Riemann surfaces, ribbon graphs and combinatorial classes

Abstract

We begin by describing the duality between arc systems and ribbon graphs embedded in a punctured surface and explaining 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 argument to prove that Witten classes are stable.