Topological equivalence and rigidity of flows on certain solvmanifolds

Given a Lie group C and a lattice Gamma in G, a one-parameter subgroup phi of G is said to be rigid if for any other one-parameter subgroup psi, the flows induced by phi and psi On Gamma\G (by right translations) are topologically orbit-equivalent only if they are affinely orbit-equivalent. It was previously known that if G is a simply connected solvable Lie group such that all the eigenvalues of Ad(g), g is an element of G, are real, then all one-parameter subgroups of G are rigid for any lattice in G. Here we consider a complementary case, in which the eigenvalues of Ad(g), g is an element of G, form the unit circle of complex numbers. Let G be the semidirect product N x M, where M and N are finite-dimensional real vector spaces and where the action of M on the normal subgroup N is such that the center of G is a lattice in M. We prove that there is a generic class of abelian lattices Gamma in G such that any semisimple one-parameter subgroup phi (namely phi such that Ad(phi t) is diagonalizable over the complex numbers for all t) is rigid for Gamma (see Theorem 1.4). We also show that, on the other hand, there are fairly high-dimensional spaces of abelian lattices for which some semisimple phi are not rigid (see Corollary 4.3); further, there are non-rigid semisimple phi for which the induced flow is ergodic.