Growth of the number of simple closed geodesics on hyperbolic surfaces

By Maryam Mirzakhani. 2008

233 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/dabc46f11276dd3a60542f477b99d54f4fd14f4f

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Citations in the corpus

Counting curves in hyperbolic surfaces (2015, background): “Many thanks are due to Maryam Mirzakhani for interesting comments and for sharing a draft of her paper [14].”
Ergodic Theory of the Earthquake Flow (2010, methodology): “Using the results of this paper, we prove an equidistribution result for the level sets of the lengths of simple closed curves in the moduli space [25].”
Geodesics with one self-intersection, and other stories (2009, background): “By the results of Mirzakhani [16] on the equidistribution of the mapping class orbit of a multicurve in measured lamination space, we get: Theorem 4.”
Lengths of closed geodesics on random surfaces of large genus (2017, unknown): “perbolic surface Xwe have Ng,[0,L](X) ∼ e L/L as L→ ∞, see for instance [Bus10] for details. More recently, the number of simple closed geodesics of length ≤ Lwas shown to be asymptotic to cX · L6g−6 [Mir08], where cX is a constant depending only on X. Similar results are known to hold for the numbe”
Masur-Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves (2019, methodology): “ed geodesic laminations with respect to the length function deﬁned by the hyperbolic metric X. The integral on the right-hand side of formula (3.6) is explicitely evaluated by Mirzakhani in the paper [Mi3]. Combined with our formula (1.13) for the Masur– Veech volume VolQg of the moduli space of qua”
A random tunnel number one 3–manifold does not fiber over the circle (2005, methodology): “Fortunately, Mirzakhani has shown that there is a Geometry & Topology, Volume 10 (2006) Geometry & Topology, Volume 10 (2006) Geometry & Topology, Volume 10 (2006) definite probability, roughly 1/5, that a randomly chosen multicurve is of this form (see Theorem 11.1 below).”