Dual Teichmuller and lamination spaces

TL;DR

This paper surveys coordinate descriptions of Teichmuller and lamination spaces for open surfaces, extending to surfaces with boundary points, and explores their structures, actions, and tropical limits, serving as an introduction to higher Teichmuller theory.

Contribution

It provides explicit coordinate descriptions and extends the theory to surfaces with boundary points, linking lamination and Teichmuller spaces as tropical limits.

Findings

Descriptions of mapping class group actions

Poisson and symplectic structures detailed

Canonical pairings between spaces constructed

Abstract

We survey explicit coordinate descriptions for two (A and X) versions of Teichmuller and lamination spaces for open 2D surfaces, and extend them to the more general set-up of surfaces with distinguished collections of points on the boundary. Main features, such as mapping class group action, Poisson and symplectic structures and others, are described in these terms. The lamination spaces are interpreted as the tropical limits of the Teichmuller ones. Canonical pairings between lamination and Teichmuller spaces are constructed. The paper could serve as an introduction to higher Teichmuller theory developed by the authors in math.AG/0311149, math.AG/0311245.