Salem numbers and arithmetic hyperbolic groups
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- by Vincent Emery, John G. Ratcliffe and Steven T. Tschantz PDF
- Trans. Amer. Math. Soc. 372 (2019), 329-355 Request permission
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.References
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Additional Information
- Vincent Emery
- Affiliation: Mathematisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern, Switzerland
- MR Author ID: 922488
- Email: vincent.emery@math.ch
- John G. Ratcliffe
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 145190
- Email: j.g.ratcliffe@vanderbilt.edu
- Steven T. Tschantz
- Affiliation: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240
- MR Author ID: 174820
- Email: steven.tschantz@vanderbilt.edu
- Received by editor(s): June 20, 2017
- Received by editor(s) in revised form: February 22, 2018
- Published electronically: March 25, 2019
- Additional Notes: The first author is supported by SNSF, Project No. PP00P2_157583.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 372 (2019), 329-355
- MSC (2010): Primary 11E10, 11F06, 11R06, 20H10, 30F40
- DOI: https://doi.org/10.1090/tran/7655
- MathSciNet review: 3968771