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arXiv · 1907.03350

Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold

Katie McKeon
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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.

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Katie McKeon. 2019-07-07. Low-lying Geodesics in an Arithmetic Hyperbolic Three-Manifold. https://arxiv.org/abs/1907.03350

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