Minimal, strongly proximal actions of locally compact group G on compact spaces X, also known as boundary actions, were introduced by Furstenberg in the study of Lie groups. In particular, the action of a semi-simple real Lie group G on homogeneous spaces G/Q where Q in G is a parabolic subgroup, are boundary actions. Countable discrete groups admit a wide variety of boundary actions. In this note we show that if X is a compact manifold with a faithful boundary action of some locally compact group H, then (under some Holder regularity assumption) the group H, the space X, and the action split into a direct product of a semi-simple Lie group G acting on G/Q and a boundary action of a discrete countable group.
Authors: A. Furman,