scholarly journals The mapping class group orbits in the framings of compact surfaces

2018 ◽  
Vol 69 (4) ◽  
pp. 1287-1302
Author(s):  
Nariya Kawazumi
Keyword(s):  
Moduli Spaces ◽  
Mapping Class Group ◽  
Quadratic Forms ◽  
London Math ◽  
Class Group ◽  
Mapping Class ◽  
Class Groups ◽  
Spin Structures ◽  
Arf Invariant ◽  
Relative Case

Abstract We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.

THE RESHETIKHIN-TURAEV REPRESENTATION OF THE MAPPING CLASS GROUP

1994 ◽  
Vol 03 (04) ◽  
pp. 547-574 ◽  
Author(s):  
GRETCHEN WRIGHT
Keyword(s):  
Mapping Class Group ◽  
Quadratic Forms ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Representation Space ◽  
Torelli Group ◽  
Spin Structures ◽  
Basis Vectors

The Reshetikhin-Turaev representation of the mapping class group of an orientable surface is computed explicitly in the case r = 4. It is then shown that the restriction of this representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. The proof makes use of an explicit correspondence between the basis vectors of the representation space, and the Z/2Z-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold.

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 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 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.

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.

Mapping Class Groups

2017 ◽  
Author(s):  
Leah Childers ◽  
Dan Margalit
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Open Problems ◽  
Dehn Twists ◽  
Group Of Homeomorphisms ◽  
Compact Orientable Surface ◽  
Group Of Symmetries

This chapter considers the mapping class group, the group of symmetries of a surface, and some of its basic properties. It first provides an overview of surfaces and the concept of homeomorphism before giving examples of homeomorphisms and defining the mapping class group as a certain quotient of the group of homeomorphisms of a surface. It then looks at Dehn twists and describes some of the relations they satisfy. It also presents a theorem stating that the mapping class group of a compact orientable surface is generated by Dehn twists and proves it. It concludes with some projects and open problems. The discussion also includes various exercises.

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.

scholarly journals ON STABILIZERS OF SOME TEICHMÜLLER DISKS IN POINTED MAPPING CLASS GROUPS

2010 ◽  
Vol 52 (3) ◽  
pp. 593-604
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Mapping Class Group ◽  
Isometric Embedding ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class Groups ◽  
Natural Projection ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Groups

AbstractWe prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

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