A norm on homology of surfaces and counting simple geodesics

By Greg McShane, Igor Rivin. International mathematics research notices, 2000

56 citations recorded by the source corpus.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2698464ac0714197d025a36296c756a049f13e87

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geodesics with one self-intersection, and other stories (2009, background): “ve length at least 2arccosh3 (this is attained for a thrice punctured sphere) – this result should be compared with related results of J. Hempel obtained in [17]. Using a diﬀerent technique (based on [27, 28]) we show thatthenumberofelementsshorterthanLintheMod(T)orbitofabouquetoftwo geodesic circle”
THE DIMENSION OF THE TORELLI GROUP (2007, background): “We start by proving the necessary facts about minimizing cycles in the integral case (see also [29]), then proceed to the rational case, and finally the real case.”
Counting curves in hyperbolic surfaces (2015, methodology): “SON AND JUAN SOUTO where Thu is the Thurston measure on the space of measured laminations ML() and C 0 >0 depends only on 0. In the case of simple multicurves Theorem 1.1 is due to McShane-Rivin [12]. Also, for simple multicurves, Mirzakhani [13] proved that the limit (1.1) exists for all g and ”
Rotation, entropy, and equilibrium states (2001, background): “In fact this rotation set is precisely the unit ball for the stable norm (or Federer norm) on H1(M,R) (see [30], [31], [35], [36] for more details).”