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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Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732728

Digital Object Identifier
doi:10.2140/gt.2014.18.491

Mathematical Reviews number (MathSciNet)
MR3159167

Zentralblatt MATH identifier
1290.32028

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

Keywords
identities hyperbolic geometry moments

Citation

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. https://projecteuclid.org/euclid.gt/1513732728


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