Key Takeaway: Random punctured tori can be studied using the cross ratio distribution of ideal hyperbolic quadrilaterals, revealing distributions of geometric quantities like geodesic length spectrum and conformal modulus.

Abstract

Earlier work introduced a geometrically natural probability measure on the group of all Möbius transformations of the hyperbolic plane so as to be able to study ‘random’ groups of Möbius 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 2π 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 Teichmüller space to the central ‘square’ punctured torus.