On the arithmetic structure of lattice actions on compact spaces

Abstract

We discuss results on engaging actions of a lattice \Gamma in a higher rank simple group G on a compact manifold M. An action is engaging if there is no loss of ergodicity in passing to lifts of the action on finite covers of M.

Suppose \mathbb{R}-rank (G)\geq 2 and \Gamma<G is a lattice. Let \Lambda be the group of lifts of the \Gamma action on M to the universal cover of M. Assume the \Gamma action on M is measure preserving and engaging. We show that the image of \Lambda under any linear representation \sigma is \mathfrak{s}-arithmetic. Also, associated to each \sigma we have a measurable quotient of the \Gamma action on M; this measurable quotient is a generalized affine action on a double coset space. Furthermore, the pushforward of the invariant measure on M is a Lebesgue measure on the quotient. The fundamental group of this quotient is closely related to the image of \pi_1(M) under \sigma.