Moduli via double pants decompositions.

Abstract

We consider (local) parameterizations of Teichmüller space Tg,n (of genus g hyperbolic surfaces with n boundary components) by lengths of 6g−6+3n geodesics. We find a large family of suitable sets of 6g−6+3n geodesics, each set forming a special structure called “admissible double pants decomposition”. For admissible double pants decompositions containing no double curves we show that the lengths of curves contained in the decomposition determine the point of Tg,n up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of Tg,n∖X where X is a union of 3g−3+n hypersurfaces. Furthermore, there exists a groupoid acting transitively on admissible double pants decompositions and generated by transformations exchanging only one curve of the decomposition. The local charts arising from different double pants decompositions compose an atlas covering the Teichmüller space. The gluings of the adjacent charts are coming from the elementary transformations of the decompositions, the gluing functions are algebraic. The same charts provide an atlas for a large part of the boundary strata in Deligne–Mumford compactification of the moduli space Mg,n.