# Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold

## TL;DR

This paper investigates the properties of closed geodesics in a specific hyperbolic three-manifold, establishing a link with quadratic forms and demonstrating the existence of infinitely many fundamental geodesics within a compact set.

## Contribution

It introduces a novel connection between geodesics, continued fractions, and quadratic forms in an arithmetic hyperbolic 3-manifold, proving the abundance of fundamental geodesics.

## Key findings

- Existence of infinitely many fundamental geodesics in a compact set
- Establishment of a correspondence between geodesics and quadratic forms
- Application of sieve theory and symbolic dynamics techniques

## 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.