Affine lamination spaces for surfaces

Abstract

In this paper we extend Thurston's space 3°&(M) of projective classes of measured laminations in the compact surface M to a space srf3?{M) of laminations with transverse structures. The main theorem is that stfSf(M) is homeomorphic to the product tR). To avoid confusion, it should be pointed out at the outset that the term affine can have two different meanings, depending on whether the transverse structure is defined in the ambient surface or just in a neighborhood of the lamination. It is the former, stronger, notion that we are interested in here. Such ambient structures can be regarded as transverse structures on singular foliations of M. The topology on stf3?{M) is defined via length functions on homologically trivial loops in M, by the following procedure. The obstruction to an lamination L e stfS?{M) being a measured lamination is a holonomy homomorphism a^ n\(M) -> R+, which measures the amount by which arcs transverse to L are stretched or shrunk as they are transported around loops in M. Since R+ is abelian, commutators in π\(M) have trivial holonomy, so the lift L of L to the universal abelian cover M of M, corresponding to the commutator subgroup of π\(M), has a transverse measure, unique up to scalar multiplication. Let srfS*{M) denote the unprojectivized^version of srf£?(M), consisting of the measured laminations L c M constructed in this way. Loops γ in M determine length functions lγ: stfS?(M) —• [0, oo), with lγ(L) as usual the infimum of the length, with respect to the transverse measure on L, of loops in M homotopic to γ. These ly 's are the coordinates of a function /: J^S^(M) ~v [0, oc)^, ^ being the set of freejiomotopy classes of loops in M. We prove / is injective, and give stfSf{M) the induced topology. Projectivizing this yields stfS?{M). We then determine the global topology of s£S?(M). The most