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The length of closed geodesics on random Riemann Surfaces

Eran Makover·Jeffrey McGowan
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Abstract

Short geodesics are important in the study of the geometry and the spectra of Riemann surfaces. Bers' theorem gives a global bound on the length of the first $3g-3$ geodesics. We use the construction of Brooks and Makover of random Riemann surfaces to investigate the distribution of short ($< \log (g)$) geodesics on a random Riemann surfaces. We calculate the expected value of the shortest geodesic, and show that if one orders prime non-intersecting geodesics by length $γ_1\le γ_2\le ... \le γ_i ,...$, then for fixed $k$, if one allows the genus to go to infinity, the length of $γ_{k}$ is independent of the genus.

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Eran Makover, Jeffrey McGowan. 2005-04-08. The length of closed geodesics on random Riemann Surfaces. https://arxiv.org/abs/math/0504175

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