scholarly journals Geometric realizations of cyclic actions on surfaces

2019 ◽  
Vol 11 (04) ◽  
pp. 929-964
Author(s):  
Shiv Parsad ◽  
Kashyap Rajeevsarathy ◽  
Bidyut Sanki
Keyword(s):  
Finite Order ◽  
Representation Formula ◽  
Orientable Surface ◽  
Mapping Class ◽  
Realization Problem ◽  
Cyclic Subgroups ◽  
Genus Formula ◽  
Mapping Classes ◽  
Nielsen Realization ◽  
Maximal Reduction

Let [Formula: see text] denote the mapping class group of the closed orientable surface [Formula: see text] of genus [Formula: see text], and let [Formula: see text] be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on [Formula: see text] that realizes [Formula: see text] as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of [Formula: see text]. Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation [Formula: see text].

Torsion

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Finite Order ◽  
Class Group ◽  
Mapping Class ◽  
Cyclic Subgroups ◽  
Cyclic Groups ◽  
Hyperbolic Orbifold ◽  
Finite Subgroups ◽  
Mapping Classes ◽  
Bonnet Theorem ◽  
Nielsen Realization

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen realization theorem for cyclic groups and detection of torsion with the symplectic representation. It then considers the problem of finding an Euler characteristic for orbifolds, to prove a Gauss–Bonnet theorem for orbifolds, and to use these results to show that there is a universal lower bound of π‎/21 for the area of any 2-dimensional orientable hyperbolic orbifold. The chapter demonstrates that, when g is greater than or equal to 2, finite subgroups have order at most 84(g − 1) and cyclic subgroups have order at most 4g + 2. It also describes finitely many conjugacy classes of finite subgroups in Mod(S) and concludes by proving that Mod(Sɡ) is generated by finitely many elements of order 2.

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 DIFFUSIVE REPRESENTATIONS FOR THE ANALYSIS AND SIMULATION OF FLARED ACOUSTIC PIPES WITH VISCO-THERMAL LOSSES

2006 ◽  
Vol 16 (04) ◽  
pp. 503-536 ◽  
Author(s):  
TH. HÉLIE ◽  
D. MATIGNON
Keyword(s):  
Finite Order ◽  
Acoustic Waves ◽  
Complex Analysis ◽  
Differential Operators ◽  
Transfer Functions ◽  
Representation Formula ◽  
Input Output ◽  
Numerical Approximations ◽  
Pseudo Differential Operators ◽  
Scattering Matrices

Acoustic waves travelling in axisymmetric pipes with visco-thermal losses at the wall obey a Webster–Lokshin model. Their simulation may be achieved by concatenating scattering matrices of elementary transfer functions associated with nearly constant parameters (e.g. curvature). These functions are computed analytically and involve diffusive pseudo-differential operators, for which we have representation formula and input-output realizations, yielding direct numerical approximations of finite order. The method is based on some involved complex analysis.

scholarly journals ISOTOPY AND HOMEOMORPHISM OF CLOSED SURFACE BRAIDS

2020 ◽  
pp. 1-10
Author(s):  
MARK GRANT ◽  
AGATA SIENICKA
Keyword(s):  
Braid Group ◽  
Class Group ◽  
Closed Surface ◽  
Orientable Surface ◽  
Mapping Class ◽  
Closed Orientable Surface ◽  
Outer Action ◽  
Positive Genus ◽  
Closed Sphere ◽  
Group Modulo

Abstract The closure of a braid in a closed orientable surface Ʃ is a link in Ʃ × S1. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its centre.

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 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 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.

scholarly journals Non-realizability of the pure braid group as area-preserving homeomorphisms

2020 ◽  
pp. 1-12
Author(s):  
LEI CHEN
Keyword(s):  
Braid Group ◽  
Natural Projection ◽  
Realization Problem ◽  
Pure Braid Group ◽  
Rotation Numbers ◽  
Marked Points ◽  
Area Preserving ◽  
Nielsen Realization

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

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