Intersection number and some metrics on Teichmüller space

Abstract

Let T (X) be the Teichmüller space of a closed surface X of genus g ≥ 2, C(X) be the space of geodesic currents on X, and L : T (X)→ C(X) be the embedding introduced by Bonahon which maps a hyperbolic metric to its corresponding Liouville current. In this paper, we compare some quantitative relations and topological behaviors between the intersection number and the Teichmüller metric, the length spectrum metric and Thurston’s asymmetric metrics on T (X), respectively.