Publicacions Matemàtiques

Primitive geodesic lengths and (almost) arithmetic progressions

J.-F. Lafont and D. B. McReynolds

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In this article we investigate when the set of primitive geodesic lengths on a Riemannian manifold have arbitrarily long arithmetic progressions. We prove that in the space of negatively curved metrics, a metric having such arithmetic progressions is quite rare. We introduce almost arithmetic progressions, a coarsification of arithmetic progressions, and prove that every negatively curved, closed Riemannian manifold has arbitrarily long almost arithmetic progressions in its primitive length spectrum. Concerning genuine arithmetic progressions, we prove that every noncompact, locally symmetric, arithmetic manifold has arbitrarily long arithmetic progressions in its primitive length spectrum. We end with a conjectural characterization of arithmeticity in terms of arithmetic progressions in the primitive length spectrum. We also suggest an approach to a well known spectral rigidity problem based on the scarcity of manifolds with arithmetic progressions.

Article information

Publ. Mat., Volume 63, Number 1 (2019), 183-218.

Received: 7 April 2017
Revised: 18 May 2018
First available in Project Euclid: 7 December 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C22: Geodesics [See also 58E10]

primitive geodesic almost arithmetic progression specification property locally symmetric space arithmetic manifold modular surface


Lafont, J.-F.; McReynolds, D. B. Primitive geodesic lengths and (almost) arithmetic progressions. Publ. Mat. 63 (2019), no. 1, 183--218. doi:10.5565/PUBLMAT6311906.

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