Counting curves in hyperbolic surfaces

Abstract

Abstract. Let Σ be a hyperbolic surface. We study the set of curves on Σ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary γ0. For example, in the particular case that Σ is a once-punctured torus, we prove that the cardinality of the set of curves of type γ0 and of at most length L is asymptotic to L 2 times a constant. 1.Throughout this paper we let Σ be a complete hyperbolic surface of finite area, with genus g and r punctures, and distinct from a thrice punctured sphere. By an immersed multicurve, or simply multicurve, in Σ we will mean an immersed compact 1-dimensional submanifold of Σ each of whose components represents (the conjugacy class of) a primitive non-peripheral element in π 1 (Σ). Two multicurves γ, γ are of the same type if they belong to the same mapping class orbit, meaning that there is a diffeomorphism φ of Σ such that γ and φ(γ ) are isotopic as immersed submanifolds. In general, isotopic geodesics are considered to be equivalent. For instance, every multicurve γ is isotopic to a geodesic multicurve and the length Σ (γ) is the length of the latter.In this paper we study the set S γ 0 = Map(Σ) · γ 0 of (isotopy classes of) multicurves of some given type γ 0 . More precisely, we are interested in the behavior, when L tends to infinity, of the number |{γ ∈ S γ 0 | Σ (γ) ≤ L}| of multicurves in Σ of type γ 0 and of at most length L. Since this number grows coarsely like a polynomial of degree 6g − 6 + 2r (see [18] for the case that γ 0 is simple and [20,21] or Corollary 3.6 below for the general case), the perhaps most grappling question is whether the limitexists. Our main result is that it does if Σ is a once-punctured torus: Theorem 1.1. Let Σ be a complete hyperbolic surface of finite volume homeomorphic to a once punctured torus and let γ 0 ⊂ Σ be a multicurve. The limit (1.1) exists and moreover we have where µ Thu is the Thurston measure on the space of measured laminations ML(Σ) and C γ 0 > 0 depends only on γ 0 .In the case of simple multicurves Theorem 1.1 is due to . 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.Remark. Recently, and independently of our work, Mirzakhani [14] has established the existence of (1.1) in complete generality. Her argument and ours are different in nature and in some sense complementary. See the remarks following the statement of Corollary 4.4 in this introduction for more on the relation between Mirzakhani's result and ours.Still in the setting of simple multicurves, the case of the torus is much more treatable than the general one because any two simple multicurves in the torus are of the same type as long as they have the same number of components. This means that, in the case of the torus, counting simple multicurves of some fixed type basically reduces to counting all simple multicurves, a much simpler problem. In fact,...

Lengths of closed geodesics on random surfaces of large genus

“…More recently, the number of simple closed geodesics of length ≤ L was shown to be asymptotic to c X · L 6g−6 [Mir08], where c X is a constant depending only on X. Similar results are known to hold for the number of curves up to a given length in a fixed mapping class group orbit [Mir16,ES16].…”

Counting square-tiled surfaces with prescribed real and imaginary foliations and connections to Mirzakhani's asymptotics for simple closed hyperbolic geodesics

“…In [Mir16], Mirzakhani showed that the same asymptotic growth holds for the mapping class group orbit of any closed curve, without restrictions on the number of intersections. Erlandsson, Parlier, and Souto, see [ES16], [Erl16], and [EPS16], extended Mirzakhani's results to general length functions of closed curves, not only hyperbolic length, that extend to the space of geodesic currents; see [EU18] for a unified discussion. More recently, Rafi and Souto, see [RS17], proved analogous counting results for mapping class group orbits of arbitrary filling geodesic currents.…”

Counting curves, and the stable length of currents

“…Here δ x stands for the Dirac measure centered at x. To see what µ Thu actually has to do with the measures ν L = ν γ 0 L recall first that by [10,Proposition 4.1] we have that the family ν L is precompact and that any limit is a multiple of the Thurston measure. Suppose then that ν Ln converges to C · µ Thu .…”

“…Corollary 1.3 is due for hyperbolic metrics to Mirzakhani [17], generalizing her work on the growth of simple closed geodesics [16]. For non-positively curved metrics, it is due to Erlandsson-Souto [10]. In fact, the results of these papers are instrumental in the proof of Theorem 1.1.…”

“…In any event, armed with Theorem 1.5, we can follow the strategy from [10] and [9] to prove a version of Theorem 1.1 replacing length by stable length: Theorem 1.6. Let Σ be a compact surface with genus g and r boundary components and assume that 3g + r > 3.…”