Counting curves, and the stable length of currents

Abstract

Let γ0 be a curve on a surface Σ of genus g and with r boundary components and let π1(Σ) X be a discrete and cocompact action on some metric space. We study the asymptotic behavior of the number of curves γ of type γ0 with translation length at most L on X. For example, as an application, we derive that for any finite generating set S, of π1(Σ) the limit lim L→∞ 1 L 6g−6+2r {γ of type γ0 with S-translation length ≤ L} exists and is positive. The main new technical tool is that the function which associates to each curve its stable length with respect to the action on X extends to a (unique) continuous and homogenous function on the space of currents. We prove that this is indeed the case for any action of a torsion free hyperbolic group.

Geodesic Currents and Counting Problems

“…The length function ℓ X on the set of curves S extends continuously to a positive homogenous function on the space C c (Σ) of compactly supported currents. In fact, many other length functions extend as well [7]. • There is some compact set K ⊂ Σ with ML(Σ) ⊂ C K (Σ).…”

“…Other important examples of positive, homogenous, and continuous functions on the space of currents arise as extensions of lengths functions, see [7]. In the particular case that the length function comes from a point X ∈ T (Σ) in Teichmüller space then we set m(X) = µ Thu λ ∈ ML ℓ X (λ) ≤ 1 .…”

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.…”

Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups

“…Recently, Erlandsson considered generating sets consisting of simple loops and obtained interesting results about intersection numbers [11,12]. However, generating sets she considers are finite and we consider infinite generating sets.…”