Tropical Lambda Lengths, Measured Laminations and Convexity

Abstract

This work uncovers the tropical analogue, for measured laminations, of the convex hull construction in decorated Teichmüller theory; namely, it is a study in coordinates of geometric degeneration to a point of Thurston’s boundary for Teichmüller 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 interest.