Equidistribution of horospheres on moduli spaces of hyperbolic surfaces

Abstract

Given a simple closed curve γ on a connected, oriented, closed surface S of negative Euler characteristic, Mirzakhani showed that the set of points in the moduli space of hyperbolic structures on S having a simple closed geodesic of length L of the same topological type as γ equidistributes with respect to a natural probability measure as L → ∞. We prove several generalizations of Mirzakhani's result and discuss some of the technical aspects ommited in her original work. The dynamics of the earthquake flow play a fundamental role in the arguments in this paper. Contents 1. Introduction 1 2. Background material 11 3. Equidistribution of horoballs 18 4. Equidistribution of horospheres 33 References 40

Length statistics of random multicurves on closed hyperbolic surfaces

“…We learned from the paper [AH19] that this kind of statistics was initially conjectured by S. Wolpert. Papers [AH19] and [AH20] established results closely related to Theorem 1.2. Proposition 4.8 below is based on a theorem stated by M. Mirzakhani but presented without a detailed proof.…”

“…Remark. While the author was finishing this note, the article of F. Arana-Herrera [AH19] appeared on the arXiv. The paper [AH19] and the current paper are devoted to similar circle of problems and use similar circle of ideas, though they were written in parallel and completely independently.…”

“…While the author was finishing this note, the article of F. Arana-Herrera [AH19] appeared on the arXiv. The paper [AH19] and the current paper are devoted to similar circle of problems and use similar circle of ideas, though they were written in parallel and completely independently. In particular, paper [AH19] proves a much more general version of our Theorem 4.1 ([AH19, Theorem 1.3]), which is one of the key ingredients allowing to attack the counting problem and the length statistics.…”

“…The paper [AH19] and the current paper are devoted to similar circle of problems and use similar circle of ideas, though they were written in parallel and completely independently. In particular, paper [AH19] proves a much more general version of our Theorem 4.1 ([AH19, Theorem 1.3]), which is one of the key ingredients allowing to attack the counting problem and the length statistics. We learned from the paper [AH19] that this kind of statistics was initially conjectured by S. Wolpert.…”

“…In particular, paper [AH19] proves a much more general version of our Theorem 4.1 ([AH19, Theorem 1.3]), which is one of the key ingredients allowing to attack the counting problem and the length statistics. We learned from the paper [AH19] that this kind of statistics was initially conjectured by S. Wolpert. Papers [AH19] and [AH20] established results closely related to Theorem 1.2.…”