Foliations and laminations on hyperbolic surfaces

Abstract

GEODESIC LAMINATIONS on surfaces have been introduced by Thurston in his study[l7] of 3-manifolds with a hyperbolic structure (i.e. a Riemannian metric of constant curvature -1). He has also noticed (unpublished) that, given a hyperbolic structure on a surface, there is a bijective correspondence between measured laminations and equivalence classes of measured foliations (a related result has been obtained by Aranson and Grines [ 11). But a general foliation with p-prong saddle singularities (p 3 3) on a surface is topologically conjugate to a measured foliation (in the sense of .[4, 161) if and only if its non-compact leaves are locally dense (see 94 below), and very few foliations (even of class Cr) satisfy this condition. The aim of the present paper is to show how Thurston’s construction can be generalized to non-measured foliations in order to yield a canonical representation of foliations with saddle singularities on a given hyperbolic surface. Let us first consider (non-singular) foliations 9 with no compact leaf on the 2-torus T2 = R2/Z2. As is well known since Denjoy[3], such a foliation, if C2, is topologically conjugate to an “irrational flow” (the projection onto T* of a foliation of R2 by lines of irrational slope). Denjoy also gave an example of a Co (even C’) foliation not conjugate to an irrational flow. Roughly speaking, his example can be obtained from an irrational flow by “opening up” a leaf f, that is replacing f by two leaves f’ and fwhose distance goes to 0 as-one goes out to infinity on either leaf, and pushing apart the other leaves to make room; the space between f’ and fis then filled in by new leaves, and the complement of these new leaves is a non-trivial compact minimal set. One way of distinguishing a Denjoy example from a C2 foliation is by considering the induced foliation 9 on the universal covering R2. Given a leaf f of @, the line defined by two points p and 4 of f has a limiting position rci> when p and 4 go to infinity in opposite directions on f. The line ru) has some irrational slope (Y, and conversely every line of slope (Y is obtained by “straightening” a leaf { If 9 is conjugate to an irrational flow, different leaves of .@ give rise to different lines. If 9 is a Denjoy example, corresponding lifts to R2 of the leaves f’ and f(and all “new” leaves in between) give rise to the same line. We call such a line “thick”. Projecting down to T2, we see that by straightening leaves we have attached to a foliation 9 an irrational flow r(F), which is a totally geodesic foliation for the canonical flat metric of T’; if 9 is not conjugate to an irrational flow, then y(9) has at least one thick leaf. Markley has shown[8] that foliations on T’ with no compact leaf can be classified up to isotopy by specifying an irrational flow on T* and a family (at most countably infinite) of thick lines; the foliation is obtained from the irrational flow by opening up these thick lines. A similar pattern works for a foliation 9 with saddle singularities on a hyperbolic