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

TL;DR

This paper constructs explicit Jenkins-Strebel representatives for all connected components of strata in Abelian and quadratic differentials, providing combinatorial descriptions and polygonal models for these differentials.

Contribution

It introduces a method to explicitly represent all strata components using Jenkins-Strebel differentials with a single cylinder and describes their combinatorial structures via generalized permutations.

Findings

Explicit Jenkins-Strebel representatives for all strata components.

Polygonal models with identified edges for almost all differentials.

Construction of generalized permutations in extended Rauzy classes.

Abstract

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.