Automorphisms, orbits, and homogeneous spaces of non-connected lie groups

Greenleaf, Frederick P.; Moskowitz, Martin; Preiss-Rothschild, Linda

doi:10.1007/bf01350782

Cited by 9 publications

(9 citation statements)

References 13 publications

(13 reference statements)

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“…If ~o is a 1-cocycle and ~ I H is trivial then ~ is itself bounded. These results depend ultimately on [4] and [6] and when combined with those of w 1, are quite specific. In a later publication, a generalization to the H bounded, G bounded situation will be given.…”

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confidence: 78%

“…The results of w 1 above are necessary in order to verify the hypothesis in making concrete applications of various results of [4], [5], [6] and [11]. We will illustrate this by the following "density type" result special cases of which are given by Theorem 3 of [5] and Corollary 5 of [6].…”

Section: Some Applications To Homogeneous Spaces Of Finite Volumementioning

confidence: 92%

“…We will illustrate this by the following "density type" result special cases of which are given by Theorem 3 of [5] and Corollary 5 of [6]. Notice that if G/H is compact the result is trivial.…”

Section: Some Applications To Homogeneous Spaces Of Finite Volumementioning

confidence: 94%

“…In a later publication, a generalization to the H bounded, G bounded situation will be given. A few of the results below were proven in a somewhat ad hoc way or were stated without proof in [4] and [6] 2. Since one of our purposes here is to place all these matters on a firm foundation, it seemed advisable to be complete.…”

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confidence: 97%

“…One observes easily that the inner automorphism aa is of bounded displacement if and only if gaB(G). If V is a real (or complex) finite dimensional vector 2 A slight error which occurs in Lemma 7 of [6] has been corrected in a mimeographed note which was generally disseminated several years ago. By analogy with [2] and [14] we call a real linear action of type E if each Qg has no eigenvalue of absolute value i except for 1 itself.…”

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confidence: 99%

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Some remarks on automorphisms of bounded displacement and bounded cocycles

Moskowitz

1978

Monatshefte f�r Mathematik

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Some time ago TI~s has considered the question of characterizing B (G), and more generally, of finding the automorphisms of bounded displacement of G, particularly when G is a connected real Lie group. Our purpose here is to extend these results to various other cases as well as to deal with the analogous questions for 1-cocycles. We concern ourselves, among other things, ~vith the question of sufficient conditions for B(G)-~ Z(G), the center, or more generally, for G to have no non-trivial automorphisms of bounded displacement. The significance of such conditions can be seen in work of the author together with F. G~EE~LEAF and L. ROTHSCHILD where, for example, the Selberg form of the Borel density theorem is considerably generalized. These conditions are therefore closely related to, but not identical with, sufficient conditions for the Zariski density of a closed subgroup H of G with G/H having finite volume, see [11]. For this reason it is enlightening to compare these results with those of [11]. On the cocycle level, we give sufficient conditions for the points with bounded orbit under a linear representation to be fixed and more generally, for a bounded 1-cocycle to be identically zero. These conditions actually play a role in [11]; they are among the sufficient conditions necessary to establish Zariski density of H in G. We also deal with certain converse questions and applications to homogeneous spaces of finite volume. For example, if G/H has finite volume and a is an automorphism of G leaving H pointwise fixed, then a has bounded displacement. If ~ is a 1-cocycle and r I H is trivial, then ~0 is itseff bounded. w 1. In [13] TITS has considered the question of characterizing B (G), and more generally, of finding the automorphisms of bounded displacement of G, particularly when G is a connected real Lie group. Our purpose here is to extend these results to various other cases as well as to deal with the analogous questions for 1-cocycles. We concern ourselves, among other things, with the question of

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confidence: 78%