scholarly journals The extended mapping class group can be generated by two torsions

2017 ◽  
Vol 26 (11) ◽  
pp. 1750061
Author(s):  
Xiaoming Du
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Oriented Surface ◽  
Genus Formula ◽  
Extended Mapping ◽  
Closed Oriented Surface

Let [Formula: see text] be the closed-oriented surface of genus [Formula: see text] and let [Formula: see text] be the extended mapping class group of [Formula: see text]. When the genus is at least 5, we prove that [Formula: see text] can be generated by two torsion elements. One of these generators is of order [Formula: see text], and the other one is of order [Formula: see text].

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 The quotient of a Kauffman bracket skein algebra by the square of an augmentation ideal

2017 ◽  
Vol 26 (05) ◽  
pp. 1750030
Author(s):  
Shunsuke Tsuji
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Augmentation Ideal ◽  
Oriented Surface ◽  
Kauffman Bracket ◽  
Genus Formula

We give an explicit basis [Formula: see text] of the quotient of the Kauffman bracket skein algebra [Formula: see text] on a surface [Formula: see text] by the square of an augmentation ideal. Moreover, we construct an embedding of the mapping class group of a compact connected oriented surface of genus [Formula: see text] into the Kauffman bracket skein algebra on the surface completed with respect to a filtration coming from the augmentation ideal.

scholarly journals On stable commutator lengths of Dehn twists along separating curves

2017 ◽  
Vol 26 (10) ◽  
pp. 1750056 ◽  
Author(s):  
Naoyuki Monden ◽  
Kazuya Yoshihara
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Upper Bounds ◽  
Mapping Class ◽  
Oriented Surface ◽  
Dehn Twists ◽  
Closed Oriented Surface

We give new upper bounds on the stable commutator lengths of Dehn twists along separating curves in the mapping class group of a closed oriented surface. The estimates of these upper bounds are [Formula: see text], where [Formula: see text] is the genus of the surface.

scholarly journals Nilpotency and triviality of mod p Morita-Mumford classes of mapping class groups of surfaces

2002 ◽  
Vol 165 ◽  
pp. 1-22 ◽  
Author(s):  
Toshiyuki Akita
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Mapping Class ◽  
Supporting Evidence ◽  
Class Groups ◽  
Oriented Surface ◽  
Closed Oriented Surface ◽  
Mod P

This paper is concerned with mod p Morita-Mumford classes of the mapping class group Γg of a closed oriented surface of genus g ≥ 2, especially triviality and nontriviality of them. It is proved that is nilpotent if n ≡ − 1 (mod p − 1), while the stable mod p Morita-Mumford class is proved to be nontrivial and not nilpotent if n ≢ −1 (mod p − 1). With these results in mind, we conjecture that vanishes whenever n ≡ − 1 (mod p − 1), and obtain a few pieces of supporting evidence.

scholarly journals A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group

2009 ◽  
Vol 146 (1) ◽  
pp. 109-118 ◽  
Author(s):  
YUSUKE KUNO
Keyword(s):  
Mapping Class Group ◽  
Homology Group ◽  
Class Group ◽  
Base Point ◽  
Mapping Class ◽  
Combinatorial Formula ◽  
Hyperelliptic Involution ◽  
Fixed Base ◽  
Genus 2 ◽  
Closed Oriented Surface

AbstractWe present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

scholarly journals A lower bound of the distortion of the Torelli group in the mapping class group with boundary components

2017 ◽  
Vol 26 (08) ◽  
pp. 1750049
Author(s):  
Erika Kuno ◽  
Genki Omori
Keyword(s):  
Lower Bound ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Torelli Group ◽  
Oriented Surface

We prove that the Torelli group of an oriented surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus–Farb–Putman’s techniques. Further we show that the distortion of the Torelli group in the level [Formula: see text] mapping class group is the same as that in the mapping class group.

scholarly journals Contractible, hyperbolic but non-CAT(0) complexes

2020 ◽  
Vol 30 (5) ◽  
pp. 1439-1463
Author(s):  
Richard C. H. Webb
Keyword(s):  
Isoperimetric Inequality ◽  
Finite Index ◽  
Mapping Class Group ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Index Subgroup ◽  
Analogous Statement ◽  
Almost All ◽  
Finite Index Subgroup

AbstractWe prove that almost all arc complexes do not admit a CAT(0) metric with finitely many shapes, in particular any finite-index subgroup of the mapping class group does not preserve such a metric on the arc complex. We also show the analogous statement for all but finitely many disc complexes of handlebodies and free splitting complexes of free groups. The obstruction is combinatorial. These complexes are all hyperbolic and contractible but despite this we show that they satisfy no combinatorial isoperimetric inequality: for any n there is a loop of length 4 that only bounds discs consisting of at least n triangles. On the other hand we show that the curve complexes satisfy a linear combinatorial isoperimetric inequality, which answers a question of Andrew Putman.

scholarly journals The abelianization of a symmetric mapping class group

2009 ◽  
Vol 147 (2) ◽  
pp. 369-388 ◽  
Author(s):  
MASATOSHI SATO
Keyword(s):  
Lower Bounds ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Oriented Surface

AbstractLet Σg,r be a compact oriented surface of genus g with r boundary components. We determine the abelianization of the symmetric mapping class group (g,r)(p2) of a double unbranched cover p2: Σ2g − 1,2r → Σg,r using the Riemann constant, Schottky theta constant, and the theta multiplier. We also give lower bounds on the order of the abelianizations of the level d mapping class group.

scholarly journals Generating the mapping class group by torsion elements of small order

2012 ◽  
Vol 154 (1) ◽  
pp. 41-62 ◽  
Author(s):  
NAOYUKI MONDEN
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Oriented Surface

AbstractWe show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated by 3 elements of order 3. Moreover, we show that the mapping class group of a closed, connected, oriented surface of genus at least three is generated and by 4 elements of order 4.

scholarly journals The Casson–Walker invariant of 3-manifolds with genus one open book decompositions

2019 ◽  
Vol 28 (06) ◽  
pp. 1950018
Author(s):  
Atsushi Mochizuki
Keyword(s):  
Mapping Class Group ◽  
Central Extension ◽  
Class Group ◽  
The Other ◽  
Mapping Class ◽  
Open Book ◽  
Open Book Decomposition ◽  
Open Book Decompositions ◽  
Framed Link ◽  
Genus One

In this paper, we give two formulae of values of the Casson–Walker invariant of 3-manifolds with genus one open book decompositions; one is a formula written in terms of a framed link of a surgery presentation of such a 3-manifold, and the other is a formula written in terms of a representation of the mapping class group of a 1-holed torus. For the former case, we compute the invariant through the combinatorial calculation of the degree 1 part of the LMO invariant. For the latter case, we construct a representation of a central extension of the mapping class group through the action of the degree 1 part of the LMO invariant on the space of Jacobi diagrams on two intervals, and compute the invariant as the trace of the representation of a monodromy of an open book decomposition.

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