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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Bottom of the length spectrum of arithmetic orbifolds
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by Mikołaj Frączyk and Lam L. Pham HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 4745-4764 Request permission

Abstract:

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $\Gamma \backslash X$ by showing the existence of a positive constant $\delta (X)>0$ such that squares of lengths of closed geodesics shorter than $\delta$ must be pairwise linearly dependent over $\mathbb {Q}$.
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Additional Information
  • Mikołaj Frączyk
  • Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
  • ORCID: 0000-0002-6780-2548
  • Email: mfraczyk@math.uchicago.edu
  • Lam L. Pham
  • Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachussetts 02453
  • MR Author ID: 1359688
  • Email: lampham@brandeis.edu
  • Received by editor(s): July 14, 2022
  • Received by editor(s) in revised form: November 4, 2022
  • Published electronically: April 19, 2023
  • Additional Notes: The second author was supported by the Zuckerman STEM Leadership Program (as Zuckerman Postdoctoral Scholar)
  • © Copyright 2023 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 4745-4764
  • MSC (2020): Primary 11D75, 11F06, 11R06, 20G30, 22E40
  • DOI: https://doi.org/10.1090/tran/8886