A Primer on Mapping Class Groups

Farb, Benson; Margalit, Dan

doi:10.1515/9781400839049

Cited by 1,053 publications

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“…A key result (see [217,237,356] Denoting as usual by π g the fundamental group of a compact complex curve C of genus g, we have in fact a more concrete description of the mapping class group (see [9,10,44,45,125,165,212,220] for general results on braid and mapping class groups):…”

Section: The Moduli Space Of Curvesmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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Section: The Moduli Space Of Curvesmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…This curve is two-sided, which means that its regular neighbourhood is an annulus. Cutting S along a, twisting one of the sides by 360 • in the direction indicated by the small arrows on the figure and gluing back gives a self-homeomorphism of S, whose isotopy class is denoted by T a and called a Dehn twist along a (see [6] for a precise definition). For i = 1, 2 we denote by V i the isotopy class of a self-homeomorphism of S obtained by sliding the puncture P once along the loop v i shown on Fig.…”

Section: (Left)mentioning

confidence: 99%

On topological type of periodic self-homeomorphisms of closed non-orientable surfaces

Gromadzki

Szepietowski

2015

RACSAM

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Let S g denote a closed non-orientable surface of genus g ≥ 3. At the beginning of 1980s E. Bujalance showed that the maximum order of a periodic self-homeomorphism of S g is equal to 2g or 2(g − 1) for g odd or even respectively, and this upper bound is attained for all g ≥ 3. In this paper we enumerate, up to topological conjugation, actions on S g of a cyclic group Z N of order N > g − 2 with prescribed type of the quotient orbifold S g /Z N . We also compute, for a fixed g and N ranging between max{g, 3(g − 2)/2} and 2g, the total numbers of different topological types of action of Z N on S g .

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“…Since the canonical map from Out + (π 1 (Σ g )) on Sp(2g, Z) is known to be surjective (c.f. [FM11], theorem 6.4), it follows that Aut + (π 1 (Σ g )) acts transitively on the set of normal subgroups of index p in π 1 (Σ g ). Using this, we have :…”

Section: The Closed Surface Group Casementioning

confidence: 95%

Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

Guerin

2017

Geom Dedicata

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We give the centralizers of irreducible representations from a finitely generated group Γ to P SL(p, C) where p is a prime number. This leads to a description of the singular locus (the set of conjugacy classes of representations whose centralizer strictly contains the center of the ambient group) of the irreducible part of the character variety χ i (Γ, P SL(p, C)). When Γ is a free group of rank l ≥ 2 or the fundamental group of a closed Riemann surface of genus g ≥ 2, we give a complete description of this locus and prove that this locus is exactly the set of algebraic singularities of the irreducible part of the character variety.

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A Primer on Mapping Class Groups

Cited by 1,053 publications

References 0 publications

Topological methods in moduli theory

Topological methods in moduli theory

On topological type of periodic self-homeomorphisms of closed non-orientable surfaces

Bad irreducible subgroups and singular locus for character varieties in $$PSL(p,\mathbb {C})$$ P S L ( p , C )

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