Salem numbers and arithmetic hyperbolic groups

@article{Emery2019SalemNA,
title={Salem numbers and arithmetic hyperbolic groups},
author={Vincent Emery and John G. Ratcliffe and Steven T. Tschantz},
journal={Transactions of the American Mathematical Society},
year={2019}
}
• Published 11 June 2015
• Mathematics
• Transactions of the American Mathematical Society
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…
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