Closed Geodesics on a Riemann Surface With Application to the Markov Spectrum

DOI10.2307/2000055zbMath0597.10024OpenAlexW4247547384MaRDI QIDQ3728986

Mark Sheingorn, Joseph Lehner, Alan F. Beardon

Publication date: 1986

Full work available at URL: https://doi.org/10.2307/2000055

zbMATH Keywords

Riemann surfaceclosed geodesicsMarkoff forms

Mathematics Subject Classification ID

Continued fractions and generalizations (11J70) Structure of modular groups and generalizations; arithmetic groups (11F06) Holomorphic modular forms of integral weight (11F11) Variational problems in applications to the theory of geodesics (problems in one independent variable) (58E10) Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) (30F35)

Related Items (9)

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 ⋮ McShane's identity, using elliptic elements ⋮ Statistical mechanics of \(2+1\) gravity from Riemann zeta function and Alexander polynomial: Exact results ⋮ Growth in free groups (and other stories) -- twelve years later ⋮ 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

• The geometry of Markoff numbers
• Characterization of simple closed geodesics on Fricke surfaces
• Diophantine approximation on hyperbolic Riemann surfaces
• Approach to Markoff's minimal forms through modular functions
• Simple closed geodesics on 𝐻⁺/Γ(3) arise from the Markov spectrum
• On the Number of Markoff Numbers Below a Given Bound
• Minimum of quadratic forms with respect to Fuchsian groups. I.

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