Abstract

We consider hyperbolic manifolds which fibre over the circle with fibre the once punctured torus. Normalising so that ∞ is a parabolic fixed point, we analyse the Ford domain of such a manifold. Using the cutting surfaces associated to this domain, we give a canonical decomposition of the manifold into ideal tetrahedra

Introduction

A well known result of Thurston [Thu86b] says that if Σ is a surface of negative Euler characteristic and ø is a pseudo-Anosov diffeomorphism of Σ to itself then the mapping torus M of ø carries a finite volume hyperbolic structure (see also McMullen [McM96], Otal [Ota01]). In the case where Σ is a once punctured torus its fundamental group is a free group of rank 2 and the mapping class group of Σ is the classical modular group Γ = PSL(2, ℤ). An automorphism ø of Σ is pseudo-Anosov if and only if, as an element of Γ, it is hyperbolic.

We consider manifolds M which are mapping tori of pseudo-Anosov diffeomorphisms of the once punctured torus. We can associate a combinatorial ideal triangulation of M by decomposing the automorphism of π1(Σ) induced by ø into elementary Nielsen moves (see for example page 328 of [Bow97]). The main problem addressed in this paper is to show that when we give M its unique hyperbolic structure then this combinatorial triangulation is realised as an ideal triangulation of M by ideal hyperbolic tetrahedra.