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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Salem numbers and arithmetic hyperbolic groups


Authors: Vincent Emery, John G. Ratcliffe and Steven T. Tschantz
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
Published electronically: March 25, 2019
MathSciNet review: 3968771
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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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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

Keywords: Arithmetic group, closed geodesic, hyperbolic lattice, quadratic form, Salem number, totally real number field
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.
Article copyright: © Copyright 2019 American Mathematical Society