Lower bound on the number of non-simple closed geodesics on surfaces

Abstract

Abstract. We give a lower bound on the number of non-simple closed curves on a hyperbolic surface, given upper bounds on both length and self-intersection number. In particular, we carefully show how to construct closed geodesics on pairs of pants, and give a lower bound on the number of curves in this case. The lower bound for arbitrary surfaces follows from the lower bound on pairs of pants. This lower bound demonstrates that as the self-intersection number K = K(L) goes from a constant to a quadratic function of L, the number of closed geodesics transitions from polynomial to exponential in L. We show upper bounds on the number of such geodesics in a subsequent paper.

Counting curves in hyperbolic surfaces

“…A lower bound of the same order of magnitude can be obtained just by adding crossings to the simple multicurves carried by some maximal recurrent train-track, but it is anyways due to Sapir [20,21]: Corollary 3.6. Let Σ be a hyperbolic surface of genus g and r punctures.…”

“…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 limit (1.1) lim L→∞ |{γ ∈ S γ 0 | Σ (γ) ≤ L}| L 6g−6+2r exists. Our main result is that it does if Σ is a once-punctured torus: Theorem 1.1.…”

Bounds on the Number of Non-Simple Closed Geodesics on a Surface

“…Words corresponding to closed geodesics. In Section 3, we build a combinatorial model for geodesics on S that is very similar to the one for geodesics on a pair of pants P described in [Sap15b]. In that paper, we assign a cyclic word w(γ) to each curve γ on P .…”

“…Moreover, we can deduce a lower bound on #O(L, K) from our previous work. In our previous paper [Sap15b], we give a lower bound on the size of G c (L, K) for a pair of pants. Since the Mod S stabilizer of a pair of pants inside a surface S is finite, this gives us the following lower bound on O(L, K) for an arbitrary surface: Theorem 1.3 (Corollary of Theorem 1.1 in [Sap15b]).…”

“…(In fact, their results give these same asymptotics when the length of a curve is measured using any geodesic current.) The author has given explicit bounds on #G c (L, K) in terms of both L and K in [Sap15b,Sap15a].…”