Teichmuller Space

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space ◽  
Hyperbolic Structure ◽  
Closed Curves ◽  
Pants Decomposition

This chapter deals with Teichmüller space Teich(S) of a surface S. It first defines Teichmüller space and a topology on Teich(S) before giving two heuristic counts of its dimension. It then describes explicit coordinates on Teich(Sɡ) coming from certain length and twist parameters for curves in a pair of pants decomposition of Sɡ; these are the Fenchel–Nielsen coordinates on Teich(Sɡ). The chapter also considers the Teichmüller space of the torus and concludes by proving the 9g – 9 theorem, which states that a hyperbolic structure on Sɡ is completely determined by the lengths assigned to 9g – 9 isotopy classes of simple closed curves in Sɡ.

scholarly journals Singular pleated surfaces and CP1–structures

1995 ◽  
Vol 37 (2) ◽  
pp. 179-190 ◽  
Author(s):  
Ser Peow Tan
Keyword(s):  
Teichmüller Space ◽  
Universal Cover ◽  
Orientable Surface ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Distinguished Point ◽  
Totally Geodesic ◽  
Closed Orientable Surface ◽  
Hyperbolic Structures ◽  
Usual Interpretation

Let Fg be a closed orientable surface of genus g > 1 and let be the Teichmuller space of Fg, i.e., the space of marked hyperbolic structures on Fg We shall also denote by the space of marked hyperbolic structures on Fgwith one distinguished point; by this, we mean a distinguished point on the universal cover gof Fg. This space is isomorphic to the space of marked complete hyperbolic structures on a genus g surface with 1 cusp which is the usual interpretation of . Choose a decomposition of Fginto pairs of pants by a collection of non–intersecting, totally geodesic simple closed curves. The Fenchel–Nielsen coordinates for relative to this decomposition are given by the lengths of the curves as well as twist parameters defined on each curve. Varying the length and twist parameters gives deformations of the marked hyperbolic structures.

scholarly journals QUASISYMMETRIC SEWING IN RIGGED TEICHMÜLLER SPACE

2006 ◽  
Vol 08 (04) ◽  
pp. 481-534 ◽  
Author(s):  
DAVID RADNELL ◽  
ERIC SCHIPPERS
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Operator Algebras ◽  
Complex Structure ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Teichmüller Spaces ◽  
Teichmuller Spaces

One of the basic geometric objects in conformal field theory (CFT) is the moduli space of Riemann surfaces whose n boundaries are "rigged" with analytic parametrizations. The fundamental operation is the sewing of such surfaces using the parametrizations to identify points. An alternative model is the moduli space of n-punctured Riemann surfaces together with local biholomorphic coordinates at the punctures. We refer to both of these moduli spaces as the "rigged Riemann moduli space".By generalizing to quasisymmetric boundary parametrizations, and defining rigged Teichmüller spaces in both the border and puncture pictures, we prove the following results: (1) The Teichmüller space of a genus-g surface bordered by n closed curves covers the rigged Riemann and rigged Teichmüller moduli spaces of surfaces of the same type, and induces complex manifold structures on them; (2) With this complex structure, the sewing operation is holomorphic; (3) The border and puncture pictures of the rigged moduli and rigged Teichmüller spaces are biholomorphically equivalent.These results are necessary in rigorously defining CFT (in the sense of G. Segal), as well as for the construction of CFT from vertex operator algebras.

COMPLEX FENCHEL-NIELSEN COORDINATES FOR QUASI-FUCHSIAN STRUCTURES

1994 ◽  
Vol 05 (02) ◽  
pp. 239-251 ◽  
Author(s):  
SER PEOW TAN
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space ◽  
Closed Curves ◽  
Oriented Surface ◽  
Hyperbolic Structures ◽  
Embedded Image ◽  
Totally Real ◽  
Real Subspace ◽  
Closed Oriented Surface

Let Fg be a closed oriented surface of genus g ≥ 2 and let [Formula: see text] be the space of marked quasi-fuchsian structures on Fg. Let [Formula: see text] be a set of non-intersecting, non-trivial simple closed curves on Fg that cuts Fg into pairs of pants components. In this note, we construct global complex coordinates for [Formula: see text] relative to [Formula: see text] giving an embedding of [Formula: see text] into [Formula: see text]. The totally real subspace of [Formula: see text] with respect to these coordinates is the Teichmüller Space [Formula: see text] of marked hyperbolic structures on Fg, the coordinates reduce to the usual Fenchel-Nielsen coordinates for [Formula: see text] relative to [Formula: see text]. Various properties of the embedded image are studied.

scholarly journals Fenchel–Nielsen coordinates on upper bounded pants decompositions

2015 ◽  
Vol 158 (3) ◽  
pp. 385-397 ◽  
Author(s):  
DRAGOMIR ŠARIĆ
Keyword(s):  
Teichmüller Space ◽  
Hyperbolic Surface ◽  
Length Spectrum ◽  
Teichmuller Space ◽  
Closed Geodesics ◽  
Infinite Type ◽  
Pants Decomposition ◽  
Length Spectrum Metric ◽  
Path Connected ◽  
Simple Closed Geodesics

AbstractLet X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

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.

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