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Braid groups and Artin groups. (English) Zbl 1230.20040
Papadopoulos, Athanase (ed.), Handbook of Teichmüller theory. Volume II. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-055-5/hbk). IRMA Lectures in Mathematics and Theoretical Physics 13, 389-451 (2009).
Braid groups have been studied for a long time and have been related to many fields in mathematics. This survey explores some of these, from the classical definition of braid groups using braid diagrams, to the one as the fundamental group of a configuration space and relating braid groups to mapping class groups. The author defines Artin and Garside groups, gives some fundamental properties and looks carefully at decision problems for braid groups. Next, the author describes the cohomology of Artin groups and the Salvetti complex to give cohomological properties. In the last two sections, the author examines the solution to the linearity problem of braid groups by Bigelow and Krammer and the profound relation between braid groups and mapping class groups, this is what the author calls geometric representations of braid groups.
For the entire collection see [Zbl 1158.30001].

MSC:
20F36 Braid groups; Artin groups
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M07 Topological methods in group theory
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
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