scholarly journals The mapping class group action on -character varieties

2020 ◽  
pp. 1-15
Author(s):  
WILLIAM M. GOLDMAN ◽  
SEAN LAWTON ◽  
EUGENE Z. XIA
Keyword(s):  
Group Action ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Character Variety ◽  
Character Varieties ◽  
Compact Orientable Surface

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

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.

scholarly journals On the Cohomology of the Mapping Class Group of the Punctured Projective Plane

2020 ◽  
Vol 71 (2) ◽  
pp. 539-555
Author(s):  
Miguel A Maldonado ◽  
Miguel A Xicoténcatl
Keyword(s):  
Exact Sequence ◽  
Configuration Space ◽  
Mapping Class Group ◽  
Homotopy Theory ◽  
Class Group ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Mapping Class ◽  
Serre Spectral Sequence ◽  
Classical Homotopy

Abstract The mapping class group $\Gamma ^k(N_g)$ of a non-orientable surface with punctures is studied via classical homotopy theory of configuration spaces. In particular, we obtain a non-orientable version of the Birman exact sequence. In the case of ${\mathbb{R}} \textrm{P}^2$, we analyze the Serre spectral sequence of a fiber bundle $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k \to X_k \to BSO(3)$ where $X_k$ is a $K(\Gamma ^k({\mathbb{R}} \textrm{P}^2),1)$ and $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$ denotes the configuration space of unordered $k$-tuples of distinct points in ${\mathbb{R}} \textrm{P}^2$. As a consequence, we express the mod-2 cohomology of $\Gamma ^k({\mathbb{R}} \textrm{P}^2)$ in terms of that of $F_k({\mathbb{R}}{\textrm{P}}^{2}) / \Sigma _k$.

scholarly journals ABELIAN COVERS OF SURFACES AND THE HOMOLOGY OF THE LEVEL L MAPPING CLASS GROUP

2011 ◽  
Vol 03 (03) ◽  
pp. 265-306 ◽  
Author(s):  
ANDREW PUTMAN
Keyword(s):  
Group Ring ◽  
Mapping Class Group ◽  
Boundary Component ◽  
Homology Group ◽  
Class Group ◽  
Marked Point ◽  
Mapping Class ◽  
Rational Group ◽  
Rational Homology ◽  
Abelian Covers

We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian ℤ/L-cover of the surface. If the surface has one marked point, then the answer is ℚτ(L), where τ(L) is the number of positive divisors of L. If the surface instead has one boundary component, then the answer is ℚ. We also perform the same calculation for the level L subgroup of the mapping class group. Set HL = H1(Σg; ℤ/L). If the surface has one marked point, then the answer is ℚ[HL], the rational group ring of HL. If the surface instead has one boundary component, then the answer is ℚ.

scholarly journals A minimal generating set of the level 2 mapping class group of a non-orientable surface

2014 ◽  
Vol 157 (2) ◽  
pp. 345-355
Author(s):  
SUSUMU HIROSE ◽  
MASATOSHI SATO
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Orientable Surface ◽  
Mapping Class ◽  
Nonorientable Surface ◽  
Generating Set ◽  
Level 2 ◽  
Minimal Generating Set

AbstractWe construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus g, and determine its abelianization for g ≥ 4.

scholarly journals Automorphisms of braid groups on orientable surfaces

2016 ◽  
Vol 25 (05) ◽  
pp. 1650022
Author(s):  
Byung Hee An
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Automorphism Groups ◽  
Orientable Surface ◽  
Braid Groups ◽  
Mapping Class ◽  
Characteristic Subgroup ◽  
Group Extensions ◽  
Extended Mapping ◽  
Orientable Surfaces

In this paper, we compute the automorphism groups [Formula: see text] and [Formula: see text] of braid groups [Formula: see text] and [Formula: see text] on every orientable surface [Formula: see text], which are isomorphic to group extensions of the extended mapping class group [Formula: see text] by the transvection subgroup except for a few cases. We also prove that [Formula: see text] is always a characteristic subgroup of [Formula: see text], unless [Formula: see text] is a twice-punctured sphere and [Formula: see text].

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