Salem numbers and arithmetic hyperbolic groups

Vincent Emery; John G. Ratcliffe; Steven T. Tschantz

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arXiv · 1506.03727

Salem numbers and arithmetic hyperbolic groups

Vincent Emery·John G. Ratcliffe·Steven T. Tschantz

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Abstract

In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.

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BibTeX RIS

Vincent Emery, John G. Ratcliffe, Steven T. Tschantz. 2015-06-11. Salem numbers and arithmetic hyperbolic groups. https://arxiv.org/abs/1506.03727

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