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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Moments of a length function on the boundary of a hyperbolic manifold

Nicholas G Vlamis

Algebraic & Geometric Topology 15 (2015) 1909–1929

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.

Basmajian's identity, identities on hyperbolic manifolds, length function, moments
Mathematical Subject Classification 2000
Primary: 51M10
Secondary: 57M50
Received: 11 February 2014
Revised: 24 November 2014
Accepted: 8 December 2014
Published: 10 September 2015
Nicholas G Vlamis
Department of Mathematics
University of Michigan
530 Church St.
Ann Arbor, MI 48109 USA