scholarly journals Effective counting of simple closed geodesics on hyperbolic surfaces

2021 ◽  
Author(s):  
Alex Eskin ◽  
Maryam Mirzakhani ◽  
Amir Mohammadi
Keyword(s):  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Simple Closed Geodesics

scholarly journals SQUARE-INTEGRABILITY OF THE MIRZAKHANI FUNCTION AND STATISTICS OF SIMPLE CLOSED GEODESICS ON HYPERBOLIC SURFACES

2020 ◽  
Vol 8 ◽  
Author(s):  
FRANCISCO ARANA-HERRERA ◽  
JAYADEV S. ATHREYA
Keyword(s):  
Moduli Space ◽  
Global Behavior ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Counting Problems ◽  
Finite Area ◽  
Geodesic Laminations ◽  
Square Integrability ◽  
Simple Closed Geodesics ◽  
Hyperbolic Geodesics

Given integers $g,n\geqslant 0$ satisfying $2-2g-n<0$ , let ${\mathcal{M}}_{g,n}$ be the moduli space of connected, oriented, complete, finite area hyperbolic surfaces of genus $g$ with $n$ cusps. We study the global behavior of the Mirzakhani function $B:{\mathcal{M}}_{g,n}\rightarrow \mathbf{R}_{{\geqslant}0}$ which assigns to $X\in {\mathcal{M}}_{g,n}$ the Thurston measure of the set of measured geodesic laminations on $X$ of hyperbolic length ${\leqslant}1$ . We improve bounds of Mirzakhani describing the behavior of this function near the cusp of ${\mathcal{M}}_{g,n}$ and deduce that $B$ is square-integrable with respect to the Weil–Petersson volume form. We relate this knowledge of $B$ to statistics of counting problems for simple closed hyperbolic geodesics.

SYSTOLIC FILLINGS OF SURFACES

2018 ◽  
Vol 98 (3) ◽  
pp. 502-511
Author(s):  
BIDYUT SANKI
Keyword(s):  
Closed Geodesic ◽  
Disjoint Union ◽  
Hyperbolic Surface ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Simple Closed Geodesics

A filling of a closed hyperbolic surface is a set of simple closed geodesics whose complement is a disjoint union of hyperbolic polygons. The systolic length is the length of a shortest essential closed geodesic on the surface. A geodesic is called systolic, if the systolic length is realised by its length. For every $g\geq 2$, we construct closed hyperbolic surfaces of genus $g$ whose systolic geodesics fill the surfaces with complements consisting of only two components. Finally, we remark that one can deform the surfaces obtained to increase the systole.

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 Graphs of systoles on hyperbolic surfaces

2019 ◽  
Vol 11 (01) ◽  
pp. 1-20 ◽  
Author(s):  
Bidyut Sanki ◽  
Siddhartha Gadgil
Keyword(s):  
Planar Graphs ◽  
Classical Result ◽  
Hyperbolic Surface ◽  
Necessary Condition ◽  
Major Result ◽  
Closed Geodesics ◽  
Hyperbolic Surfaces ◽  
Fat Graphs ◽  
Admissible Graph

Given a hyperbolic surface, the set of all closed geodesics whose length is minimal forms a graph on the surface, in fact a so-called fat graph, which we call the systolic graph. We study which fat graphs are systolic graphs for some surface (we call these admissible).There is a natural necessary condition on such graphs, which we call combinatorial admissibility. Our first main result is that this condition is also sufficient.It follows that a sub-graph of an admissible graph is admissible. Our second major result is that there are infinitely many minimal non-admissible fat graphs (in contrast, for instance, to the classical result that there are only two minimal non-planar graphs).

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.

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