Counting problems for geodesics on arithmetic hyperbolic surfaces
Geometric Topology
2017-02-28 v1
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 nonnegative real numbers.
Cite
@article{arxiv.1702.08062,
title = {Counting problems for geodesics on arithmetic hyperbolic surfaces},
author = {Benjamin Linowitz},
journal= {arXiv preprint arXiv:1702.08062},
year = {2017}
}