Counting Problems for Geodesics on Arithmetic Hyperbolic Surfaces

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.

Publisher

American Mathematical Society

Publication Date

3-1-2018

Publication Title

Proceedings of the American Mathematical Society

Department

Mathematics

Document Type

Article

DOI

https://dx.doi.org/10.1090/proc/13782

Keywords

Quaternion algebras, Commensurability, Theorem

Language

English

Format

text

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