[Paper Review] Amoebas, tropical varieties and compactification of Teichmuller spaces

Daniele Alessandrini|ArXiv.org|May 12, 2005

Polynomial and algebraic computation13 references3 citations

TL;DR

This paper introduces a novel tropical-geometric approach to compactifying Teichmüller spaces using amoebas and Maslov dequantization, constructing the boundary as a polyhedral subset of a sphere whose cone forms a tropical variety. The method establishes a natural piecewise linear structure on the boundary, equivalent to Thurston's original construction, and provides a theoretical framework for tropicalizing polynomial relations among trace functions.

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.

Motivation & Objective

• To develop a new compactification method for Teichmüller spaces using tropical geometry and amoebas.
• To provide a geometric and algebraic justification for the piecewise linear structure on the Thurston boundary.
• To show that polynomial relations among trace functions on Teichmüller space naturally induce tropical relations on the boundary.
• To generalize the Morgan–Shalen compactification by replacing infinite points with tropical-like boundary points via valuation theory.

Proposed method

• The paper uses Maslov dequantization to deform real algebraic varieties into tropical semifields, enabling the study of their asymptotic behavior.
• It defines the amoeba of a variety as the image under the logarithmic map, and uses its asymptotic behavior to construct a boundary via valuations.
• The boundary is constructed as the image of a map $ U_{\mathbb{R}} $ from real-valued valuations on the coordinate ring, restricted to non-zero values on a generating set.
• The compactification is achieved by removing points with zero coordinates and replacing them with new points defined via valuation limits, forming a closed subset of the sphere.
• The cone over this boundary is shown to be a tropical variety, specifically a subpolyhedron of a tropical hypersurface.
• The method relies on the theory of valuations and quasi-valuating sequences to characterize ideal points and their images in the boundary.

Experimental results

Research questions

• RQ1 How can the boundary of Teichmüller space be naturally described using tropical geometry and amoebas?
• RQ2 What is the relationship between the piecewise linear structure on the Thurston boundary and the polyhedral structure of tropical varieties?
• RQ3 Can polynomial relations among trace functions on Teichmüller space be systematically transformed into tropical relations on the boundary?
• RQ4 How does this new compactification compare to the classical Morgan–Shalen construction in terms of structure and geometric interpretation?

Key findings

• The boundary of Teichmüller space is shown to be homeomorphic to the image of a map $ U_{\mathbb{R}} $ from real-valued valuations, establishing a direct link between valuation theory and compactification.
• The piecewise linear structure on the boundary is naturally inherited from the polyhedral structure of tropical varieties, confirming its equivalence to Thurston’s original PL structure.
• Every polynomial relation among trace functions on Teichmüller space induces a corresponding tropical relation among intersection forms on the boundary, providing a theoretical foundation for known results.
• The compactification is realized as a closed subset of the sphere, with the cone over the boundary being a tropical variety, specifically a subpolyhedron of a tropical hypersurface.
• The construction generalizes the Morgan–Shalen compactification by replacing the focus on 'points at infinity' with a valuation-based replacement of null-coordinate points.
• For irreducible varieties, the boundary $ B(V^\prime) $ is exactly the image $ U_{\mathbb{R}}(S_{\mathbb{R}}^\prime) $, and this characterization extends to reducible cases via union over irreducible components.