Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials

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January 2008, 2(1): 139-185. doi: 10.3934/jmd.2008.2.139

Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials

1.	IRMAR, Université Rennes 1, Campus de Beaulieu, 35042 Rennes cedex, France

Received October 2007 Revised November 2007 Published November 2007

Abstract
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Moduli spaces of Abelian and quadratic differentials are stratiﬁed by multiplicities of zeroes; connected components of the strata correspond to ergodic components of the Teichmüller geodesic ﬂow. It is known that the strata are not necessarily connected; the connected components were recently classiﬁed by M. Kontsevich and the author and by E. Lanneau. The strata can be also viewed as families of ﬂat metrics with conical singularities and with $\mathbb Z$/$2 \mathbb Z$-holonomy.
For every connected component of each stratum of Abelian and quadratic differentials we construct an explicit representative which is a Jenkins–Strebel differential with a single cylinder. By an elementary variation of this construction we represent almost every Abelian (quadratic) differential in the corresponding connected component of the stratum as a polygon with identiﬁed pairs of edges, where combinatorics of identiﬁcations is explicitly described.
Speciﬁcally, the combinatorics is expressed in terms of a generalized permutation. For any component of any stratum of Abelian and quadratic differentials we construct a generalized permutation in the corresponding extended Rauzy class.

Keywords: Teichmüller geodesic flow, interval-exchange transformation, Jenkins-Strebel differential, moduli space of quadratic differentials, Rauzy class., spin structure.

Mathematics Subject Classification: 32G15, 30F30, 30F60, 37E05, 37E35, 37G9.

Citation: Anton Zorich. Explicit Jenkins-Strebel representatives of all strata of Abelian and quadratic differentials. Journal of Modern Dynamics, 2008, 2 (1) : 139-185. doi: 10.3934/jmd.2008.2.139

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