  Moduli spaces of Abelian and quadratic differentials are stratified by
multiplicities of zeroes; connected components of the strata correspond to
ergodic components of the Teichmuller 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
Z/2Z-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 (linear involution) in the
corresponding extended Rauzy class.

Geometric Topology;Algebraic Geometry;Dynamical Systems