scholarly journals Hyperbolic structures from Sol on pseudo-Anosov mapping tori

2016 ◽  
Vol 20 (1) ◽  
pp. 437-468
Author(s):  
Kenji Kozai
Keyword(s):  
Hyperbolic Structures ◽  
Mapping Tori

scholarly journals HYPERBOLIC 3-MANIFOLDS AND CLUSTER ALGEBRAS

2017 ◽  
Vol 235 ◽  
pp. 1-25
Author(s):  
KENTARO NAGAO ◽  
YUJI TERASHIMA ◽  
MASAHITO YAMAZAKI
Keyword(s):  
Cluster Algebras ◽  
Hyperbolic Structures ◽  
Mapping Tori ◽  
Mapping Classes

We advocate the use of cluster algebras and their $y$ -variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster $y$ -variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.

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.

scholarly journals Ideal simplices and double-simplices, their non-orientable hyperbolic manifolds, cone manifolds and orbifolds with Dehn type surgeries and graphic analysis

2021 ◽  
Vol 112 (1) ◽  
Author(s):  
E. Molnár ◽  
I. Prok ◽  
J. Szirmai
Keyword(s):  
Acta Math ◽  
Hyperbolic Manifolds ◽  
Space Forms ◽  
Hyperbolic Structures ◽  
Princeton University ◽  
Regular Ideal ◽  
Graphic Analysis ◽  
Novi Sad ◽  
Memorial Volume ◽  
János Bolyai

AbstractIn connection with our works in Molnár (On isometries of space forms. Colloquia Math Soc János Bolyai 56 (1989). Differential geometry and its applications, Eger (Hungary), North-Holland Co., Amsterdam, 1992), Molnár (Acta Math Hung 59(1–2):175–216, 1992), Molnár (Beiträge zur Algebra und Geometrie 38/2:261–288, 1997) and Molnár et al. (in: Prékopa, Molnár (eds) Non-Euclidean geometries, János Bolyai memorial volume mathematics and its applications, Springer, Berlin, 2006), Molnár et al. (Symmetry Cult Sci 22(3–4):435–459, 2011) our computer program (Prok in Period Polytech Ser Mech Eng 36(3–4):299–316, 1992) found 5079 equivariance classes for combinatorial face pairings of the double-simplex. From this list we have chosen those 7 classes which can form charts for hyperbolic manifolds by double-simplices with ideal vertices. In such a way we have obtained the orientable manifold of Thurston (The geometry and topology of 3-manifolds (Lecture notes), Princeton University, Princeton, 1978), that of Fomenko–Matveev–Weeks (Fomenko and Matveev in Uspehi Mat Nauk 43:5–22, 1988; Weeks in Hyperbolic structures on three-manifolds. Ph.D. dissertation, Princeton, 1985) and a nonorientable manifold $$M_{c^2}$$ M c 2 with double simplex $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 , seemingly known by Adams (J Lond Math Soc (2) 38:555–565, 1988), Adams and Sherman (Discret Comput Geom 6:135–153, 1991), Francis (Three-manifolds obtainable from two and three tetrahedra. Master Thesis, William College, 1987) as a 2-cusped one. This last one is represented for us in 5 non-equivariant double-simplex pairings. In this paper we are going to determine the possible Dehn type surgeries of $$M_{c^2}={\widetilde{{\mathcal {D}}}}_1$$ M c 2 = D ~ 1 , leading to compact hyperbolic cone manifolds and multiple tilings, especially orbifolds (simple tilings) with new fundamental domain to $${\widetilde{{\mathcal {D}}}}_1$$ D ~ 1 . Except the starting regular ideal double simplex, we do not get further surgery manifold. We compute volumes for starting examples and limit cases by Lobachevsky method. Our procedure will be illustrated by surgeries of the simpler analogue, the Gieseking manifold (1912) on the base of our previous work (Molnár et al. in Publ Math Debr, 2020), leading to new compact cone manifolds and orbifolds as well. Our new graphic analysis and tables inform you about more details. This paper is partly a survey discussing as new results on Gieseking manifold and on $$M_{c^2}$$ M c 2 as well, their cone manifolds and orbifolds which were partly published in Molnár et al. (Novi Sad J Math 29(3):187–197, 1999) and Molnár et al. (in: Karáné, Sachs, Schipp (eds) Proceedings of “Internationale Tagung über geometrie, algebra und analysis”, Strommer Gyula Nemzeti Emlékkonferencia, Balatonfüred-Budapest, Hungary, 1999), updated now to Memory of Professor Gyula Strommer. Our intention is to illustrate interactions of Algebra, Analysis and Geometry via algorithmic and computational methods in a classical field of Geometry and of Mathematics, in general.

scholarly journals Ranks of mapping tori via the curve complex

2019 ◽  
Vol 2019 (748) ◽  
pp. 153-172 ◽  
Author(s):  
Ian Biringer ◽  
Juan Souto
Keyword(s):  
Fundamental Group ◽  
Orientable Surface ◽  
Curve Complex ◽  
Mapping Torus ◽  
Closed Orientable Surface ◽  
Mapping Tori

Abstract We show that if ϕ is a homeomorphism of a closed, orientable surface of genus g, and ϕ has large translation distance in the curve complex, then the fundamental group of the mapping torus {M_{\phi}} has rank {2g+1} .

scholarly journals Toric actions in cosymplectic geometry

2019 ◽  
Vol 31 (4) ◽  
pp. 907-915 ◽  
Author(s):  
Giovanni Bazzoni ◽  
Oliver Goertsches
Keyword(s):  
Symplectic Manifolds ◽  
Cosymplectic Manifolds ◽  
Mapping Tori

Abstract We show that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.

On the Reducibility and the η-Invariant of Periodic Automorphisms of Genus 2 Surface

1997 ◽  
Vol 06 (06) ◽  
pp. 827-831 ◽  
Author(s):  
Takayuki Morifuji
Keyword(s):  
Finite Order ◽  
Mapping Class Group ◽  
Class Group ◽  
Finite Subgroup ◽  
Mapping Class ◽  
Mapping Tori ◽  
Genus 2

We give a characterization for the reducibility of elements of any finite subgroup of the mapping class group of genus 2 surface in terms of the η-invariant of finite order mapping tori.

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