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Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds.

Mikhail Belolipetsky1, Matilde Lalín2, Plinio G P Murillo3

  • 1IMPA, Estrada Dona Castorina, 110, Rio de Janeiro, 22460-320 Brazil.

Bulletin of the Brazilian Mathematical Society = Boletim Da Sociedade Brasileira De Matematica
|June 1, 2022
PubMed
Summary

This study quantifies the link between closed geodesics in arithmetic hyperbolic orbifolds and Salem numbers. It reveals that 3-dimensional orbifolds define specific Salem numbers, offering new insights into their distribution and growth properties.

Keywords:
Arithmetic 3-orbifoldClosed geodesicSalem number

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Area of Science:

  • Number Theory
  • Geometry
  • Dynamical Systems

Background:

  • The lengths of closed geodesics in arithmetic hyperbolic orbifolds are known to be related to Salem numbers.
  • A quantitative understanding of this relationship is lacking.

Purpose of the Study:

  • To initiate a quantitative study of the relationship between closed geodesics and Salem numbers in arithmetic hyperbolic orbifolds.
  • To establish bounds for Salem numbers associated with these geometric structures.

Main Methods:

  • Analysis of arithmetic 3-dimensional hyperbolic orbifolds.
  • Comparison of defined Salem numbers with the total count of such numbers.
  • Application of the Marklof gap conjecture for compact orbifolds.

Main Results:

  • Any non-compact arithmetic 3-dimensional orbifold defines a specific quantity of square-rootable Salem numbers of degree 4.
  • This quantity is shown to be asymptotically related to the total number of such Salem numbers.
  • Lower bounds for strong exponential growth in the geodesic spectrum of even-dimensional orbifolds are obtained.

Conclusions:

  • The study provides a quantitative framework for understanding Salem numbers in the context of hyperbolic orbifolds.
  • Results are extended to compact orbifolds under the assumption of the gap conjecture.
  • New lower bounds are established for the growth of mean multiplicities in geodesic spectra.