scholarly journals Stable subgroups and Morse subgroups in mapping class groups

2019 ◽  
Vol 29 (05) ◽  
pp. 893-903 ◽  
Author(s):  
Heejoung Kim
Keyword(s):  
Class Group ◽  
Hyperbolic Group ◽  
Mapping Class Groups ◽  
Natural Generalization ◽  
Mapping Class ◽  
Infinite Index ◽  
Class Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Cube Complexes

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor [M. Durham and S. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol.  15(5) (2015) 2839–2859] defined stability and proved stability is equivalent to convex cocompactness in mapping class groups. Another natural generalization of quasiconvexity is given by the notion of a Morse or strongly quasiconvex subgroup of a finitely generated group, studied recently by Tran [H. Tran, On strongly quasiconvex subgroups, To Appear in Geom. Topol., preprint (2017), arXiv:1707.05581 ] and Genevois [A. Genevois, Hyperbolicities in CAT (0) cube complexes, preprint (2017), arXiv:1709.08843 ]. In general, a subgroup is stable if and only if the subgroup is Morse and hyperbolic. In this paper, we prove that two properties of being Morse and stable coincide for a subgroup of infinite index in the mapping class group of an oriented, connected, finite type surface with negative Euler characteristic.

scholarly journals Cube Complexes, Subgroups of Mapping Class Groups and Nilpotent Genus

2017 ◽  
Author(s):  
Martin R. Bridson
Keyword(s):  
Finite Type ◽  
Mapping Class Groups ◽  
The Other ◽  
Mapping Class ◽  
Class Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Groups Of Automorphisms ◽  
Cube Complexes ◽  
Finitely Presented

Based on a lecture at PCMI this chapter is structured around two sets of results, one concerning groups of automorphisms of surfaces and the other concerning the nilpotent genus of groups. The first set of results exemplifies the theme that even the nicest of groups can harbour a diverse array of complicated finitely presented subgroups: we shall see that the finitely presented subgroups of the mapping class groups of surfaces of finite type can be much wilder than had been previously recognised. The second set of results fits into the quest to understand which properties of a finitely generated group can be detected by examining the group’s finite and nilpotent quotients and which cannot.

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 The First Integral Cohomology of Pure Mapping Class Groups

2020 ◽  
Author(s):  
Javier Aramayona ◽  
Priyam Patel ◽  
Nicholas G Vlamis
Keyword(s):  
Cohomology Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
First Integral ◽  
Orientable Surface ◽  
Mapping Class ◽  
Class Groups ◽  
Simplicial Homology ◽  
Integral Cohomology ◽  
Orientable Surfaces

Abstract It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.

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.

THE MAPPING CLASS GROUP OF (S3, K), WHERE K IS A TWO-BRIDGE KNOT OR LINK

2010 ◽  
Vol 19 (07) ◽  
pp. 881-892
Author(s):  
THERESA JEEVANJEE
Keyword(s):  
Rational Number ◽  
Class Group ◽  
Quotient Space ◽  
Mapping Class Groups ◽  
Lens Space ◽  
Mapping Class ◽  
Class Groups ◽  
Exceptional Set ◽  
Generators And Relations ◽  
Underlying Space

Let K be a two-bridge knot or link in S3. Then K is also denoted as the four-plat, b(p, q) to indicate its association with some rational number p/q. The lens space L = L(p, q) admits an isometry τ of order two, such that the quotient space L modulo the involution τ is an orbifold whose exceptional set is K and whose underlying space is S3. In this paper, the mapping class groups of these orbifolds are classified. While these groups can be found as a result of Mostow's Rigidity Theorem, this paper calculates the generators and relations of the groups and the proof does not rely on this strong theorem for the majority of cases.

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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