2019 Primitive geodesic lengths and (almost) arithmetic progressions
J.-F. Lafont, D. B. McReynolds
Publ. Mat. 63(1): 183-218 (2019). DOI: 10.5565/PUBLMAT6311906


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.


Download Citation

J.-F. Lafont. D. B. McReynolds. "Primitive geodesic lengths and (almost) arithmetic progressions." Publ. Mat. 63 (1) 183 - 218, 2019. https://doi.org/10.5565/PUBLMAT6311906


Received: 7 April 2017; Revised: 18 May 2018; Published: 2019
First available in Project Euclid: 7 December 2018

zbMATH: 07040966
MathSciNet: MR3908791
Digital Object Identifier: 10.5565/PUBLMAT6311906

Primary: 53C22

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

Rights: Copyright © 2019 Universitat Autònoma de Barcelona, Departament de Matemàtiques


This article is only available to subscribers.
It is not available for individual sale.

Vol.63 • No. 1 • 2019
Back to Top