Random Ideal Hyperbolic Quadrilaterals, the Cross Ratio Distribution and Punctured Tori

TL;DR

This paper studies the probability distribution of geometric features of random punctured tori, derived from ideal hyperbolic quadrilaterals, extending previous work on random M"obius groups to these surfaces.

Contribution

It identifies the distribution of the cross ratio of ideal quadrilaterals and computes distributions of geometric quantities of random punctured tori, including length spectrum and conformal modulus.

Findings

Distribution of cross ratios for ideal quadrilaterals is characterized.

Distributions of geodesic length spectrum and conformal modulus are calculated.

Distribution of Teichm"uller space distance to the square torus is derived.

Abstract

Earlier work introduced a geometrically natural probability measure on the group of all M\"obius transformations of the hyperbolic plane so as to be able to study "random" groups of M\"obius transformations, and in particular random two-generator groups. Here we extend these results to consider random punctured tori. These Riemann surfaces have finite hyperbolic area $2\pi$ and fundamental group the free group of rank 2. They can be obtained by pairing (identifying) the opposite sides of an ideal hyperbolic quadrilateral. There is a natural distribution on ideal quadrilateral given by the cross ratio of their vertices. We identify this distribution and then calculate the distributions of various geometric quantities associated with random punctured tori such as the base of the geodesic length spectrum and the conformal modulus, along with more subtle things such as the distribution of the distance in Teichm\"uller space to the central "square" punctured torus.