Comparison theorems for closed geodesics on negatively curved surfaces

TL;DR

This paper provides new asymptotic estimates and statistical results comparing word length and geodesic length of closed geodesics on negatively curved surfaces, improving existing formulas and connecting to recent research.

Contribution

It introduces novel asymptotic comparison theorems and statistical limit results for closed geodesics on variable negative curvature surfaces.

Findings

Established a central limit theorem for geodesic lengths.

Derived a local limit theorem with precise asymptotics.

Improved an existing asymptotic formula by R. Sharp and co-authors.

Abstract

In this note we present new asymptotic estimates comparing the word length and geodesic length of closed geodesics on surfaces with (variable) negative sectional curvatures. In particular, we provide an averaged comparison of these two important quantities and obtain precise statistical results, including a central limit theorem and a local limit theorem. Further, as a corollary we also improve an asymptotic formula of R. Sharp and the second author. Finally, we relate our results to recent work of Gekhtman, Taylor and Tiozzo.