scholarly journals Embeddings of braid groups into mapping class groups and their homology

2012 ◽  
pp. 173-191 ◽  
Author(s):  
Carl-Friedrich Bödigheimer ◽  
Ulrike Tillmann
Keyword(s):  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Class Groups

scholarly journals Right-angled Artin groups as normal subgroups of mapping class groups

2021 ◽  
Vol 157 (8) ◽  
pp. 1807-1852
Author(s):  
Matt Clay ◽  
Johanna Mangahas ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Groups ◽  
Normal Subgroups ◽  
Class Groups ◽  
Right Angled Artin Groups ◽  
Proper Normal Subgroup

We construct the first examples of normal subgroups of mapping class groups that are isomorphic to non-free right-angled Artin groups. Our construction also gives normal, non-free right-angled Artin subgroups of other groups, such as braid groups and pure braid groups, as well as many subgroups of the mapping class group, such as the Torelli subgroup. Our work recovers and generalizes the seminal result of Dahmani–Guirardel–Osin, which gives free, purely pseudo-Anosov normal subgroups of mapping class groups. We give two applications of our methods: (1) we produce an explicit proper normal subgroup of the mapping class group that is not contained in any level $m$ congruence subgroup and (2) we produce an explicit example of a pseudo-Anosov mapping class with the property that all of its even powers have free normal closure and its odd powers normally generate the entire mapping class group. The technical theorem at the heart of our work is a new version of the windmill apparatus of Dahmani–Guirardel–Osin, which is tailored to the setting of group actions on the projection complexes of Bestvina–Bromberg–Fujiwara.

scholarly journals Linearity of some low-complexity mapping class groups

2020 ◽  
Vol 32 (2) ◽  
pp. 279-286
Author(s):  
Ignat Soroko
Keyword(s):  
Mapping Class Groups ◽  
Low Complexity ◽  
Braid Groups ◽  
Mapping Class ◽  
Artin Group ◽  
Class Groups ◽  
Orientable Surfaces

AbstractBy analyzing known presentations of the pure mapping groups of orientable surfaces of genus g with b boundary components and n punctures in the cases when {g=0} with b and n arbitrary, and when {g=1} and {b+n} is at most 3, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type {D_{4}}. As a corollary, we conclude that the pure mapping class groups are linear in these cases.

Braid Groups

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Closed Surfaces ◽  
Pictorial Representations ◽  
Marked Points

This chapter introduces the reader to Artin's classical braid groups Bₙ. The group Bₙ is isomorphic to the mapping class group of a disk with n marked points. Since disks are planar, the braid groups lend themselves to special pictorial representations. This gives the theory of braid groups its own special flavor within the theory of mapping class groups. The chapter begins with a discussion of three equivalent ways of thinking about the braid group, focusing on Artin's classical definition, fundamental groups of configuration spaces, and the mapping class group of a punctured disk. It then presents some classical facts about the algebraic structure of the braid group, after which a new proof of the Birman–Hilden theorem is given to relate the braid groups to the mapping class groups of closed surfaces.

Brunnian braids over the 2-sphere and Artin combed form

2021 ◽  
pp. 2150012
Author(s):  
Duzhin Fedor ◽  
Loh Sher En Jessica
Keyword(s):  
Exact Sequence ◽  
Word Problem ◽  
Open Problem ◽  
Homotopy Group ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Homotopy Groups ◽  
Class Groups ◽  
Homotopy Groups Of Spheres

Finding homotopy group of spheres is an old open problem in topology. Berrick et al. derive in [A. J. Berrick, F. Cohen, Y. L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006)] an exact sequence that relates Brunnian braids to homotopy groups of spheres. We give an interpretation of this exact sequence based on the combed form for braids over the sphere developed in [R. Gillette and J. V. Buskirk, The word problem and consequences for the braid groups and mapping class groups of the two-sphere, Trans. Amer. Math. Soc. 131 (1968) 277–296] with the aim of helping one to visualize the sequence and to do calculations based on it.

scholarly journals Introduction to Representations of Braid Groups

2015 ◽  
Vol 49 (1) ◽  
pp. 1-38
Author(s):  
Camilo Arias Abad
Keyword(s):  
Representation Theory ◽  
Mapping Class Groups ◽  
Braid Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Los Grupos

Estas notas fueron preparadas para un minicurso enseñado en la escuela Cimpa Algebraic and geometric aspects of representation theory, en Curitiba, Brazil en Marzo de 2013. El propósito del curso es presentar una introducción al estudio de las representaciones de los grupos de trenzas. Tres clases generales de representaciones son consideradas: representaciones homológicas de mapping class groups, representaciones de monodromía de la connección de Knizhnik-Zamolodchikov, y soluciones de la equación de Yang- Baxter en términos de quasi-triangular bialgebras. Algunas de las notables relaciones entre estas construcciones son descritas.

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