Geodesic
Moments of a length function on the boundary of a hyperbolic manifold
Vlamis, Nicholas G1
1Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109 USA
Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 1909-1929
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

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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.

DOI : 10.2140/agt.2015.15.1909
Keywords: Basmajian's identity, identities on hyperbolic manifolds, length function, moments

Vlamis, Nicholas G  1

1 Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI 48109 USA 
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Vlamis, Nicholas G. Moments of a length function on the boundary of a hyperbolic manifold. Algebraic and Geometric Topology, Tome 15 (2015) no. 4, pp. 1909-1929. doi: 10.2140/agt.2015.15.1909

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