On Mirzakhani's twist torus conjecture

TL;DR

This paper investigates Mirzakhani's conjecture on the statistical distribution of twist tori in hyperbolic surface moduli space, demonstrating equidistribution for some families and singularity for others.

Contribution

It proves Mirzakhani's conjecture for specific families of twist tori and identifies multiple expanding families with distinct limiting distributions.

Findings

Certain twist tori equidistribute to Lebesgue measure

Other families have mutually singular limiting distributions

Results advance understanding of hyperbolic surface moduli space

Abstract

We address a conjecture of Mirzakhani about the statistical behavior of certain expanding families of ``twist tori'' in the moduli space of hyperbolic surfaces, showing that they equidistribute to a certain Lebesgue-class measure along almost all sequences. We also identify a number of other expanding families of twist tori whose limiting distributions are mutually singular to Lebesgue.