Side parameter for the torus with a single cone point

2015 ◽  
Vol 24 (10) ◽  
pp. 1540005 ◽  
Author(s):  
Hirotaka Akiyoshi
Keyword(s):  
Hyperbolic Plane ◽  
Kleinian Groups ◽  
Cone Point ◽  
Hyperbolic Structures ◽  
Single Cone ◽  
Punctured Torus

We introduce the side parameter map from the space of cone hyperbolic structures on the torus with a single cone point to the hyperbolic plane ℍ2, based on the idea in the Jorgensen theory for punctured torus Kleinian groups. Then, we show that the side parameter map is a homeomorphism.

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 Hyperbolic manifolds and degenerating handle additions

1993 ◽  
Vol 55 (1) ◽  
pp. 72-89 ◽  
Author(s):  
Martin Scharlemann ◽  
Ying-Qing Wu
Keyword(s):  
Hyperbolic Manifolds ◽  
Combinatorial Proof ◽  
Dehn Filling ◽  
Hyperbolic Structures ◽  
Punctured Torus ◽  
First Main Theorem

AbstractA 2-handle addition on the boundary of a hyperbolic 3-manifold M is called degenerating if the resulting manifold is not hyperbolic. There are examples that some manifolds admit infinitely many degenerating handle additions. But most of them are not ‘basic’. (See Section 1 for definitions). Our first main theorem shows that there are only finitely many basic degenerating handle additions. We also study the case that one of the handle additions produces a reducible manifold, and another produces a ∂-reducible manifold, showing that in this case either the two attaching curves are disjoint, or they can be isotoped into a once-punctured torus. A byproduct is a combinatorial proof of a similar known result about degenerating hyperbolic structures by Dehn filling.

scholarly journals Shapes of hyperbolic triangles and once-punctured torus groups

2021 ◽  
Author(s):  
Sang-hyun Kim ◽  
Thomas Koberda ◽  
Jaejeong Lee ◽  
Ken’ichi Ohshika ◽  
Ser Peow Tan ◽  
...  
Keyword(s):  
Fundamental Groups ◽  
Hyperbolic Metrics ◽  
Analogous Question ◽  
Single Cone ◽  
Punctured Torus ◽  
Fixed Area ◽  
Dense Set ◽  
Finitely Presented ◽  
Torsion Free ◽  
Holonomy Map

AbstractLet $$\Delta $$ Δ be a hyperbolic triangle with a fixed area $$\varphi $$ φ . We prove that for all but countably many $$\varphi $$ φ , generic choices of $$\Delta $$ Δ have the property that the group generated by the $$\pi $$ π -rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $$\varphi \in (0,\pi ){\setminus }\mathbb {Q}\pi $$ φ ∈ ( 0 , π ) \ Q π , a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $$\mathfrak {C}_\theta $$ C θ of singular hyperbolic metrics on a torus with a single cone point of angle $$\theta =2(\pi -\varphi )$$ θ = 2 ( π - φ ) , and answer an analogous question for the holonomy map $$\rho _\xi $$ ρ ξ of such a hyperbolic structure $$\xi $$ ξ . In an appendix by Gao, concrete examples of $$\theta $$ θ and $$\xi \in \mathfrak {C}_\theta $$ ξ ∈ C θ are given where the image of each $$\rho _\xi $$ ρ ξ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3-manifolds.

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

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