Abstract
Some time agoTits has considered the question of characterizingB(G), and more generally, of finding the automorphisms of bounded displacement ofG, particularly whenG 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 sufficient conditions forB(G)=Z(G), the center, or more generally, forG to have no non-trivial automorphisms of bounded displacement. The significance of such conditions can be seen in work of the author together withF. Greenleaf andL. 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 subgroupH ofG withG/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 ofH inG. We also deal with certain converse questions and applications to homogeneous spaces of finite volume. For example, ifG/H has finite volume and α is an automorphism ofG leavingH pointwise fixed, then α has bounded displacement. If ϕ is a 1-cocycle and ϕ/H is trivial, then ϕ is itself bounded.
Similar content being viewed by others
References
Bourbaki, N.: Groupes et Algèbres de Lie, Ch. 1. Paris: Hermann. 1960.
Dixmier, J.: L'application exponentielle dans les groupes de Lie resolubles. Bull. Math. Soc. Fr.85, 113–121 (1957).
Garland, H., andM. Goto: Lattices and the adjoint group of a Lie group. Trans. Amer. Math. Soc.124, 450–460 (1966).
Greenleaf, F., M. Moskowitz, andL. Rothschild: Unbounded conjugacy classes in Lie groups and the location of central measures. Acta Math. Scand.132, 3–4, 225–243 (1974).
Greenleaf, F., M. Moskowitz andL. Rothschild: Compactness of certain homogeneous spaces of finite volume. Amer. J. Math.97, 248–259 (1975).
Greenleaf, F., M. Moskowitz, andL. Rothschild: Automorphisms, orbits, and homogeneous spaces of non-connected Lie groups. Math. Ann.212, 145–155 (1974).
Grosser, S., O. Loos, andM. Moskowitz: Über Automorphismengruppen lokal-kompakter Gruppen und Derivationen von Lie-Gruppen. Math. Z.114, 321–339 (1970).
Hochschild, G.: The automorphism group of a Lie group. Trans. Amer. Math. Soc.72, 209–216 (1952).
Hochschild, G., andG. D. Mostow: Extensions of representations of Lie groups and Lie algebras. Amer. J. Math.79, 924–942 (1957).
Kaplansky, I.: An Introduction to Differential Algebra. Paris: Hermann. 1957.
Moskowitz, M.: On the density theorems of Borel and Fürstenberg. Arkiv. Math. (To appear.)
Moskowitz, M.: Homological algebra in locally compact abelian groups. Trans. Amer. Math. Soc.127, 361–404 (1967).
Raghunathan, M. S.: Discrete Subgroups of Lie Groups. Berlin-Heidelberg-New York: Springer. 1972.
Saito, M.: Sur certains groupes de Lie resolubles. Scientific papers of the School of General Education. Univ. of Tokyo,7, 1–11 (1957).
Tits, J.: Automorphismes de déplacement bornés de groupes de Lie. Topology3 (suppl. 1) 97–107 (1964).
Wang, S. P.: On density properties of certain subgroups of locally compact groups. Duke Math. J.43, 561–578 (1976).
Author information
Authors and Affiliations
Additional information
Research partially supported by NSF Grant=MPS 75-08268.
Rights and permissions
About this article
Cite this article
Moskowitz, M. Some remarks on automorphisms of bounded displacement and bounded cocycles. Monatshefte für Mathematik 85, 323–336 (1978). https://doi.org/10.1007/BF01305961
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01305961