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.

scholarly journals Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

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 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 On the classification of mapping class actions on Thurston's asymmetric metric

2013 ◽  
Vol 155 (3) ◽  
pp. 499-515 ◽  
Author(s):  
L. LIU ◽  
A. PAPADOPOULOS ◽  
W. SU ◽  
G. THÉRET
Keyword(s):  
Finite Type ◽  
Mapping Class Group ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Actions ◽  
The One

AbstractWe study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping class elements. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil–Petersson metric.

scholarly journals The action of the mapping class group on metrics of positive scalar curvature

2021 ◽  
Author(s):  
Georg Frenck
Keyword(s):  
Scalar Curvature ◽  
Mapping Class Group ◽  
Class Group ◽  
Index Theorem ◽  
Positive Scalar Curvature ◽  
Mapping Class ◽  
High Dimensional ◽  
Positive Scalar ◽  
Simply Connected

AbstractWe present a rigidity theorem for the action of the mapping class group $$\pi _0({\mathrm{Diff}}(M))$$ π 0 ( Diff ( M ) ) on the space $$\mathcal {R}^+(M)$$ R + ( M ) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional $${\mathrm{Spin}}$$ Spin -manifolds.

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.

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.

Generating the Mapping Class Group

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Exact Sequence ◽  
Simplicial Complex ◽  
Mapping Class Group ◽  
Class Group ◽  
Mapping Class ◽  
Detailed Structure ◽  
Inductive Step ◽  
Closed Curves ◽  
Combinatorial Object ◽  
Generating Sets

This chapter considers the Dehn–Lickorish theorem, which states that when g is greater than or equal to 0, the mapping class group Mod(Sɡ) is generated by finitely many Dehn twists about nonseparating simple closed curves. The theorem is proved by induction on genus, and the Birman exact sequence is introduced as the key step for the induction. The key to the inductive step is to prove that the complex of curves C(Sɡ) is connected when g is greater than or equal to 2. The simplicial complex C(Sɡ) is a useful combinatorial object that encodes intersection patterns of simple closed curves in Sɡ. More detailed structure of C(Sɡ) is then used to find various explicit generating sets for Mod(Sɡ), including those due to Lickorish and to Humphries.

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