids

Boyd, Rachael

doi:10.1112/plms.12335

Cited by 6 publications

(13 citation statements)

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“…Homological stability can also be formulated for sequences of spaces. There are many important examples of groups and spaces for which homological stability is known to hold, such as symmetric groups [Nak60], general linear groups [Cha80,Maa79,vdK80], mapping class groups of surfaces [Har85,RW16] and 3-manifolds [HW10], automorphism groups of free groups [HV98,HV04], diffeomorphism groups of high-dimensional manifolds [GRW18], configuration spaces [Chu12,RW13], Coxeter groups [Hep16], Artin monoids [Boyd20], and many more. In almost all cases, homological stability is one of the strongest things we know about the homology of these families.…”

Section: Introductionmentioning

confidence: 99%

On the homology of the Temperley-Lieb algebras

Boyd¹,

Hepworth²

2020

Preprint

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This paper studies the homology and cohomology of the Temperley-Lieb algebra TL n (a), interpreted as appropriate Tor and Ext groups. Our main result applies under the common assumption that a = v + v −1 for some unit v in the ground ring, and states that the homology and cohomology vanish up to and including degree (n − 2). To achieve this we simultaneously prove homological stability and compute the stable homology. We show that our vanishing range is sharp when n is even.Our methods are inspired by the tools and techniques of homological stability for families of groups. We construct and exploit a chain complex of 'planar injective words' that is analogous to the complex of injective words used to prove stability for the symmetric groups. However, in this algebraic setting we encounter a novel difficulty: TL n (a) is not flat over TL m (a) for m < n, so that Shapiro's lemma is unavailable. We resolve this difficulty by constructing what we call 'inductive resolutions' of the relevant modules.We believe that these results, together with the second author's work on Iwahori-Hecke algebras, are the first time the techniques of homological stability have been applied to algebras that are not group algebras.

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Section: Introductionmentioning

confidence: 99%

On the homology of the Temperley-Lieb algebras

Boyd¹,

Hepworth²

2020

Preprint

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“…Discussion: Homological stability for Coxeter groups and Artin monoids. The present paper builds strongly on previous work of the author [Hep16], which proved homological stability for families of Coxeter groups, and of Boyd [Boyd20], which proved homological stability for families of Artin monoids. These papers demonstrated that one can do all of the normal work of a homological stability proof purely in terms of a Coxeter or Artin-type presentation, rather than in terms of a concrete model of the group or monoid being studied.…”

Section: Comparison With Work Of Benson-erdmann-mikaelian the Cohomol...mentioning

confidence: 56%

“…are isomorphisms when n is sufficiently large compared to d. Homological stability can similarly be formulated for sequences of topological groups, and for families of spaces that are not necessarily classifying spaces of groups. Examples of families for which homological stability holds include symmetric groups [Nak60], general linear groups [Qui73,Cha80,vdK80], mapping class groups of surfaces and 3-manifolds [Har85,RW16,Wah13,HW10], diffeomorphism groups of highly connected manifolds [GRW18], automorphism groups of free groups [HV04,HV98], families of Coxeter groups [Hep16] and Artin monoids [Boyd20], configuration spaces of manifolds [Chu12], [RW13], and a great many others besides.…”

Section: Homological Stability a Family Of Discrete Groupsmentioning

confidence: 99%

“…In both [Hep16] and [Boyd20], the results apply to families of groups or monoids obtained from sequences of Coxeter diagrams that 'grow a tail' of type A n−1 as n increases. These families are very general, but include as the basic case the families of type A, B and D. So one may ask whether Theorem 1.1 can be extended to apply to any of these more general families.…”

Section: Comparison With Work Of Benson-erdmann-mikaelian the Cohomol...mentioning

confidence: 99%

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Homological stability for Iwahori–Hecke algebras

Hepworth

2022

Journal of Topology

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We show that the Iwahori-Hecke algebras H n of type A n−1 satisfy homological stability, where homology is interpreted as an appropriate Tor group. Our result precisely recovers Nakaoka's homological stability result for the symmetric groups in the case that the defining parameter is equal to 1. We believe that this paper, and our joint work with Boyd on Temperley-Lieb algebras, are the first time that the techniques of homological stability have been applied to algebras that are not group algebras. * (1, 1) over the group algebra RG, where 1 denotes the trivial representation. This formulation shows that the homology of a group depends only on the group algebra. We can therefore say that a family of algebras

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“…The Deligne complex D Γ is built from cosets of finite type special subgroups of A Γ . In the first author's work [Boyd20], an analogue to these cosets for the Artin monoid is defined and studied, in the setting of homological stability. This has inspired the current work, in which we imitate the construction of the Deligne complex for the Artin monoid to produce a new cube complex, D + Γ , which we call the monoid Deligne complex.…”

Section: Introductionmentioning

confidence: 99%

A Deligne complex for Artin Monoids

Boyd,

Charney,

Morris-Wright

2020

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In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis [CD95a] to study the K(π, 1) conjecture for these groups. Using a notion of Artin monoid cosets introduced by the first author in [Boyd20], we construct a version of the Deligne complex for Artin monoids.We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.

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Homological stability for Artin monoids

Cited by 6 publications

References 28 publications

On the homology of the Temperley-Lieb algebras

On the homology of the Temperley-Lieb algebras

Homological stability for Iwahori–Hecke algebras

A Deligne complex for Artin Monoids

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