Configurations of curves and geodesics on surfaces (original) (raw)
Related papers
Geometriae Dedicata, 2001
We study simple closed geodesics on a hyperbolic surface of genus g with b geodesic boundary components and c cusps. We show that the number of such geodesics of length at most L is of order L 6g+2b+2c−6. This answers a long-standing open question.
Geodesics with one self-intersection, and other stories
Advances in Mathematics, 2012
In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic γ with a single double point is asymptotic to L dim(Teichmuller space of S.). Since closed geodesics with one double point fall into a finite number of Mod(S) orbits, we get the same asympotic estimate for the number of such geodesics of length bounded by L, and systems of curves, where one curve has a self-intersection, or there are two curves intersecting once. We also use our (elementary) methods to do a more precise study of geodesics with a single double point on a punctured torus, including an extension of McShane's identity to such geodesics. In the second part of the paper we study the question of when a covering of the boundary of an oriented surface S can be extended to a covering of the surface S itself, we obtain a complete answer to that question, and also to the question of when we can further require the extension to be a regular covering of S. We also analyze the question of the minimal index of a subgroup in a surface group which does not contain a given element. We show that we have a linear bound for the index of an arbitrary subgroup, a cubic bound for the index of a normal subgroup, but also poly-log bounds for each fixed level in the lower central series (using elementary arithmetic considerations)-the results hold for free groups and fundamental groups of closed surfaces.
Hyperbolic 3-Manifolds With Nonintersecting Closed Geodesics
Geometriae Dedicata, 2003
A hyperbolic 3-manifold is said to have the spd-property if all its closed geodesics are simple and pairwise disjoint. For a 3-manifold which supports a geometrically finite hyperbolic structure we show the following dichotomy: either the generic hyperbolic structure has the spd-property or no hyperbolic structure has the spd-property. Both cases are shown to occur. In particular, we prove that the generic hyperbolic structure on the interior of a handlebody (or a surface cross an interval) of negative Euler characteristic has the spd-property. Simplicity and disjointness are consequences of a variational result for hyperbolic surfaces. Namely, the intersection angle between closed geodesics varies nontrivially under deformation of a hyperbolic surface.
Shortest Length geodesics on closed hyperbolic surfaces
Given a hyperbolic surface, the set of all closed geodesics whose length is minimal form a graph on the surface, in fact a so called fat graph, which we call the systolic graph. The central question that we study in this paper is: which fat graphs are systolic graphs for some surface - we call such graphs admissible. This is motivated in part by the observation that we can naturally decompose the moduli space of hyperbolic surfaces based on the associated systolic graphs. A systolic graph has a metric on it, so that all cycles on the graph that correspond to geodesics are of the same length and all other cycles have length greater than these. This can be formulated as a simple condition in terms of equations and inequations for sums of lengths of edges which we call combinatorial admissibility. Our first main result is that admissibility is equivalent to combinatorial admissibility. This is proved using properties of negative curvature, specifically that polygonal curves with long e...
Non-simple geodesics in hyperbolic 3-manifolds
Mathematical Proceedings of the Cambridge Philosophical Society, 1994
Chinburg 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.
Some remarks on simple closed geodesics of surfaces with ends
2009
If a non-compact complete surface M is not homeomorphic to a subset of the plane or of the projective plane, then it has infinitely many simple closed geodesics . In this paper, we consider simple closed geodesics on a surface homeomorphic to such a subset.
Cleanliness of geodesics in hyperbolic 3-manifolds
Pacific Journal of Mathematics, 2004
In this paper, we investigate geodesics in cusped hyperbolic 3-manifolds. We derive conditions guaranteeing the existence of geodesics avoiding the cusps and use these geodesics to show that in "almost all" finite volume hyperbolic 3-manifolds, infinitely many horoballs in the universal cover corresponding to a cusp are visible in a fundamental domain of the cusp when viewed from infinity.
Simple closed geodesics in hyperbolic 3-manifolds
Bulletin of the London Mathematical Society, 1999
The question of which Riemannian manifolds admit simple closed geodesics is still a mystery. It is not known whether all closed Riemannian manifolds contain simpleclosed geodesics. For closed manifolds with nontrivial fundamental group, a simple closed geodesic can always be found by taking the shortest homotopically nontrivial closed geodesic. When the manifold is closed but simply connected, the question is open for dimensions three and above. In dimension two, it is known by the theorem of Lusternik ...
Geodesic systems of tunnels in hyperbolic 3–manifolds
Algebraic & Geometric Topology, 2014
It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite-volume hyperbolic 3-manifold. In this paper, we address the generalization of this question to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite-volume hyperbolic 3-manifolds with a single cusp, with a system of n tunnels, n 1 of which come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number n manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a .1I n/-compression body with a system of n core tunnels, n 1 of which self-intersect. 57M50; 57M27, 30F40