Dessins for modular operads and the Grothendieck–Teichmüller group

Abstract

A part of Grothendieck’s program for studying the Galois group $G_{\mathbf{Q}}$ of the field of all algebraic numbers $\overline{\mathbf{Q}}$ emerged from his insight that one should lift its action upon $\overline{\mathbf{Q}}$ to the action of $G_{\mathbf{Q}}$ upon the (appropriately defined) profinite completion of ${\pi }_1({\mathbf{P}}^1\{0,1,\infty \})$. The latter admits a good combinatorial encoding via finite graphs, “dessins d’enfant”.

This part was actively developing during the last decades, starting with foundational works of A. Belyi, V. Drinfeld and Y. Ihara.

This chapter concerns another part of Grothendieck’s program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labeled points. Our main goal is to show that dual graphs of such curves may play the role of “modular dessins” in an appropriate operadic context.