Dessins for Modular Operad and Grothendieck-Teichmuller Group

TL;DR

This paper explores the extension of Grothendieck's program to moduli spaces of genus zero stable curves, proposing that dual graphs of these curves serve as 'modular dessins' within an operadic framework.

Contribution

It introduces the concept of 'modular dessins' derived from dual graphs of stable curves, expanding the combinatorial tools used in Grothendieck's program.

Findings

Dual graphs of stable curves can act as modular dessins.

Operadic structures can encode these modular dessins.

Extends the combinatorial approach to a broader geometric setting.

Abstract

A part of Grothendieck's program for studying the Galois group $G_{\mathbb Q}$ of the field of all algebraic numbers $\overline{\mathbb Q}$ emerged from his insight that one should lift its action upon $\overline{\mathbb Q}$ to the action of $G_{\mathbb Q}$ upon the (appropriately defined) profinite completion of $\pi_1({\mathbb P}^1 \setminus \{0,1, \infty\})$. 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. 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.