Siegel-Veech constants in H(2)

Details Siegel-Veech constants in H(2) Siegel-Veech constants in H(2)

2005-03-30

arxiv.org/abs/math/0503718v3

Geom. Topol. 10 (2006) 1157-1172

Mathematics

Geometric Topology

This is the version published by Geometry & Topology on 12 September 2006 (V3: typesetting correction)

Scientific paper

10.2140/gt.2006.10.1157

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length generically has quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics called a Siegel--Veech constant which is shared by almost all surfaces in each moduli space of translation surfaces. Square-tiled surfaces are specific translation surfaces which have their own quadratic asymptotics for the number of cylinders of closed geodesics. It is an interesting question whether, as n tends to infinity, the Siegel--Veech constants of square-tiled surfaces with n tiles tend to the generic constants of the ambient moduli space. We prove that this is the case in the moduli space H(2) of translation surfaces of genus two with one singularity.

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