Moments of a length function on the boundary of a hyperbolic manifold

Vlamis, Nicholas G

doi:10.2140/agt.2015.15.1909

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Moments of a length function on the boundary of a hyperbolic manifold

Nicholas G Vlamis

Algebraic & Geometric Topology 15 (2015) 1909–1929

DOI: 10.2140/agt.2015.15.1909

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

Mathematical Subject Classification 2000

Primary: 51M10

Secondary: 57M50

References

Publication

Received: 11 February 2014

Revised: 24 November 2014

Accepted: 8 December 2014

Published: 10 September 2015

Authors

Nicholas G Vlamis
Department of Mathematics University of Michigan 530 Church St. Ann Arbor, MI 48109 USA
http://nickvlamis.com

Volume 15, issue 4 (2015)