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Topological equivalence of flows on homogeneous spaces, and divergence of one-parameter subgroups of Lie groups


Author: Diego Benardete
Journal: Trans. Amer. Math. Soc. 306 (1988), 499-527
MSC: Primary 58F25; Secondary 22E40, 58F10
MathSciNet review: 933304
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Abstract: Let $ \Gamma $ and $ \Gamma ' $ be lattices, and $ \phi $ and $ \phi ' $ 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/\Gamma $ and $ G' /\Gamma ' $ 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' )$, $ \operatorname{ad} (x)$ and $ \operatorname{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 $ \operatorname{PSL} (2,\,R)$ such that $ \Gamma H$ is closed in $ G$. Moreover, in condition (c), the induced flow of $ \phi $ on $ G/\Gamma $ is assumed to be ergodic.

Theorem A depends on Theorem B, which concerns divergence properties of one-parameter subgroups. We say $ \phi $ is isolated if and only if for any $ \phi ' $ which recurrently approaches $ \phi $ for positive and negative time, $ \phi $ equals $ \phi ' $ up to sense-preserving reparameterization. Theorem B(a) states that if $ G$ is one-connected and nilpotent, or one-connected and solvable with exp: $ L(G) \to G$ a diffeomrophism, then every $ \phi $ of $ G$ is isolated. Let $ G$ be connected and semisimple and $ \phi (t) = \exp (tX)$. Then Theorem B(b) states that $ \phi $ is isolated, if $ [X,\,Y] = 0$ and $ \operatorname{ad} (Y)$ being semisimple imply that $ \operatorname{ad} (Y)$ has some eigenvalue not pure imaginary and not zero. If $ G$ has finite center, $ \phi $ is isolated if there is no compact connected subgroup in the centralizer of $ \phi $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0933304-3
Keywords: Topological equivalence, flows, homogeneous space, one-parameter subgroups, Lie groups, discrete subgroups, lattice
Article copyright: © Copyright 1988 American Mathematical Society