Mirzakhani's frequencies of simple closed geodesics on hyperbolic surfaces in large genus and with many cusps

Abstract

We present a proof of a conjecture proposed by V. Delecroix, E. Goujard, P. Zograf, and A. Zorich, which describes the large genus asymptotic behaviours of the ratio of frequencies of separating over nonseparating simple closed geodesics on a closed hyperbolic surface of genus $g$ with $n$ cusps. We explicitly give the function $f(\frac{n}{g})$ in the conjecture. The moderate behaviour of the frequencies with respect to the growth rate of the number of cusps compared to that of the genus drastically contrasts with the behaviour of other geometric quantities and exhibits the topological nature of the frequencies.