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 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 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 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 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.

FOUR-VARIABLE LINK-INVARIANTS FOR THICKENED SURFACES

1996 ◽  
Vol 05 (01) ◽  
pp. 1-21
Author(s):  
PAOLO COTTA-RAMUSINO ◽  
MAURIZIO RINALDI
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Two Dimensional ◽  
Oriented Surface ◽  
Link Invariants ◽  
Homfly Polynomials

For a given two-dimensional compact oriented surface Σ with boundary, we define invariants with 4 variables for links in Σ×[0, 1]. These invariants restrict to the two-variable HOMFLY polynomials when Σ=D2. Some comments are made about the action of the mapping class group of Σ on these invariants.

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 Separating subgroups of mapping class groups in homological representations

2020 ◽  
pp. 1-15
Author(s):  
Asaf Hadari
Keyword(s):  
Automorphism Group ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class Groups ◽  
Representation Formula ◽  
Closed Surface ◽  
Mapping Class ◽  
Finite Cover ◽  
Class Groups ◽  
Genus Formula

Let [Formula: see text] be either the mapping class group of a closed surface of genus [Formula: see text], or the automorphism group of a free group of rank [Formula: see text]. Given any homological representation [Formula: see text] of [Formula: see text] corresponding to a finite cover, and any term [Formula: see text] of the Johnson filtration, we show that [Formula: see text] has finite index in [Formula: see text], the Torelli subgroup of [Formula: see text]. Since [Formula: see text] for [Formula: see text], this implies for instance that no such representation is faithful.

scholarly journals Roots of Dehn twists on nonorientable surfaces

2019 ◽  
Vol 28 (12) ◽  
pp. 1950077
Author(s):  
Anna Parlak ◽  
Michał Stukow
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Maximal Degree ◽  
Simple Arithmetic ◽  
Nonorientable Surface ◽  
Dehn Twists ◽  
Nonorientable Surfaces ◽  
Genus Formula ◽  
Orientable Surfaces

Margalit and Schleimer observed that Dehn twists on orientable surfaces have nontrivial roots. We investigate the problem of roots of a Dehn twist [Formula: see text] about a nonseparating circle [Formula: see text] in the mapping class group [Formula: see text] of a nonorientable surface [Formula: see text] of genus [Formula: see text]. We explore the existence of roots and, following the work of McCullough, Rajeevsarathy and Monden, give a simple arithmetic description of their conjugacy classes. We also study roots of maximal degree and prove that if we fix an odd integer [Formula: see text], then for each sufficiently large [Formula: see text], [Formula: see text] has a root of degree [Formula: see text] in [Formula: see text]. Moreover, for any possible degree [Formula: see text], we provide explicit expressions for a particular type of roots of Dehn twists about nonseparating circles in [Formula: see text].

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.

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