Simple closed geodesics on 𝐻⁺/Γ(3) arise from the Markov spectrum

DOI10.1090/S0273-0979-1984-15307-2zbMath0556.10015OpenAlexW1525294786MaRDI QIDQ3220654

Mark Sheingorn, Joseph Lehner

Publication date: 1984

Published in: Bulletin of the American Mathematical Society (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1090/s0273-0979-1984-15307-2

zbMATH Keywords

horocyclesimple closed geodesicsfour punctured sphere

Mathematics Subject Classification ID

Structure of modular groups and generalizations; arithmetic groups (11F06) Fuchsian groups and their generalizations (group-theoretic aspects) (20H10) Geodesic flows in symplectic geometry and contact geometry (53D25) Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) (37D40) Homogeneous approximation to one number (11J04) Riemann surfaces (30F99)

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Diophantine approximation on hyperbolic Riemann surfaces ⋮ The hyperbolic geometry of Markov's theorem on Diophantine approximation and quadratic forms ⋮ On the generalized Hurwitz equation and the Baragar-Umeda equation ⋮ Diophantine approximation on hyperbolic orbifolds ⋮ Length spectra of the Hecke triangle groups ⋮ Systoles of arithmetic surfaces and Markoff spectrum ⋮ Markov spectra for modular billiards ⋮ CLASSIFYING LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$ ⋮ A simple proof of the Markoff conjecture for prime powers ⋮ The sum of Lagrange numbers ⋮ Closed Geodesics on a Riemann Surface With Application to the Markov Spectrum ⋮ Parametrizing simple closed geodesy on Γ³\ℋ ⋮ Geometry of Riemann surfaces based on closed geodesics ⋮ The Markov spectra for Fuchsian groups ⋮ Geometry and Markoff’s spectrum for ℚ(𝕚), I ⋮ Characterization of simple closed geodesics on Fricke surfaces

Cites Work

• Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen
• On the Number of Markoff Numbers Below a Given Bound