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The Quarterly Journal of Mathematics Advance Access originally published online on May 24, 2007
The Quarterly Journal of Mathematics 2007 58(3):281-295; doi:10.1093/qmath/ham014
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© 2007. Published by Oxford University Press. All rights reserved. For permissions, please email: journals.permissions@oxfordjournals.org

DISTRIBUTION OF ANGLES BETWEEN GEODESIC RAYS ASSOCIATED WITH HYPERBOLIC LATTICE POINTS

Florin P. Boca{dagger}

1 Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA
2 Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy, PO Box 1-764, RO-014700 Bucharest Romania

{dagger} E-mail: fboca{at}math.uiuc.edu

Received 30 August 2006; revised 2 January 2007
  Abstract

For every two points z0, z1 in the upper-half plane H, consider all elements {gamma} in the principal congruence group {Gamma}(N), acting on H by fractional linear transformations, such that the hyperbolic distance between z1 and {gamma} z0 is at most R > 0. We study the distribution of angles between the geodesic rays [z1, {gamma} z0] as R -> {infty}, proving that the limiting distribution exists independently of N and explicitly computing it. When z1 = z0, this is found to be the uniform distribution on the interval [–{pi}/2, {pi}/2].


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