Amoebas, tropical varieties and compactification of Teichmuller spaces

TL;DR

This paper explores the compactification of Teichmuller spaces using tropical geometry, describing boundaries via tropical varieties and revealing natural PL structures and relations among trace functions.

Contribution

It introduces a tropical geometric framework for compactifying Teichmuller spaces, connecting boundary properties with tropical hypersurfaces and PL structures.

Findings

Boundary of Teichmuller space has a natural PL structure.

Tropical relations among trace functions correspond to polynomial relations.

The boundary can be described as a subpolyhedron of a tropical hypersurface.

Abstract

In this paper we try to look at the compactification of Teichmuller spaces from a tropical viewpoint. We describe a general construction for the compactification of algebraic varieties, using their amoebas, and we describe the boundary via tropical varieties. When we apply this construction to the Teichmuller spaces we see that they can be mapped in a real algebraic hypersurface in such a way that the cone over the boundary is a subpolyhedron of a tropical hypersurface. We want to show how some properties of the boundary becomes straightforward if looked at from this point of view. For example there is a PL structure that appears naturally on the boundary, and it is shown to be equivalent to the one defined by Thurston. Also we may see easily that every polynomial relation among trace functions on Teichmuller space may be turned automatically in a tropical relation among intersection forms over the boundary.