Teichm\"uller theory via random simple closed curves

TL;DR

This paper establishes a proper embedding of Teichmüller space into the space of geodesic currents using a map based on random simple closed geodesics, linking hyperbolic geometry and geodesic currents.

Contribution

It introduces a novel embedding of Teichmüller space into geodesic currents via a map involving random simple closed geodesics, supported by a new intersection number formula.

Findings

The map $\sigma : T_g o C_g$ is a proper embedding.

The intersection number formula is expressed in Dehn coordinates.

The approach connects hyperbolic surfaces with geodesic currents through probabilistic methods.

Abstract

We show the map $\sigma : T_g \to C_g$ sending a compact hyperbolic surface $X$ to a random simple closed geodesic on $X$ determines a proper embedding of Teichm\"uller space into the space of geodesic currents. The proof depends on a formula for the intersection number $i(C,C')$ of a pair of multicurves, expressed in terms of Dehn coordinates on $ML_g(\mathbb{Z})$.