Teichmuller geometry of moduli space, II: M(S) seen from far away

TL;DR

This paper constructs a geometric model of the moduli space of Riemann surfaces and analyzes its asymptotic geometry by describing the tangent cone at infinity using a simplicial complex and the complex of curves.

Contribution

It introduces a new metric simplicial complex model for the moduli space and computes its tangent cone at infinity, linking it to the complex of curves and Minsky's product regions theorem.

Findings

The tangent cone at infinity is the topological cone on the quotient of the complex of curves.

The model is almost isometric to the moduli space.

The approach uses Minsky's product regions theorem.

Abstract

We construct a metric simplicial complex which is an almost isometric model of the moduli space M(S) of Riemann surfaces. We then use this model to compute the "tangent cone at infinity" of M(S): it is the topological cone on the quotient of the complex of curves C(S) by the mapping class group of S, endowed with an explicitly described metric. The main ingredient is Minsky's product regions theorem.