Dual Teichm\" uller spaces

TL;DR

This paper provides elementary geometric descriptions of Teichmüller spaces for decorated and holed surfaces, constructs explicit coordinates, and explores their asymptotic relationships and quantisation, accessible without prior advanced knowledge.

Contribution

It introduces explicit global coordinates for Teichmüller spaces and measured lamination spaces, and demonstrates their asymptotic isomorphism, with brief discussion on quantisation.

Findings

Explicit global coordinates on Teichmüller spaces

Asymptotic isomorphism between lamination and Teichmüller spaces

Discussion on quantisation of Teichmüller spaces

Abstract

We describe in elementary geometrical terms Teichm\" uller spaces of decorated and holed surfaces. We construct explicit global coordinates on them as well as on the spaces of measured laminations with compact and closed support respectively. We show explicitly that the latter spaces are asymptotically isomorphic to the former. We discuss briefly quantisation of Teichm\" uller spaces and some other application of the constructed approach. The paper does not require any preliminary knowledge of the subject above the Poincar\' e uniformisation theorem.