Riemann surfaces, ribbon graphs and combinatorial classes

TL;DR

This survey explores the duality between arc systems and ribbon graphs on punctured surfaces, methods to cellularize moduli spaces, and the stability of Witten classes, connecting combinatorial, geometric, and topological perspectives.

Contribution

It provides a comprehensive overview of dualities, cellularizations, and stability results related to moduli spaces of curves, integrating various geometric and combinatorial approaches.

Findings

Duality between arc systems and ribbon graphs explained.

Cellularization of moduli space via Jenkins-Strebel differentials and hyperbolic geometry.

Witten classes are proven to be stable.

Abstract

This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we exhibit a simple direct argument to prove that Witten classes are stable.