Geodesics with one self-intersection, and other stories

By Igor Rivin. 2009

63 citations recorded by the source corpus.

Abstract In this paper, 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 asymptotic 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5b9672a7d847dc8d6a14e5ff0653cdf4fd793de4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Maryam Mirzakhani
2. V. Erlandsson
3. Juan Souto
4. A. Wright
5. Bram Petri
6. Peter Zograf
7. D. Stanford
8. Kefeng Liu

In Mathematical Physics

1. Maryam Mirzakhani
2. V. Erlandsson
3. Juan Souto
4. A. Wright
5. Kefeng Liu
6. Peter Zograf
7. Hao Xu
8. D. Stanford

Citations in the corpus

• Asymptotic growth and least common multiples in groups (2009, unknown): “, word growth investigates the asymptotic behavior of the function wΓ,X(n) = |{γ ∈ Γ : ||γ|”
• Counting curves in hyperbolic surfaces (2015, methodology): “imple multicurves Theorem 1.1 is due to McShane-Rivin [12]. Also, for simple multicurves, Mirzakhani [13] proved that the limit (1.1) exists for all g and r. Building on the work of Mirzakhani, Rivin [19] established the existence of the limit (1.1) for multicurves with a single self-intersection. R”
• Counting Mapping Class group orbits on hyperbolic surfaces (2016, methodology): “X),...,ℓβm(X)}. See §1.6 (8). 2. In this theorem γ′ is might not be simple and they can intersect. For the proof in the case of simple closed curves and curves with one self intersection see [M4] and [R2]. We remark that we do not rely on the proof of (1.1) and this paper gives a different proof of t”