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 ON STABILIZERS OF SOME TEICHMÜLLER DISKS IN POINTED MAPPING CLASS GROUPS

2010 ◽  
Vol 52 (3) ◽  
pp. 593-604
Author(s):  
C. ZHANG
Keyword(s):  
Riemann Surface ◽  
Mapping Class Group ◽  
Isometric Embedding ◽  
Class Group ◽  
Teichmüller Space ◽  
Mapping Class Groups ◽  
Natural Projection ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Class Groups

AbstractWe prove that for each Riemann surface of finite analytic type (p, n) with p ≥ 2, there exist uncountably many Teichmüller disks Δ in the Teichmüller space T(S), where S = - {a point a}, with these properties: (1) the natural projection j: T(S) → T() defined by forgetting a induces an isometric embedding of each Δ into T(); and (2) the stabilizer of each Teichmüller disk Δ in the a-pointed mapping class group of S is trivial.

scholarly journals Statistical Hyperbolicity for Harmonic Measure

2020 ◽  
Author(s):  
Aitor Azemar ◽  
Vaibhav Gadre ◽  
Luke Jeffreys
Keyword(s):  
Random Walks ◽  
Harmonic Measure ◽  
Mapping Class Group ◽  
Class Group ◽  
Probability Distributions ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Teichmüller Metric ◽  
Harmonic Measures

Abstract We consider harmonic measures that arise from random walks on the mapping class group determined by probability distributions that have finite first moments with respect to the Teichmüller metric and whose supports generate nonelementary subgroups. We prove that Teichmüller space with the Teichmüller metric is statistically hyperbolic for such a harmonic measure.

scholarly journals Torelli groups, extended Johnson homomorphisms, and new cycles on the moduli space of curves

2008 ◽  
Vol 144 (3) ◽  
pp. 651-671 ◽  
Author(s):  
S. MORITA ◽  
R. C. PENNER
Keyword(s):  
Moduli Space ◽  
Mapping Class Group ◽  
Class Group ◽  
Main Tool ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Teichmuller Space ◽  
Moduli Space Of Curves ◽  
Torelli Group ◽  
Invariant Ideal

AbstractInfinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given, and finite presentations of the classical Torelli groups acting trivially on homology modulo N are derived for all N. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit canonical 1-cocycle of the Teichmüller space. The main tool for these results is the known mapping class group invariant ideal cell decomposition of the Teichmüller space.This new 1-cocycle is mapping class group equivariant, so various contractions of its powers yield various combinatorial (co)cycles of the moduli space of curves, which are also new. Our combinatorial construction can be related to former works of Kawazumi and the first-named author with the consequence that the algebra generated by the cohomology classes represented by the new cocycles is precisely the tautological algebra of the moduli space.There is finally a discussion of prospects for similarly finding cocycle lifts of the higher Johnson homomorphisms.

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