Topological Equivalence of Flows on Homogeneous Spaces, and Divergence of One-parameter Subgroups of Lie Groups

Let F and r' be lattices, and (p and 4> one-parameter subgroups of the connected Lie groups G and G'. If one of the following conditions (a), (b), or (c) hold, Theorem A states that if the induced flows on the homogeneous spaces G/T and G'/Y' are topologically equivalent, then they are topologically equivalent by an affine map. (a) G and G' are one-connected and nilpotent. (b) G and G' are one-connected and solvable, and for all X in L(G) and X' in L(G'), ad(i) and ad(X') have only real eigenvalues, (c) G and G are centerless and semisimple with no compact direct factor and no direct factor H isomorphic to PSL(2, R) such that TH is closed in G. Moreover, in condition (c), the induced flow of on G/T is assumed to be ergodic. Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say ' which recurrently approaches G a diffeomorphism, then every 4> of G is isolated. Let G be connected and semisimple and (t) = exp(iX). Then Theorem B(b) states that .