Cutting directed ribbon graphs and recursion for volumes of the combinatorial moduli spaces

TL;DR

This paper introduces a canonical decomposition method for directed metric ribbon graphs, enabling a recursion scheme to compute volumes of moduli spaces and applications to counting Grothendieck dessins d'enfants.

Contribution

It presents a novel acyclic decomposition of directed ribbon graphs, providing a new recursion approach for moduli space volumes and applications in combinatorics.

Findings

Decomposition into one-vertex ribbon graphs is canonical and based on surgeries.

Recursion scheme explicitly derived for four-valent metric ribbon graphs.

Applications to counting Grothendieck dessins d'enfants.

Abstract

In this paper we study metric ribbon graphs, in particular, directed metric ribbon graphs. These ribbon graphs are dual to bipartite maps and appear in the context of Abelian differentials. We prove that it is possible to decompose a directed ribbon graph into a family of ribbon graphs with one vertex, by performing surgeries along appropriate multi curves. The decomposition is canonical and we call it acyclic decomposition due to a condition on the stable graphs that encode the surgeries. This result provides a recursion scheme for volumes of moduli space of directed metric ribbon graphs, we give explicitly the recursion in the case of four valent metric ribbon graphs. In a particular case, we give applications to count of Grothendieck dessins d'enfants.