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Abramovich, Caporaso and Payne (2014) have constructed functorial tropicalization maps from the Berkovich analytification of the moduli spaces of stable curves, $\overline{M}_{g,n}$, to the moduli spaces of extended tropical curves, $\overline{M}^{trop}_{g,n}$, that are compatible with the natural forgetful, clutching and glueing maps on both sides.

Grothendieck in his Esquisse sketched a program for studying the action of the absolute Galois group, Gal($\overline{\mathbb{Q}}/\mathbb{Q}$), on the "Lego-Tëichmuller tower" consisting of profinite completions of fundamental groupoids of moduli spaces of smooth curves, $M_{g,n}$, connected via the same forgetful, clutching and glueing maps.

What implications if any does the tropicalization result have on Grothendieck's program? Does the absolute Galois action transfer to the corresponding "tropical tower" of $\overline{M}^{trop}_{g,n}$ in any way at all, and thus be studied combinatorially?

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