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Algebraic hulls and smooth orbit equivalence

Let (Mi,,ui), i = 1, 2, be two manifolds with quasi-invariant measures, and let Hi c Diff(Mi) be connected Lie groups. If there is a measure class preserving diffeomorphism 0: M1 - > M2 which is a bijection of Hl-orbits and H2-orbits then we say that the actions are smoothly orbit equivalent. If the Hi-actions determine foliations g on the manifolds MiS then the map 0 is just a diffeomorphism between the foliations. These phenomena, and the corresponding phenomena arising in situations in which the (Mi, Ai) are just Borel (or topological) Hi-spaces and the map 0 is a measure class preserving Borel isomorphism (or homeomorphism), have been studied, using a variety of techniques, independently by several authors [B, CFW, D1, D2, K, PnZ, W1, W2, Z2, Z4]. Most of the results obtained require some strengthening of the hypotheses, such as finiteness and invariance of the measures and amenability or semisimplicity (in higher rank) of the groups acting. One of the results that we want to describe in this paper fits in this geometric setting and provides an obstruction, in terms of a geometric invariant of the actions, to the foliations being diffeomorphic. Recall that, if H acts ergodically and by diffeomorphisms on the n-dimensional manifold M, the algebraic hull of the H-action is the unique (up to conjugacy) smallest algebraic subgroup L c GL(n, R) such that there exists a measurable H-invariant reduction to L of the frame bundle on M on which H acts by automorphisms. (For an analytic definition see §2 and for the general context see [Z5, 9.2].) Then the normal algebraic hull of the H-action on M will be the projection of L in the direction normal to the orbits.