Transactions of the American Mathematical Society

Abstract:

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the fundamental group $\pi _1(X)$ on the universal covering $\tilde {X}$ is of the first kind. We first prove that any geodesic lamination on $X$ is nowhere dense. Given a fixed geodesic pants decomposition of $X$ we define a family of train tracks on $X$ such that any geodesic lamination on $X$ is weakly carried by at least one train track. The set of measured laminations on $X$ carried by a train track is in a one to one correspondence with the set of edge weight systems on the train track. Furthermore, the above correspondence is a homeomorphism when we equipped the measured laminations (weakly carried by a train track) with the weak* topology and the edge weight systems with the topology of pointwise (weak) convergence.

The space $ML_b(X)$ of bounded measured laminations appears prominently when studying the Teichmüller space $T(X)$ of $X$. If $X$ has a bounded pants decomposition, a measured lamination on $X$ weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The space $ML_b(X)$ is equipped with the uniform weak* topology. The correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism for the uniform weak* topology on $ML_b(X)$ and the topology induced by supremum norm on the edge weight system.