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 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].

On the Reducibility and the η-Invariant of Periodic Automorphisms of Genus 2 Surface

1997 ◽  
Vol 06 (06) ◽  
pp. 827-831 ◽  
Author(s):  
Takayuki Morifuji
Keyword(s):  
Finite Order ◽  
Mapping Class Group ◽  
Class Group ◽  
Finite Subgroup ◽  
Mapping Class ◽  
Mapping Tori ◽  
Genus 2

We give a characterization for the reducibility of elements of any finite subgroup of the mapping class group of genus 2 surface in terms of the η-invariant of finite order mapping tori.

scholarly journals The stratum of random mapping classes

2017 ◽  
Vol 38 (7) ◽  
pp. 2666-2682 ◽  
Author(s):  
VAIBHAV GADRE ◽  
JOSEPH MAHER
Keyword(s):  
Random Walks ◽  
Mapping Class Group ◽  
Class Group ◽  
Sample Path ◽  
Mapping Class ◽  
Elementary Subgroup ◽  
Random Mapping ◽  
Mapping Classes ◽  
First Moment

We consider random walks on the mapping class group that have finite first moment with respect to the word metric, whose support generates a non-elementary subgroup and contains a pseudo-Anosov map whose invariant Teichmüller geodesic is in the principal stratum. For such random walks, we show that mapping classes along almost every infinite sample path are eventually pseudo-Anosov, with invariant Teichmüller geodesics in the principal stratum. This provides an answer to a question of Kapovich and Pfaff [Internat. J. Algebra Comput.25, 2015 (5) 745–798].

scholarly journals An effective algebraic detection of the Nielsen–Thurston classification of mapping classes

2014 ◽  
Vol 07 (01) ◽  
pp. 1-21 ◽  
Author(s):  
Thomas Koberda ◽  
Johanna Mangahas
Keyword(s):  
Mapping Class Group ◽  
Word Length ◽  
Class Group ◽  
Mapping Class ◽  
Finite Cover ◽  
Generating Set ◽  
Reduction System ◽  
Conjugacy Separability ◽  
Mapping Classes

In this paper, we propose two algorithms for determining the Nielsen–Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod (S), then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π1(S) which allows us to lift the action of ψ to a finite cover [Formula: see text] of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on [Formula: see text] to determine the Nielsen–Thurston classification of ψ.

Is quantum simulation of turbulence within reach?

2014 ◽  
Vol 12 (07n08) ◽  
pp. 1560008 ◽  
Author(s):  
Mario Rasetti
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Fock Space ◽  
Quantum Simulation ◽  
Mapping Class ◽  
3D Flow ◽  
Topological Features ◽  
Mapping Classes ◽  
Flow Domain ◽  
Volume Preserving

The possibility of approaching quantum simulation of "topological turbulence" is discussed. In the latter, the topological features of turbulent flow are ascribed to a purely group-theoretic and topological view through the group SDiff of volume-preserving diffeomorphisms and the Mapping Class Group for the 2D surfaces that foliate the 3D flow domain. A class of representations of the group of mapping classes for compact boundaryless surfaces is explicitly constructed in terms of the Witt–Virasoro Hopf-coalgebra and implemented in terms of su(1,1). This leads to a complex, nonlinear Hamiltonian naturally associated with the fluid diffeomorphisms that could lend itself to the quantum simulation of turbulence in Fock space.

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