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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
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.
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.
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