# 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.

## Repository Citation

Linowitz, Benjamin. 2018. "Counting Problems for Geodesics on Arithmetic Hyperbolic Surfaces." Proceedings of the American Mathematical Society 146(3): 1347-1361.

## 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