Dessins for Modular Operad and Grothendieck-Teichmüller Group

Abstract

A part of Grothendieck's program for studying the Galois group G_ℚ of the field of all algebraic numbers ℚ emerged from his insight that one should lift its action upon ℚ to the action of G_ℚ upon the (appropriately defined) profinite completion of π₁(ℙ¹∖{0,1,∞}). 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. Our brief note concerns another part of Grothendieck program, in which its geometric environment is extended to moduli spaces of algebraic curves, more specifically, stable curves of genus zero with marked/labelled 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.