#### Volume 10, issue 2 (2006)

 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Editorial Procedure Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Author Index To Appear Other MSP Journals
Siegel–Veech constants in $\mathcal{H}(2)$

### Samuel Lelièvre

Geometry & Topology 10 (2006) 1157–1172
 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 $\mathsc{ℋ}\left(2\right)$ of translation surfaces of genus two with one singularity.

##### Keywords
abelian differentials, moduli space, geodesics
Primary: 30F30
Secondary: 53C22
##### Publication
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/