scholarly journals Curves with prescribed symmetry and associated representations of mapping class groups

2021 ◽  
Author(s):  
Marco Boggi ◽  
Eduard Looijenga
Keyword(s):  
Algebraic Curve ◽  
Class Group ◽  
Mapping Class ◽  
Zariski Closure ◽  
Topological Properties ◽  
Connected Component ◽  
Class Groups ◽  
Almost Simple Group ◽  
Linear Representations ◽  
A Finite Group

AbstractLet C be a complex smooth projective algebraic curve endowed with an action of a finite group G such that the quotient curve has genus at least 3. We prove that if the G-curve C is very general for these properties, then the natural map from the group algebra $${{\mathbb {Q}}}G$$ Q G to the algebra of $${{\mathbb {Q}}}$$ Q -endomorphisms of its Jacobian is an isomorphism. We use this to obtain (topological) properties regarding certain virtual linear representations of a mapping class group. For example, we show that the connected component of the Zariski closure of such a representation often acts $${{\mathbb {Q}}}$$ Q -irreducibly in a G-isogeny space of $$H^1(C; {{\mathbb {Q}}})$$ H 1 ( C ; Q ) and with image a $${{\mathbb {Q}}}$$ Q -almost simple group.

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 group representations from Drinfeld doubles of finite groups

2020 ◽  
Vol 29 (05) ◽  
pp. 2050033
Author(s):  
Jens Fjelstad ◽  
Jürgen Fuchs
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Group Representations ◽  
Torelli Group ◽  
Group Data ◽  
A Finite Group ◽  
Marked Points

We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group [Formula: see text], focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular, we show that they have finite images, and that for surfaces of genus at least [Formula: see text] their restriction to the Torelli group is non-trivial if and only if [Formula: see text] is non-abelian.

An infinite family of braid group representations in automorphism groups of free groups

2020 ◽  
Vol 29 (10) ◽  
pp. 2042007
Author(s):  
Wonjun Chang ◽  
Byung Chun Kim ◽  
Yongjin Song
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Free Groups ◽  
Infinite Family ◽  
Braid Groups ◽  
Mapping Class ◽  
Group Representations ◽  
Class Groups ◽  
Branched Coverings

The [Formula: see text]-fold ([Formula: see text]) branched coverings on a disk give an infinite family of nongeometric embeddings of braid groups into mapping class groups. We, in this paper, give new explicit expressions of these braid group representations into automorphism groups of free groups in terms of the actions on the generators of free groups. We also give a systematic way of constructing and expressing these braid group representations in terms of a new gadget, called covering groupoid. We prove that each generator [Formula: see text] of braid group inside mapping class group induced by [Formula: see text]-fold covering is the product of [Formula: see text] Dehn twists on the surface.

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.

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