Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold
Number Theory
2019-07-09 v1
Abstract
We examine closed geodesics in the quotient of hyperbolic three space by the discrete group of isometries SL(2,Z[i]). There is a correspondence between closed geodesics in the manifold, the complex continued fractions originally studied by Hurwitz, and binary quadratic forms over the Gaussian integers. According to this correspondence, a geodesic is called fundamental if the associated binary quadratic form is. Using techniques from sieve theory, symbolic dynamics, and the theory of expander graphs, we show the existence of a compact set in the manifold containing infinitely many fundamental geodesics.
Cite
@article{arxiv.1907.03350,
title = {Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold},
author = {Katie McKeon},
journal= {arXiv preprint arXiv:1907.03350},
year = {2019}
}