Curves, Surfaces, and Hyperbolic Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Homotopy Class ◽  
Hyperbolic Geometry ◽  
Hyperbolic Plane ◽  
Closed Surface ◽  
Intersection Numbers ◽  
Hyperbolic Surfaces ◽  
Closed Curves

This chapter explains the basics of working with simple closed curves, focusing on the case of the closed surface Sɡ of genus g. When g is greater than or equal to 2, hyperbolic geometry enters as a useful tool since each homotopy class of simple closed curves has a unique geodesic representative. The chapter begins by recalling some basic results about surfaces and hyperbolic geometry, with particular emphasis on the boundary of the hyperbolic plane and hyperbolic surfaces. It then considers simple closed curves in a surface S, along with geodesics and intersection numbers. It also discusses the bigon criterion, homotopy versus isotopy for simple closed curves, and arcs. Finally, it describes the change of coordinates principle and three facts about homeomorphisms.

scholarly journals Dehn’s algorithm for simple diagrams

2015 ◽  
Vol 24 (14) ◽  
pp. 1550074
Author(s):  
Charles Frohman ◽  
Joanna Kania-Bartoszynska
Keyword(s):  
Conjugacy Class ◽  
Disjoint Union ◽  
Closed Surface ◽  
Intersection Numbers ◽  
Closed Curves ◽  
Class A ◽  
Positive Genus ◽  
Monotonically Decrease ◽  
Standard Presentation ◽  
Closed Oriented Surface

Dehn gave an algorithm for deciding if two cyclic words in the standard presentation of the fundamental group of a closed oriented surface of positive genus represent the same conjugacy class. A simple diagram on a surface is a disjoint union of simple closed curves none of which bound a disk. If [Formula: see text] is a once punctured closed surface of negative Euler characteristic, simple diagrams are classified up to isotopy by their geometric intersection numbers with the edges of an ideal triangulation of [Formula: see text]. Simple diagrams on the unpunctured surface [Formula: see text] can be represented by simple diagrams on [Formula: see text]. The weight of a simple diagram is the sum of its geometric intersection numbers with the edges of the triangulation. We show that you can pass from any representative to a least weight representative via a sequence of elementary moves, that monotonically decrease weights. This leads to a geometric analog of Dehn’s algorithm for simple diagrams.

scholarly journals The primitivity index function for a free group, and untangling closed curves on hyperbolic surfaces.With the appendix by Khalid Bou–Rabee

2017 ◽  
Vol 166 (1) ◽  
pp. 83-121
Author(s):  
NEHA GUPTA ◽  
ILYA KAPOVICH
Keyword(s):  
Lower Bounds ◽  
Free Group ◽  
Growth Function ◽  
Residual Finiteness ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Finitely Generated ◽  
Closed Curves ◽  
Index Function ◽  
Finite Covers

AbstractMotivated by the results of Scott and Patel about “untangling” closed geodesics in finite covers of hyperbolic surfaces, we introduce and study primitivity, simplicity and non-filling index functions for finitely generated free groups. We obtain lower bounds for these functions and relate these free group results back to the setting of hyperbolic surfaces. An appendix by Khalid Bou–Rabee connects the primitivity index functionfprim(n,FN) to the residual finiteness growth function forFN.

scholarly journals Short closed geodesics with self-intersections

2020 ◽  
Vol 169 (3) ◽  
pp. 623-638
Author(s):  
VIVEKA ERLANDSSON ◽  
HUGO PARLIER
Keyword(s):  
Linear Function ◽  
Hyperbolic Surface ◽  
Minimal Length ◽  
The Self ◽  
Closed Geodesics ◽  
Intersection Numbers ◽  
Hyperbolic Surfaces ◽  
Fixed Integer

AbstractOur main point of focus is the set of closed geodesics on hyperbolic surfaces. For any fixed integer k, we are interested in the set of all closed geodesics with at least k (but possibly more) self-intersections. Among these, we consider those of minimal length and investigate their self-intersection numbers. We prove that their intersection numbers are upper bounded by a universal linear function in k (which holds for any hyperbolic surface). Moreover, in the presence of cusps, we get bounds which imply that the self-intersection numbers behave asymptotically like k for growing k.

Complete systems of surfaces in 3-manifolds

1997 ◽  
Vol 122 (1) ◽  
pp. 185-191 ◽  
Author(s):  
FENGCHUN LEI
Keyword(s):  
Finite Number ◽  
Boundary Component ◽  
Pairwise Disjoint ◽  
Complete System ◽  
Closed Surface ◽  
The Other ◽  
Closed Curves ◽  
Orientable Surfaces

A complete system (CS) [Jscr ]={J1, ..., Jn} on a connected closed surface F is a collection of pairwise disjoint simple closed curves on F such that the surface obtained by cutting F open along [Jscr ] is a 2-sphere with 2n-holes. Two CSs on F are equivalent if each can be obtained from the other via finite number of slides (defined in Section 1) and isotopies. Let M be a 3-manifold and F a boundary component of M of genus n. A CS of surfaces for M is a CS on F which bounds n pairwise disjoint incompressible orientable surfaces in M. When [Jscr ] is a CS of discs on the boundary of a handlebody V, it is well known that any CS on F which is equivalent to [Jscr ] is also a CS of discs for V. Our first result says that the same thing happens for a CS of surfaces for M, that is, if [Jscr ] is a CS of surfaces for M, then any CS equivalent to [Jscr ] is also a CS of surfaces for M. The following theorem is our main result on CS of surfaces in 3-manifolds:

scholarly journals On the Convergence of the Elastic Flow in the Hyperbolic Plane

2020 ◽  
Vol 5 (1) ◽  
pp. 40-77
Author(s):  
Marius Müller ◽  
Adrian Spener
Keyword(s):  
Hyperbolic Space ◽  
Elastic Energy ◽  
Hyperbolic Plane ◽  
Blow Up ◽  
Type Inequality ◽  
Gradient Flow ◽  
Global Minimizer ◽  
Infinite Time ◽  
Closed Curves ◽  
Convergence Results

AbstractWe examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

ON SACKS–UHLENBECK'S EXISTENCE THEOREM FOR HARMONIC MAPS VIA EXPONENTIALLY HARMONIC MAPS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250105 ◽  
Author(s):  
TOSHIAKI OMORI
Keyword(s):  
Existence Theorem ◽  
Homotopy Class ◽  
Existence Result ◽  
Blow Up ◽  
Harmonic Maps ◽  
Closed Surface ◽  
Closed Manifold ◽  
Blow Up Analysis

The blow-up analysis for a sequence of exponentially harmonic maps from a closed surface is studied to reestablish an existence result of harmonic maps from a closed surface into a closed manifold whose 2-dimensional homotopy class vanishes.

scholarly journals Diameter, width and thickness in the hyperbolic plane

2021 ◽  
Vol 112 (3) ◽  
Author(s):  
Ákos G. Horváth
Keyword(s):  
Convex Body ◽  
Hyperbolic Geometry ◽  
Hyperbolic Plane ◽  
Convex Set ◽  
Constant Width ◽  
Plane Convex ◽  
Plane Convex Body ◽  
Constant Diameter

AbstractIn hyperbolic geometry there are several concepts to measure the breadth or width of a convex set. In the first part of the paper we collect them and compare their properties. Than we introduce a new concept to measure the width and thickness of a convex body. Correspondingly, we define three classes of bodies, bodies of constant with, bodies of constant diameter and bodies having the constant shadow property, respectively. We prove that the property of constant diameter follows to the fulfilment of constant shadow property, and both of them are stronger as the property of constant width. In the last part of this paper, we introduce the thickness of a constant body and prove a variant of Blaschke’s theorem on the larger circle inscribed to a plane-convex body of given thickness and diameter.

scholarly journals Minimizing intersection points of curves under virtual homotopy

2020 ◽  
Vol 29 (03) ◽  
pp. 2050007
Author(s):  
Vladimir Chernov ◽  
David Freund ◽  
Rustam Sadykov
Keyword(s):  
Poisson Bracket ◽  
Homotopy Class ◽  
Finite Collection ◽  
Virtual Link ◽  
Closed Curves ◽  
Oriented Surface ◽  
Single Curve ◽  
Intersection Points

A flat virtual link is a finite collection of oriented closed curves [Formula: see text] on an oriented surface [Formula: see text] considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves [Formula: see text], we show that the minimal number of intersection points of curves in the virtual homotopy class of [Formula: see text] equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.

Pseudo-Anosov Theory

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Braid Groups ◽  
Intersection Numbers ◽  
Branched Covers ◽  
Algebraic Integers ◽  
Dehn Twists ◽  
Mapping Classes ◽  
Dynamical Properties

This chapter focuses on the construction as well as the algebraic and dynamical properties of pseudo-Anosov homeomorphisms. It first presents five different constructions of pseudo-Anosov mapping classes: branched covers, constructions via Dehn twists, homological criterion, Kra's construction, and a construction for braid groups. It then proves a few fundamental facts concerning stretch factors of pseudo-Anosov homeomorphisms, focusing on the theorem that pseudo-Anosov stretch factors are algebraic integers. It also considers the spectrum of pseudo-Anosov stretch factors, along with the special properties of those measured foliations that are the stable (or unstable) foliations of some pseudo-Anosov homeomorphism. Finally, it describes the orbits of a pseudo-Anosov homeomorphism as well as lengths of curves and intersection numbers under iteration.

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