scholarly journals A Calculation of the Orbifold Euler Number of the Moduli Space of Curves by a New Cell Decomposition of the Teichmüller Space

2000 ◽  
Vol 23 (1) ◽  
pp. 87-100 ◽  
Author(s):  
Satoshi NAKAMURA
Keyword(s):  
Moduli Space ◽  
Euler Number ◽  
Teichmüller Space ◽  
Cell Decomposition ◽  
Teichmuller Space ◽  
Moduli Space Of Curves

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.

NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE

1998 ◽  
Vol 09 (01) ◽  
pp. 1-45 ◽  
Author(s):  
JØRGEN ELLEGAARD ANDERSEN
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Teichmüller Space ◽  
Closed Surface ◽  
Teichmuller Space ◽  
Open Dense Subset ◽  
Complex Structures ◽  
Thurston Boundary ◽  
Simply Connected ◽  
Singular Metric

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on ℳ(Σ), the moduli space of flat G-connections, for G any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold ℳ′(Σ), the open dense subset of smooth points of ℳ(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on ℳ′(Σ) converges to the polarization associated to (F, g). In particular we see that the Jeffrey–Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuiuty at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on ℳ′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.

A Hilbert manifold structure on the Weil–Petersson class Teichmüller space of bordered Riemann surfaces

2015 ◽  
Vol 17 (04) ◽  
pp. 1550016 ◽  
Author(s):  
David Radnell ◽  
Eric Schippers ◽  
Wolfgang Staubach
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Banach Manifold ◽  
Holomorphic Maps ◽  
Discontinuous Group ◽  
Hilbert Manifold ◽  
Conformal Field ◽  
Compact Riemann Surfaces

We consider bordered Riemann surfaces which are biholomorphic to compact Riemann surfaces of genus g with n regions biholomorphic to the disk removed. We define a refined Teichmüller space of such Riemann surfaces (which we refer to as the WP-class Teichmüller space) and demonstrate that in the case that 2g + 2 - n > 0, this refined Teichmüller space is a Hilbert manifold. The inclusion map from the refined Teichmüller space into the usual Teichmüller space (which is a Banach manifold) is holomorphic. We also show that the rigged moduli space of Riemann surfaces with non-overlapping holomorphic maps, appearing in conformal field theory, is a complex Hilbert manifold. This result requires an analytic reformulation of the moduli space, by enlarging the set of non-overlapping mappings to a class of maps intermediate between analytically extendible maps and quasiconformally extendible maps. Finally, we show that the rigged moduli space is the quotient of the refined Teichmüller space by a properly discontinuous group of biholomorphisms.

THE BERGMAN METRIC ON TEICHMÜLLER SPACE

2004 ◽  
Vol 15 (10) ◽  
pp. 1085-1091 ◽  
Author(s):  
BO-YONG CHEN
Keyword(s):  
Simple Proof ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Bergman Metric

We give a simple proof of a theorem of McMullen on Kähler hyperbolicity of moduli space of Riemann surfaces by using the Bergman metric on Teichmüller space.

scholarly journals Higher Complex Structures

2019 ◽  
Author(s):  
Vladimir Fock ◽  
Alexander Thomas
Keyword(s):  
Moduli Space ◽  
Geometric Structure ◽  
Hilbert Scheme ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Complex Structures ◽  
Punctual Hilbert Scheme

Abstract We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so-called higher complex structure, we use the punctual Hilbert scheme of the plane. The moduli space of higher complex structures is defined and is shown to be a generalization of the classical Teichmüller space. We give arguments for the conjectural isomorphism between the moduli space of higher complex structures and Hitchin’s component.

scholarly journals Dynamical degrees of Hurwitz correspondences

2018 ◽  
Vol 40 (7) ◽  
pp. 1968-1990 ◽  
Author(s):  
ROHINI RAMADAS
Keyword(s):  
Moduli Space ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Critical Set ◽  
Branched Covering ◽  
Numerical Invariants ◽  
Dynamical Degrees

Let $\unicode[STIX]{x1D719}$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\unicode[STIX]{x1D719}$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$—of the moduli space ${\mathcal{M}}_{0,\mathbf{P}}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $\unicode[STIX]{x1D719}$ at and near points of its post-critical set.

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