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.

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 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.

A characterization of biholomorphic automorphisms of Teichmüller space

2012 ◽  
Vol 154 (1) ◽  
pp. 71-83
Author(s):  
RYOSUKE MINEYAMA ◽  
HIDEKI MIYACHI
Keyword(s):  
Riemann Surface ◽  
Finite Type ◽  
Teichmüller Space ◽  
Teichmuller Space ◽  
Alternative Approach

AbstractIn this paper, we give an alternative approach to Royden–Earle–Kra–Markovic's characterization of biholomorphic automorphisms of Teichmüller space of Riemann surface of analytically finite type.

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 Pleating coordinates for the maskit embedding of the teichmüller space of punctured tori

Topology ◽  
1993 ◽  
Vol 32 (4) ◽  
pp. 719-749 ◽  
Author(s):  
Linda Keen ◽  
Caroline Series
Keyword(s):  
Teichmüller Space ◽  
Teichmuller Space
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