Train tracks and measured laminations on infinite surfaces

Abstract

Let X X be an infinite Riemann surface equipped with its conformal hyperbolic metric such that the action of the fundamental group π 1 ( X ) \pi _1(X) on the universal covering X ~ \tilde {X} is of the first kind. We first prove that any geodesic lamination on X X is nowhere dense. Given a fixed geodesic pants decomposition of X X we define a family of train tracks on X X such that any geodesic lamination on X X is weakly carried by at least one train track. The set of measured laminations on X 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 M L b ( X ) ML_b(X) of bounded measured laminations appears prominently when studying the Teichmüller space T ( X ) T(X) of X X . If X X has a bounded pants decomposition, a measured lamination on X 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 M L b ( X ) 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 M L b ( X ) ML_b(X) and the topology induced by supremum norm on the edge weight system.

Quadratic differentials and foliations on infinite Riemann surfaces

“…Proof. The map A(X) → ML(X) which assigns to ϕ ∈ A(X) the geodesic lamination µ ϕ obtained by straightening the horizontal foliation is welldefined and injective (see [37,Theorem 5.2]). We established above that the image of A(X) is in ML f (X).…”

“…Let F be an integrable partial foliation of X that represents µ ∈ ML f (X). If γ is a simple closed geodesic in X, then the height h F (γ) of γ with respect to F is equal to the intersection i(µ, [γ]), where [γ] is the homotopy class of γ (see [37] and [25]).…”

“…Proof. In [37], it is established that a homeomorphism g : X → Y between two Riemann surfaces whose fundamental groups are of the first kind induces a natural bijection ĝ : G(X) → G(Y ). The push forward of the measures from G(X) to G(Y ) under the bijection ĝ : G(X) → G(Y ) induces a bijection between the space of measures on G(X) and G(Y ).…”