Abstract
We present an approach for counting the Teichmüller volumes of the moduli spaces of Abelian differentials on a Riemann surface of genus g. We show that the volumes can be counted by means of counting the “integer points” in the corresponding moduli space. The “integer points” are represented by square tiled surfaces — the flat surfaces tiled by unit squares. Such tilings have several conical singularities with 8, 12,... adjacent unit squares. Counting the leading term in the asymptotics of the number of tilings having at most N unit squares, we get the volumes of the corresponding strata of the moduli spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. Eskin, H. Masur. (2001) Asymptotic formulas on flat surfaces. Ergodic Theory Dynamic. Systems 21, no. 2, 443–478.
A. Eskin, A. Okounkov. (2000) Asymptotics of number of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials, to appear in Inventiones Mathematicae. Electronic version on http://www.xxx.lanl.gov.math.AG/0006171.
G. Forni. (2002) Deviation of ergodic averages for area-preserving flows on surfaces of higher genus, Ann. of Math. 155
M. Kontsevich. (1997) Lyapunov exponents and Hodge theory. In: “The mathematical beauty of physics”, (Saclay, 1996). Adv. Ser. Math. Phys. 24, World Sci. Publishing, River Edge, NJ, pp. 318–322.
M. Kontsevich and A. Zorich. (2001) Connected components of the moduli spaces of Abelian differentials with prescribed singularities, preprint.
H. Masur. (1982) Interval exchange transformations and measured foliations. Ann. of Math. 115, 169–200.
W. Veech. (1982) Gauss measures for transformations on the space of interval exchange maps. Ann. of Math. 115, 201–242.
A. Zorich. (1999) How do the leaves of a closed 1-form wind around a surface? In: “Pseudoperiodic topology”, 135-178, Amer. Math. Soc. Transi. Ser. 2, 197, Amer. Math. Soc., Providence, RI.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Zorich, A. (2002). Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differentials. In: Burger, M., Iozzi, A. (eds) Rigidity in Dynamics and Geometry. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04743-9_25
Download citation
DOI: https://doi.org/10.1007/978-3-662-04743-9_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-07751-7
Online ISBN: 978-3-662-04743-9
eBook Packages: Springer Book Archive