http://arxiv.org/abs/1406.7772

Tropically compactify moduli via Gromov-Hausdorff collapse
Yuji Odaka
(Submitted on 30 Jun 2014)

We compactify the moduli variety of compact Riemann surfaces (resp., of abelian varieties) by attaching moduli of graphs (resp., of tori) as boundary. The compactifications patch together to form a big connected moduli space of which M_g for all g are open subsets (the same for A_g).
The "degenerations" as tropical varieties are obtained via Gromov-Hausdorff collapse by fixing diameters of Kahler-Einstein metrics. This phenomenon can be seen as an "archemidean" analogue of the tropicalisation of Berkovich analytification of M_g studied by [Abramovich-Caporaso-Payne]. We also study topologies of the boundaries a little.