scholarly journals The geometry of ideal 2-dimensional simplicial complexes

2001 ◽  
Vol 43 (1) ◽  
pp. 39-66 ◽  
Author(s):  
Charalampos Charitos ◽  
Athanase Papadopoulos
Keyword(s):  
Teichmüller Space ◽  
Simplicial Complexes ◽  
Teichmuller Space ◽  
Geometric Structures ◽  
Hyperbolic Structures

In this paper, we study geometric structures on 2-dimensional simplicial complexes. In particular, we consider hyperbolic structures and measured foliations on these simplicial complexes. We describe the spaces of such structures and we relate the two resulting spaces in a manner which is analogous to Thurston's compactification of the Teichmüller space of a surface.

scholarly journals Geodesically Complete Hyperbolic Structures

2017 ◽  
Vol 166 (2) ◽  
pp. 219-242
Author(s):  
ARA BASMAJIAN ◽  
DRAGOMIR ŠARIĆ
Keyword(s):  
Teichmüller Space ◽  
Hyperbolic Surface ◽  
Teichmuller Space ◽  
Geometric Proof ◽  
Hyperbolic Structures ◽  
Infinite Type ◽  
Length Spectrum Metric ◽  
Geodesically Complete ◽  
The Relationship ◽  
Convex Core

AbstractIn the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

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 COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

2020 ◽  
Vol 8 ◽  
Author(s):  
DAVID DUMAS ◽  
ANNA LENZHEN ◽  
KASRA RAFI ◽  
JING TAO
Keyword(s):  
Finite Type ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Global Behavior ◽  
Coarse Geometry ◽  
Hyperbolic Structures ◽  
Punctured Torus ◽  
Thurston Metric ◽  
Detailed Picture

We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem.

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.

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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