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.

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.

A note on the normal subgroups of mapping class groups

1986 ◽  
Vol 99 (1) ◽  
pp. 79-87 ◽  
Author(s):  
D. D. Long
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Orientable Surface ◽  
Faithful Representation ◽  
Mapping Class ◽  
Normal Subgroups ◽  
Class Groups ◽  
Closed Orientable Surface

0. If Fg is a closed, orientable surface of genus g, then the mapping class group of Fg is the group whose elements are orientation preserving self homeomorphisms of Fg modulo isotopy. We shall denote this group by Mg. Recall that a group is said to be linear if it admits a faithful representation as a group of matrices (where the entries for this purpose will be in some field).

scholarly journals Braid groups, mapping class groups and their homology with twisted coefficients

2021 ◽  
pp. 1-18
Author(s):  
ANDREA BIANCHI
Keyword(s):  
Braid Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Orientable Surface ◽  
Braid Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Constant Coefficients

Abstract We consider the Birman–Hilden inclusion $\phi\colon\Br_{2g+1}\to\Gamma_{g,1}$ of the braid group into the mapping class group of an orientable surface with boundary, and prove that $\phi$ is stably trivial in homology with twisted coefficients in the symplectic representation $H_1(\Sigma_{g,1})$ of the mapping class group; this generalises a result of Song and Tillmann regarding homology with constant coefficients. Furthermore we show that the stable homology of the braid group with coefficients in $\phi^*(H_1(\Sigma_{g,1}))$ has only 4-torsion.

scholarly journals Big mapping class groups are not acylindrically hyperbolic

2018 ◽  
Vol 68 (1) ◽  
pp. 71-76 ◽  
Author(s):  
Juliette Bavard ◽  
Anthony Genevois
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Infinite Type

AbstractWe give a criterion to prove that some groups are not acylindrically hyperbolic. As an application, we prove that the mapping class group of an infinite type surface is not acylindrically hyperbolic.

scholarly journals IRREDUCIBILITY OF SOME QUANTUM REPRESENTATIONS OF MAPPING CLASS GROUPS

2001 ◽  
Vol 10 (05) ◽  
pp. 763-767 ◽  
Author(s):  
JUSTIN ROBERTS
Keyword(s):  
Mapping Class Group ◽  
Prime Order ◽  
Class Group ◽  
Mapping Class Groups ◽  
Closed Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Root Of Unity

The SU(2) TQFT representation of the mapping class group of a closed surface of genus g, at a root of unity of prime order, is shown to be irreducible. Some examples of reducible representations are also given.

scholarly journals The geometry of the curve graph of a right-angled Artin group

2014 ◽  
Vol 24 (02) ◽  
pp. 121-169 ◽  
Author(s):  
Sang-Hyun Kim ◽  
Thomas Koberda
Keyword(s):  
Mapping Class Groups ◽  
Mapping Class ◽  
Artin Group ◽  
Artin Groups ◽  
Class Groups ◽  
Curve Graph ◽  
Right Angled Artin Groups ◽  
Central Result ◽  
The Right ◽  
Image Theorem

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph, respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result, we are able to develop a Nielsen–Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minsky's Bounded Geodesic Image Theorem and their distance formula.

scholarly journals Mapping class groups of highly connected $$(4k+2)$$-manifolds

2020 ◽  
Vol 26 (5) ◽  
Author(s):  
Manuel Krannich
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups

AbstractWe compute the mapping class group of the manifolds $$\sharp ^g(S^{2k+1}\times S^{2k+1})$$ ♯ g ( S 2 k + 1 × S 2 k + 1 ) for $$k>0$$ k > 0 in terms of the automorphism group of the middle homology and the group of homotopy $$(4k+3)$$ ( 4 k + 3 ) -spheres. We furthermore identify its Torelli subgroup, determine the abelianisations, and relate our results to the group of homotopy equivalences of these manifolds.

scholarly journals Generating the mapping class groups by torsions

2017 ◽  
Vol 26 (07) ◽  
pp. 1750037
Author(s):  
Xiaoming Du
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Class Groups ◽  
Oriented Surface ◽  
Genus Formula ◽  
Closed Oriented Surface

Let [Formula: see text] be a closed oriented surface of genus [Formula: see text] and let [Formula: see text] be the mapping class group. When the genus is at least 3, [Formula: see text] can be generated by torsion elements. We prove the following results: For [Formula: see text], [Formula: see text] can be generated by four torsion elements. Three generators are involutions and the fourth one is an order three element. [Formula: see text] can be generated by five torsion elements. Four generators are involutions and the fifth one is an order three element.

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