Geometry & Topology

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

Martin Bridgeman and Ser Peow Tan

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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.

Article information

Geom. Topol., Volume 18, Number 1 (2014), 491-520.

Received: 6 March 2013
Revised: 27 June 2013
Accepted: 14 August 2013
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q45: Hyperbolic and Kobayashi hyperbolic manifolds
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

identities hyperbolic geometry moments


Bridgeman, Martin; Tan, Ser Peow. Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold. Geom. Topol. 18 (2014), no. 1, 491--520. doi:10.2140/gt.2014.18.491.

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