Effective mapping class group dynamics III: Counting filling closed curves on surfaces

Abstract

We prove a quantitative estimate with a power saving error term for the number of filling closed geodesics of a given topological type and length ≤ L on an arbitrary closed, orientable, negatively curved surface. More generally, we prove estimates of the same kind for the number of free homotopy classes of filling closed curves of a given topological type on a closed, orientable surface whose geometric intersection number with respect to a given filling geodesic current is ≤ L. The proofs rely on recent progress made on the study of the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface, and introduce a novel method for addressing counting problems of mapping class group orbits that naturally yields power saving error terms. This method is also applied to study counting problems of mapping class group orbits of Teichmüller space with respect to Thurston's asymmetric metric.

Counting problems from the viewpoint of ergodic theory: from primitive integer points to simple closed curves

“…Even more recently, in [Ara21b], novel methods were introduced by the author to prove analogous effective estimates for countings of filling closed geodesics of a given topological type. These methods rely on recent progress made in the prequels [Ara20] and [Ara21a] on the study of the effective dynamics of the mapping class group on Teichmüller space and the space of closed curves of a closed, orientable surface.…”

“…Theorem 6.9. [Ara21b] Let X ∈ M g be a hyperbolic structure on S g and γ be a filling closed curve on S g . Then, for every L > 0,…”