Volume 10, issue 2 (2006)

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Siegel–Veech constants in $\mathcal{H}(2)$

Samuel Lelièvre

Geometry & Topology 10 (2006) 1157–1172

arXiv: math/0503718


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 was proved by Eskin and Masur to generically have 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 the Siegel–Veech constant of a given moduli space can be recovered as a limit of individual constants of square-tiled surfaces in this moduli space. We prove that this is the case in the moduli space (2) of translation surfaces of genus two with one singularity.

abelian differentials, moduli space, geodesics
Mathematical Subject Classification 2000
Primary: 30F30
Secondary: 53C22
Forward citations
Received: 30 March 2005
Revised: 23 February 2006
Accepted: 24 May 2006
Published: 12 September 2006
Proposed: Walter Neumann
Seconded: Benson Farb, Tobias Colding
Samuel Lelièvre
Mathematics Institute
University of Warwick
Coventry CV4 7AL