scholarly journals Parametrizing simple closed geodesy on Γ3\ℋ

2003 ◽  
Vol 74 (1) ◽  
pp. 43-60 ◽  
Author(s):  
Thomas A. Schmidt ◽  
Mark Sheingorn
Keyword(s):  
Lower Bound ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Closed Curves ◽  
Bounded Interval ◽  
Fundamental Domains ◽  
Simple Closed Geodesics

AbstractWe exhibit a canonical geometric pairing of the simple closed curves of the degree three cover of the modular surface, Γ3\ℋ, with the proper single self-intersecting geodesics of Crisp and Moran. This leads to a pairing of fundamental domains for Γ3with Markoff triples.The routes of the simple closed geodesics are directly related to the above. We give two parametrizations of these. Combining with work of Cohn, we achieve a listing of all simple closed geodesics of length within any bounded interval. Our method is direct, avoiding the determination of geodesic lengths below the chosen lower bound.

scholarly journals Counting One-Sided Simple Closed Geodesics on Fuchsian Thrice Punctured Projective Planes

2018 ◽  
Vol 2020 (13) ◽  
pp. 3886-3901
Author(s):  
Michael Magee
Keyword(s):  
Asymptotic Formula ◽  
Projective Plane ◽  
Projective Planes ◽  
Hyperbolic Metric ◽  
Closed Geodesics ◽  
Real Projective Plane ◽  
Closed Curves ◽  
The Hyperbolic Metric ◽  
Simple Closed Geodesics

Abstract We prove that there is a true asymptotic formula for the number of one-sided simple closed curves of length $\leq L$ on any Fuchsian real projective plane with three points removed. The exponent of growth is independent of the hyperbolic metric, and it is noninteger, in contrast to counting results of Mirzakhani for simple closed curves on orientable Fuchsian surfaces.

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

2007 ◽  
Vol 03 (03) ◽  
pp. 475-501 ◽  
Author(s):  
THOMAS A. SCHMIDT ◽  
MARK SHEINGORN
Keyword(s):  
Quadratic Forms ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Explicit Formulas ◽  
Indefinite Quadratic ◽  
Positive Integers ◽  
Simple Closed Geodesics ◽  
Hall Ray

The Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. In our previous work, we classified the low height-achieving non-simple geodesics of [Formula: see text] into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized.

CLASSIFYING LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$

2007 ◽  
Vol 03 (03) ◽  
pp. 421-438 ◽  
Author(s):  
THOMAS A. SCHMIDT ◽  
MARK SHEINGORN
Keyword(s):  
Closed Geodesics ◽  
Modular Surface ◽  
Positive Integers ◽  
Simple Closed Geodesics ◽  
Hall Ray

We show that low height-achieving non-simple geodesics on a low-index cover of the modular surface can be classified into seven types, according to the topology of highest arcs. The lowest geodesics of the signature (0;2,2,2,∞)-orbifold [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. As a step in resolving this conjecture, we characterize the geometry on [Formula: see text] of geodesic arcs with heights strictly between 3 and 6. Of these, one type of geodesic arc cannot realize the height of any geodesic.

SHARP POINCARÉ–SOBOLEV INEQUALITIES AND THE SHORTEST LENGTH OF SIMPLE CLOSED GEODESICS ON A TOPOLOGICAL TWO SPHERE

2004 ◽  
Vol 06 (05) ◽  
pp. 781-792 ◽  
Author(s):  
MEIJUN ZHU
Keyword(s):  
Riemannian Manifold ◽  
Sobolev Inequalities ◽  
Square Root ◽  
Two Dimensional ◽  
Closed Geodesics ◽  
Sharp Constants ◽  
Simple Closed Geodesics

We show that the sharp constants of Poincaré–Sobolev inequalities for any smooth two dimensional Riemannian manifold are less than or equal to [Formula: see text]. For a smooth topological two sphere M2, the sharp constants are [Formula: see text] if and only if M2 is isometric to two sphere S2 with the standard metric. In the same spirit, we show that for certain special smooth topological sphere the ratio between the shortest length of simple closed geodesics and the square root of its area is less than or equals to [Formula: see text].

scholarly journals Non-simple geodesics in hyperbolic 3-manifolds

1994 ◽  
Vol 116 (2) ◽  
pp. 339-351
Author(s):  
Kerry N. Jones ◽  
Alan W. Reid
Keyword(s):  
Closed Geodesic ◽  
Closed Geodesics ◽  
Totally Geodesic ◽  
Geodesic Surfaces ◽  
Simple Closed Geodesics

AbstractChinburg and Reid have recently constructed examples of hyperbolic 3-manifolds in which every closed geodesic is simple. These examples are constructed in a highly non-generic way and it is of interest to understand in the general case the geometry of and structure of the set of closed geodesics in hyperbolic 3-manifolds. For hyperbolic 3-manifolds which contain immersed totally geodesic surfaces there are always non-simple closed geodesics. Here we construct examples of manifolds with non-simple closed geodesics and no totally geodesic surfaces.

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

A quadratic parabolic group

1975 ◽  
Vol 77 (2) ◽  
pp. 281-288 ◽  
Author(s):  
Robert Riley
Keyword(s):  
Normal Form ◽  
Half Space ◽  
Transformation Group ◽  
Universal Covering ◽  
Covering Space ◽  
Fundamental Domains ◽  
Universal Covering Space ◽  
Other Information ◽  
Figure Eight Knot

When k is a 2-bridge knot with group πK, there are parabolic representations (p-reps) θ: πK → PSL(): = PSL(2, ). The most obvious problem that this suggests is the determination of a presentation for an image group πKθ. We shall settle the easiest outstanding case in section 2 below, viz. k the figure-eight knot 41, which has the 2-bridge normal form (5, 3). We shall prove that the (two equivalent) p-reps θ for this knot are isomorphisms of πK on πKθ. Furthermore, the universal covering space of S3\k can be realized as Poincaré's upper half space 3, and πKθ is a group of hyperbolic isometries of 3 which is also the deck transformation group of the covering 3 → S3\k. The group πKθ is a subgroup of two closely related groups that we study in section 3. We shall give fundamental domains, presentations, and other information for all these groups.

Sign in / Sign up

Export Citation Format

Share Document