Tropicalized Lambda Lengths, Measured Laminations and Convexity

TL;DR

This paper develops a tropical analogue of measured laminations related to Teichmüller theory, simplifying previous studies, and introduces new classes of laminations and an explicit algorithm for convex hull construction in Minkowski space.

Contribution

It introduces a tropical framework for measured laminations, simplifies existing theories, and presents an explicit algorithm for convex hull construction with new lamination classes.

Findings

Tropical analogue for measured laminations is established.

A new class of measured laminations relative to ideal cell decompositions is discovered.

An explicit algorithm for tropical convex hull construction is formulated.

Abstract

This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for Teichmueller space. This may offer a paradigm for the extension of the basic cell decomposition of Riemann's moduli space to other contexts for general moduli spaces of flat connections on a surface. In any case, this discussion drastically simplifies aspects of previous related studies as is explained. Furthermore, a new class of measured laminations relative to an ideal cell decomposition of a surface is discovered in the limit. Finally, the tropical analogue of the convex hull construction in Minkowski space is formulated as an explicit algorithm that serially simplifies a triangulation with respect to a fixed lamination and has its own independent applications.