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Counting problems for geodesics on arithmetic hyperbolic surfaces

Author: Benjamin Linowitz
Journal: Proc. Amer. Math. Soc. 146 (2018), 1347-1361
MSC (2010): Primary 57M50
Published electronically: September 14, 2017
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Abstract: It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well-known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of non-negative real numbers.

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Additional Information

Benjamin Linowitz
Affiliation: Department of Mathematics, Oberlin College, Oberlin, Ohio 44074

Received by editor(s): February 28, 2017
Received by editor(s) in revised form: April 15, 2017
Published electronically: September 14, 2017
Communicated by: David Futer
Article copyright: © Copyright 2017 American Mathematical Society

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