Moduli via double pants decompositions

TL;DR

This paper introduces a new coordinate system for Teichmuller space based on lengths of geodesics from admissible double pants decompositions, providing a structured atlas that extends to boundary strata of moduli space.

Contribution

It develops a novel local parametrization of Teichmuller space using admissible double pants decompositions, with algebraic gluing functions and a transitive groupoid action.

Findings

Lengths of curves in the decomposition determine points in Teichmuller space up to finitely many choices.

The coordinate system forms an atlas covering Teichmuller space and parts of the Deligne-Mumford boundary.

The transformations between charts are algebraic and generated by elementary exchanges of curves.

Abstract

We consider (local) parametrizations of Teichmuller space $T_{g,n}$ (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 $T_{g,n}$ up to finitely many choices. Moreover, these lengths provide a local coordinate in a neighborhood of all points of $T_{g,n}\setminus X$, 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 Teichmuller 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.