Volume 18, issue 1 (2014)

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Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold

Martin Bridgeman and Ser Peow Tan

Geometry & Topology 18 (2014) 491–520

In this paper we consider geodesic flow on finite-volume hyperbolic manifolds with non-empty totally geodesic boundary. We analyse the time for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the zeroth and first moments correspond to two cases of known identities for the orthospectrum. We also show that the second moment is given by the average time for the geodesic flow to hit the boundary. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case.

identities, hyperbolic geometry, moments
Mathematical Subject Classification 2010
Primary: 32Q45
Secondary: 32G15
Received: 6 March 2013
Revised: 27 June 2013
Accepted: 14 August 2013
Published: 29 January 2014
Proposed: Danny Calegari
Seconded: Benson Farb, Jean-Pierre Otal
Martin Bridgeman
Department of Mathematics
Boston College
Chestnut Hill, MA 02467
Ser Peow Tan
Department of Mathematics
National University of Singapore
2 Science Drive
Singapore 117543