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When Artin groups are sufficiently large...

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If you have a question about this talk, please contact David Craven.

An Artin group is a group with a presentation with generators x1,x2,...,xn, and relations that xixjxi… and xjxixj… are equal, where there are mi,j terms in the first expression and mj,i in the second, for mi,jN ∪ {∞}, mi,j ≥ 2, which can be described naturally by a Coxeter matrix or graph.

This family of groups contains a wide range of groups, including braid groups, free groups, free abelian groups and much else, and its members exhibit a wide range of behaviour. Many problems remain open for the family as a whole, including the word problem, but are solved for particular subfamilies. The groups of finite type (mapping onto finite Coxeter groups), right-angled type (with each mi,j ∈ {2,∞}), large and extra-large type (with each mi,j ≥ 3 or 4), FC type (every complete subgraph of the Coxeter graph corresponds to a finite type subgroup) have been particularly studied.

After introducing Artin groups and surveying what is known, I will describe recent work with Derek Holt and (sometimes) Laura Ciobanu, dealing with a big collection of Artin groups, containing all the large groups, which we call ‘sufficiently large’. For those Artin groups we have elementary descriptions of the sets of geodesic and shortlex geodesic words, and can reduce any input word to either form. So we can solve the word problem, and prove the groups shortlex automatic. And, following Appel and Schupp we can solve the conjugacy problem in extra-large groups in cubic time.

For many of the large Artin groups, including all extra-large groups, we can deduce the rapid decay property and verify the Baum-Connes conjecture. And although our methods are quite different from those of Godelle and Dehornoy for spherical-type groups, we can pool our resources and derive a weak form of hyperbolicity for many, many Artin groups.

I’ll explain some background for the problems we attach, and outline their solution.

This talk is part of the Algebra Seminar series.

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