McShane-Rivin norm balls and simple-length multiplicities

TL;DR

This paper establishes new bounds on the number of simple closed geodesics of a given length on hyperbolic once-punctured tori, using McShane--Rivin norm ball estimates, and improves Markoff number bounds.

Contribution

It introduces novel estimates for McShane--Rivin norm balls that lead to sharper bounds on geodesic counts and Markoff number fibers, revealing geometric obstructions.

Findings

Number of simple closed geodesics of length L ≥ 2 is at most C_X(log L)^2.

For the modular torus, Markoff number fibers are bounded by C(log log(3m))^2.

New geometric insights into McShane--Rivin norm balls and obstructions to flatness.

Abstract

We use normal-turn estimates for McShane--Rivin norm balls to prove that, for every complete finite-area hyperbolic once-punctured torus $X$, the number of simple closed geodesics of length exactly $L\geq 2$ is at most $C_X(\log L)^2$. For the modular torus, this gives $$ \#\lambda_M^{-1}(m)\leq C(\log\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic Markoff-fiber bounds. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions.