Asymptotic average multiplicities of the hyperbolic length spectrum

Determine the asymptotic (average) behavior of the multiplicities in the length spectrum for hyperbolic metrics on closed surfaces, quantifying how the multiplicity of lengths grows with the length and establishing precise asymptotic formulas or bounds.

Background

For negatively curved Riemannian metrics, the length spectrum generically has multiplicity one, and Margulis’ prime geodesic theorem provides asymptotics for primitive closed geodesics. In contrast, for hyperbolic metrics on surfaces the multiplicity of the length set is known to be unbounded. Buser established explicit lower bounds for multiplicities, and numerical and theoretical work has explored related phenomena.

Despite these advances, the paper emphasizes that the asymptotic average multiplicities for hyperbolic surfaces are largely unknown, marking an open quantitative problem about the distribution and average growth of multiplicities in the hyperbolic length spectrum.