# Train tracks and measured laminations on infinite surfaces

## TL;DR

This paper studies geodesic laminations and train tracks on infinite hyperbolic surfaces, establishing parametrizations, topological properties, and homeomorphisms for bounded measured laminations within the Teichmüller space.

## Contribution

It introduces a parametrization of measured laminations on infinite surfaces via train tracks and edge weight systems, and characterizes the topology of bounded measured laminations.

## Key findings

- Any geodesic lamination on the surface is nowhere dense.
- Measured laminations carried by train tracks correspond to edge weight systems with specific topologies.
- The homeomorphism between bounded measured laminations and edge weight systems is established under the uniform weak* topology.

## Abstract

Let $X$ be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the covering group $\pi_1(X)$ on $\tilde{X}$ is of the first kind-i.e., the surface $X$ is equal to its convex core. 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 of $X$ is weakly carried by at least one train track. Then we parametrize all measured laminations on $X$ carried by a train track by the corresponding edge weight systems on the train track. Furthermore, we show that the weak* topology on the measured laminations weakly carried by a train track corresponds to a pointwise (weak) convergence of the edge weight systems. When one considers the Teichm\"uller space $T(X)$ of the Riemann surface $X$, it is natural to restrict the attention to the space $ML_b(X)$ of bounded measured laminations. When $X$ has a bounded geometry, we prove that a measured lamination weakly carried by a train track is bounded if and only if the corresponding edge weight system has a finite supremum norm. The Teichm\"uller space considerations lead to a natural uniform weak* topology on the space of bounded measured laminations on $X$. We prove that the correspondence between bounded measured laminations weakly carried by a train track and their edge weight systems is a homeomorphism when $ML_b(X)$ is equipped with the uniform weak* topology and the edge weight system is equipped with the topology induced by the supremum norm.