Homotopically trivial actions on aspherical spaces and topological rigidity of free actions

Let rho: G --> Homeo(X) be a homotopically trivial action of a compact commutative Lie group on a connected, finitistic, aspherical topological space. We associate with rho a certain set of homotopical invariants. Using them we introduce the notion of pi(1)-freeness, pi(1)-conjugacy and pi(1)-effectiveness. We check that rho is free if and only if it is pi(1)-free. Applying the rigidity theorems of ET. Farrell and L. Jones we prove that pi(1)-conjugate, homotopically trivial, smooth, and free actions of G on appropriate aspherical manifolds are topologically conjugate. Using this we show that the number of topological conjugacy classes of free and smooth Z(k)-actions that are homotopic to a given free Z(k)-action on a closed infrasolvmanifold M it is not greather than k(rankZ(pi 1(M))Zk).