Tropical Geometric Compactification of Moduli, I - $M_g$ case -

TL;DR

This paper introduces a tropical geometric compactification of the moduli space of Riemann surfaces by attaching boundary components represented by metrized graphs, forming a connected space that extends the classical moduli space.

Contribution

It constructs a new compactification of the moduli space of Riemann surfaces using metrized graphs, providing a tropical geometric perspective and connecting to Berkovich analytification.

Findings

The compactification includes boundary points represented by metrized graphs.

The boundary components form a connected space extending the classical moduli space.

The construction relates to tropicalization and Gromov-Hausdorff collapse phenomena.

Abstract

We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli space in which $\sqcup_{g} M_{g}$ is open dense. The metrized graphs, which are often studied as "tropical curves", are obtained as Gromov-Hausdorff collapse by fixing diameters of the hyperbolic metrics of the Riemann surfaces. This phenomenon can be also seen as an archemidean analogue of the tropicalization of Berkovich analytification of $M_{g}$ (cf., [ACP]).