Foliations and Noncompact Transformation Groups

Introduction. Let G be a Lie group and M a compact C manifold. In [2] Anosov actions of G on M are defined and proved to be structurally stable. In this announcement we are concerned with the foliation ^ of M defined by the orbits of G. Under the assumption that G is connected, # is C stable (3). If G is connected and nilpotent, G has a compact orbit (4). If G is merely solvable, however, there may be no compact orbit. In fact it can happen that no foliation C° close to 9r has a compact leaf (8). Upper bounds for the number of compact orbits of given type are found (9). In (7) we discuss the intersection of certain nilpotent subgroups of a Lie group S with conjugates of a uniform discrete subgroup of S.