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### Open Access Journals

JMD

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.

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.

JMD

A cyclic cover of the complex projective line branched at four
appropriate points has a natural structure of a square-tiled surface.
We describe the combinatorics of such a square-tiled surface, the
geometry of the corresponding Teichmüller curve, and compute the
Lyapunov exponents of the determinant bundle over the Teichmüller
curve with respect to the geodesic flow. This paper includes a new
example (announced by G. Forni and C. Matheus in [17]
of a Teichmüller curve of a square-tiled cyclic cover in a stratum
of Abelian differentials in genus four with a maximally degenerate
Kontsevich--Zorich spectrum (the only known example in genus three
found previously by Forni also corresponds to a square-tiled cyclic
cover [15].
We present several new examples of Teichmüller curves in
strata of holomorphic and meromorphic quadratic differentials with
a maximally degenerate Kontsevich--Zorich spectrum.
Presumably, these examples cover all possible Teichmüller curves
with maximally degenerate spectra. We prove that this is indeed the case
within the class of square-tiled cyclic covers.

JMD

A cyclic cover over $CP^1$ branched at four points inherits a natural flat
structure from the "pillow" flat structure on the basic sphere. We give an
explicit formula for all individual Lyapunov exponents of the Hodge bundle over
the corresponding arithmetic Teichmüller curve. The key technical element is
evaluation of degrees of line subbundles of the Hodge bundle, corresponding to
eigenspaces of the induced action of deck transformations.

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