Siegel–Veech constants in ℋ(2)

Lelièvre, Samuel

doi:10.2140/gt.2006.10.1157

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

Samuel Lelièvre

Geometry & Topology 10 (2006) 1157–1172

DOI: 10.2140/gt.2006.10.1157

arXiv: math/0503718

Abstract

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 $ℋ (2)$ of translation surfaces of genus two with one singularity.

Keywords

abelian differentials, moduli space, geodesics

Mathematical Subject Classification 2000

Primary: 30F30

Secondary: 53C22

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Publication

Received: 30 March 2005

Revised: 23 February 2006

Accepted: 24 May 2006

Published: 12 September 2006

Proposed: Walter Neumann

Seconded: Benson Farb, Tobias Colding

Authors

Samuel Lelièvre
Mathematics Institute University of Warwick Coventry CV4 7AL UK
http://carva.org/samuel.lelievre/

Volume 10, issue 2 (2006)