GEODESIC DISCS AND Sp(2n,Z)

Abstract. These are (enhanced) notes of an expository lecture given at the conference ‘Outerspace and Teichüller Space ’ held at CIRM in Luminy in February 2007. The purpose is to describe the relationship between, on the one hand, a pseudo-Anosov diffeomor-phism and its associated Teichmüller disc and, on the other hand, the induced symplectic automorphism of first homology and an in-variant plane with algebraic coordinates. At the end we attempt to define analogues in the Culler-Vogtman outer space. A normal form for pseudo-Anosovs. LetM be a closed, connected, orientable, differentiable surface of genus g ≥ 1. An orientation pre-serving diffeomorphism φ: M → M is called pseudo-Anosov iff no power of φ fixes the free homotopy class of a simple closed curve γ on M. That is, φn ([γ]) 6 = [γ] for all n ∈ N and for all simple closed γ. Alternatively, by a theorem of Thurston [Thr88], φ is pseudo-Anosov iff there exists (a) a finite set F ⊂ S, (b) a diffeomorphism φ ′:M →M isotopic to φ such that φ′|F = id, (c) an atlas {µα: Uα → R 2} on the punctured surface M \ F such that µα ◦ µ −1 β (x) = (−1) ǫ · x + v. for some v = vαβ ∈ R 2 and ǫ = ǫαβ ∈ {0, 1}, and (d) a real number λ> 1 and µα ◦ φ ′ ◦ µ−1β (x1, x2) = (λ · x1, λ −1 · x2). In [Thr88] conditions (c) and (d) are described in terms of a pair of transverse measured foliations. An atlas {µα} satisfying (c) is called an F-structure [Vch86]. In the special case that ǫαβ = 0 for all α and β, the atlas is called a translation structure [GtkJdg96]