Tropical Geometric Compactification of Moduli, I  $M_g$ case 
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 GromovHausdorff 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]).
 Publication:

arXiv eprints
 Pub Date:
 June 2014
 DOI:
 10.48550/arXiv.1406.7772
 arXiv:
 arXiv:1406.7772
 Bibcode:
 2014arXiv1406.7772O
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics;
 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry
 EPrint:
 v2: Revision of the FORMER half (curve, $M_g$ case) of version 1. v3: revised exposition of v2. Resubmitted