Cauchy Inequality and the Space of Measured Laminations, I

1.1. This is the first of two papers addressing a Cauchy type inequality for the geometric intersection number between two 1-dimensional submanifolds in a surface. As a consequence, we reestablish some of the basic results in Thurston's theory of measured laminations. In this paper, we consider surfaces with non-empty boundary using ideal triangulations. In the sequel, we establish the inequality for closed surfaces using Dehn-Thurston coordinates. 1.2. Let us begin with a brief review of Thurston's theory (see [Bo], [FLP], [Mo], [PH], [Th1], [Th2] and others). Given a compact orientable surface Σ with possibly non-empty boundary, a curve system on Σ is a proper 1-dimensional submanifold so that each component of it is not null homotopic and not relatively homotopic into the boundary ∂Σ. The space of all isotopy classes of curve systems on Σ is denoted by CS(Σ). This space was introduced by Max Dehn in 1938 [De] who called it the arithmetic field of the topological surface. Given two classes α and β in CS(Σ), their geometric intersection number I(α, β), is defined to be min{ |a ∩ b| : a ∈ α, b ∈ β}. Thurston observed that the pairing I(,): CS(Σ) × CS(Σ) → Z behaves like a non-degenerate " bilinear " form in the sense that (1) given any α in CS(Σ) there is β in CS(Σ) so that their intersection number I(α, β) is non-zero, and (2) I(k 1 α 1 , k 2 α 2) = k 1 k 2 I(α 1 , α 2) for k i ∈ Z ≥0 , α i ∈ CS(Σ) where k i α i is the collection of k i copies of α i. In linear algebra, given a non-degenerate quadratic form ω on a lattice L of rank r, one can form a completion of (L, ω) by canonically embedding L into R r so that the form w extends continuously on R r. Thurston's construction is the exact analogy. Thurston's space of measured laminations on the surface Σ, denoted by M L(Σ) is defined to be the completion of the pair (CS(Σ), I) in the following sense. Given α in CS(Σ), let π(α) be the map sending β to I(α, β). This gives an embedding of π : CS(Σ) → R CS(Σ) where the target has the the product topology. The space M L(Σ) is define to be the closure of Q >0 × …