Continuous Quotients for Lattice Actions on Compact Spaces

Let Γ < SLn(Z) be a subgroup of finite index, where n≥5. Suppose Γ acts continuously on a manifold M , where π1(M) = Z n, preserving a measure that is positive on open sets. Further assume that the induced Γ action on H(M) is non-trivial. We show there exists a finite index subgroup Γ′ < Γ and a Γ′ equivariant continuous map ψ :M→Tn that induces an isomorphism on fundamental group. We prove more general results providing continuous quotients in cases where π1(M) surjects onto a finitely generated torsion free nilpotent group. We also give some new examples of manifolds with Γ actions.