Volume 15, issue 4 (2015)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Moments of a length function on the boundary of a hyperbolic manifold

Nicholas G Vlamis

Algebraic & Geometric Topology 15 (2015) 1909–1929
Abstract

In this paper we will study the statistics of the unit geodesic flow normal to the boundary of a hyperbolic manifold with nonempty 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.

Keywords
Basmajian's identity, identities on hyperbolic manifolds, length function, moments
Primary: 51M10
Secondary: 57M50