Equivariant images of projective space under the action of SL (n, ℤ)

Abstract

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ),n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→Xis a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.