Moments of a length function on the boundary of a hyperbolic manifold

TL;DR

This paper investigates the statistical properties of the normal geodesic flow on hyperbolic manifolds with boundary, deriving formulas for moments related to boundary hitting times and connecting them to known geometric identities.

Contribution

It introduces explicit formulas for the moments of boundary hitting times in hyperbolic manifolds, linking them to orthospectra and classical identities like Basmajian's and Bridgeman's.

Findings

First moment relates to average boundary hitting time and Bridgeman's identity.

Zeroth moment recovers Basmajian's identity.

Explicit formulas involve dilogarithms in dimension two and moment generating functions in dimension three.

Abstract

In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with non-empty totally geodesic boundary. Viewing the time it takes this flow to hit the boundary as a random variable, we derive a formula for its moments in terms of the orthospectrum. The first moment gives the average time for the normal flow acting on the boundary to again reach the boundary, which we connect to Bridgeman's identity (in the surface case), and the zeroth moment recovers Basmajian's identity. Furthermore, we are able to give explicit formulae for the first moment in the surface case as well as for manifolds of odd dimension. In dimension two, the summation terms are dilogarithms. In dimension three, we are able to find the moment generating function for this length function.