Counting Closed Geodesics in Rank 1 $\mathrm{SL}\left(2,\mathbb{R}\right)$-orbit Closures

TL;DR

This paper establishes bounds on the intersection numbers of triangulations and the count of closed geodesics within rank 1 $ ext{SL}(2, ext{R})$-orbit closures, revealing exponential growth constraints related to geodesic length and saddle connections.

Contribution

It provides new exponential bounds on the number of closed geodesics and intersection counts in rank 1 orbit closures in moduli space, advancing understanding of their geometric structure.

Findings

Exponential bounds on closed geodesics of length at most R

Bounds on intersection numbers along Teichmüller geodesics

Quantitative relation between geodesic length and saddle connection regions

Abstract

We obtain bounds on the numbers of intersections between triangulations as the conformal structure of a surface varies along a Teichm{\"u}ller geodesic contained in an $\mathrm{SL}\left(2,\mathbb{R}\right)$-orbit closure of rank 1 in the moduli space of Abelian differentials. For $0 \leq \theta \leq 1$, we obtain an exponential bound on the number of closed geodesics in the orbit closure, of length at most $R$, that spend at least $\theta$-fraction of their length in a region with short saddle connections.