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The structure of Euclidean Artin groups
- Edited by Peter H. Kropholler, University of Southampton, Ian J. Leary, University of Southampton, Conchita Martínez-Pérez, Universidad de Zaragoza, Brita E. A. Nucinkis, Royal Holloway, University of London
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- Book:
- Geometric and Cohomological Group Theory
- Published online:
- 11 October 2017
- Print publication:
- 19 October 2017, pp 82-114
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Summary
Abstract
The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several long-standing conjectures about these groups, proving for the first time that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finitedimensional classifying space. This article surveys our results and the techniques we use to prove them.
2010 Mathematics Subject Classification: 20F36, 20F55Key words and phrases: euclidean Coxeter groups, euclidean Artin groups, Garside structures, dual presentations.
The reflection groups that act geometrically on spheres and euclidean spaces are all described by presentations of an exceptionally simple form and general Coxeter groups are defined by analogy. These spherical and euclidean Coxeter groups have long been classified and their presentations are encoded in the well-known Dynkin diagrams and extended Dynkin diagrams, respectively. Artin groups are defined by modified versions of these Coxeter presentations, and they were initially introduced to describe the fundamental group of a space constructed from the complement of the hyperplanes in a complexified version of the reflection arrangement for the corresponding spherical or euclidean Coxeter group. The most basic example of a Coxeter group is the symmetric group and the corresponding Artin group is the braid group, the fundamental group of a quotient of the complement of a complex hyperplane arrangement called the braid arrangement.
The spherical Artin groups, that is the Artin groups corresponding to the Coxeter groups acting geometrically on spheres, have been well understood ever since Artin groups themselves were introduced by Pierre Deligne [Del72] and by Brieskorn and Saito [BS72] in adjacent articles in the Inventiones in 1972. One might have expected the euclidean Artin groups to be the next class of Artin groups whose structure was well-understood, but this was not to be. Despite the centrality of euclidean Coxeter groups in Coxeter theory and Lie theory more generally, euclidean Artin groups have remained fundamentally mysterious, with a few minor exceptions, for the past forty years.
Constructing non-positively curved spaces and groups
- Edited by Martin R. Bridson, University of Oxford, Peter H. Kropholler, University of Glasgow, Ian J. Leary, Ohio State University
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- Book:
- Geometric and Cohomological Methods in Group Theory
- Published online:
- 07 September 2011
- Print publication:
- 29 October 2009, pp 162-224
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The theory of non-positively curved spaces and groups is tremendously powerful and has enormous consequences for the groups and spaces involved. Nevertheless, our ability to construct examples to which the theory can be applied has been severely limited by an inability to test – in real time – whether a random finite piecewise Euclidean complex is non-positively curved. In this article I focus on the question of how to construct examples of non-positively curved spaces and groups, highlighting in particular the boundary between what is currently do-able and what is not yet feasible. Since this is intended primarily as a survey, the key ideas are merely sketched with references pointing the interested reader to the original articles.
Over the past decade or so, the consequences of non-positive curvature for geometric group theorists have been thoroughly investigated, most prominently in the book by Bridson and Haefliger [26]. See also the recent review article by Kleiner in the Bulletin of the AMS [59] and the related books by Ballmann [4], Ballmann-Gromov-Schroeder [5] and the original long article by Gromov [48]. In this article I focus not on the consequences of the theory, but rather on the question of how to construct examples to which it applies. The structure of the article roughly follows the structure of the lectures I gave during the Durham symposium with the four parts corresponding to the four talks.