Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T09:42:37.576Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures of groups of homeomorphisms and Auslander's conjecture

Published online by Cambridge University Press:  14 October 2010

Shmuel Friedland
Shmuel Friedland
Affiliation:
Department of Mathematics, Statistics and Computer Science (M/C 249), University of Illinois at Chicago, 851 South Morgan Street, Chicago, Illinois 60607-7045, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We give sufficient conditions for a group of homeomorphisms of a compact Hausdorff space to have an invariant probability measure. For a complex projective space CPn we give a necessary condition for a subgroup of Aut(CPn) to have an invariant probability measure. We discuss two approaches to Auslander's conjecture.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 6 , December 1995 , pp. 1075 - 1089
Copyright
Copyright © Cambridge University Press 1995

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[Aus]Auslander, L.. The structure of compact locally affine manifolds. Topology 3 (1964), 131–139.CrossRefGoogle Scholar
[B-T]Banach, S. and Tarski, A.. Sur la decomposition des ensembles de points en partes respectivement congruentes. Fund. Math. 6 (1924), 244–277.CrossRefGoogle Scholar
[Bea]Beardon, A. F.. The Geometry of Discrete Groups. Springer-Verlag, New York, 1982.Google Scholar
[D-G]Drumm, T. and Goldman, W.. Complete flat Lorentz 3-manifolds with free fundamental group. Int. J. Math. 1 (1990), 149–161.CrossRefGoogle Scholar
[D-S]Dunford, N. and Schwartz, J.T.. Linear Opertors, Part I, Interscience, New York, 1967.Google Scholar
[F-G]Fried, D. and Goldman, W.. Three-dimensional affine crystallographic groups. Adv. Math. 48 (1983), 1–49.CrossRefGoogle Scholar
[Fri1]Friedland, S.. Properly discontinuous groups on certain matrix homogeneous spaces. Preprint, July 1994.Google Scholar
[Fri2]Friedland, S.. The Patterson-Sullivan measure for discrete groups of Lie groups. In preparation.Google Scholar
[Fur]Furstenberg, H.. A note on Borel's density theorem. Proc. Amer. Math. Soc. 55 (1976), 209–212.Google Scholar
[Gan]Gantmacher, F. R.. The Theory of Matrices, I. Chelsea, New York, 1959.Google Scholar
[G-K]Goldman, W. and Kamishima, Y.. The fundamental groups a compact flat space form is virtually polycyclic. J. Differential Geometry 19 (1984), 233–240.CrossRefGoogle Scholar
[Gre]Greenleaf, F. P.. Invariant Means on Topological Groups and Their Applications. Van Nostrand Reinhold, New York, 1969.Google Scholar
[G-H]Griffiths, P. and Harris, J.. Principles of Algebraic Geometry. Wiley Interscience, New York, 1978.Google Scholar
[Gun]Gunther, P.. Einige Satze iiber das Volumenelement eines Riemannschen Raumes. Publ. Math. Debrecen 7 (1960), 78–92.CrossRefGoogle Scholar
[Mai]Malcev, A. I.. On certain classes of infinite solvable groups. Mat. Sb. 28(70) (1951), 567–588.Google Scholar
[Mar1]Margulis, G. A.. Free properly discontinuous groups of affine transformations. Dokl. Akad. Nauk SSSR 272 (1983), 937–940.Google Scholar
[Mar2]Margulis, G. A.. Complete affine locally flat manifolds with a free fundamental group. J. Sov. Math. 134 (1987), 129–133.CrossRefGoogle Scholar
[Mar3]Margulis, G. A.. On the Zariski closure of the linear part of a properly discontinuous group of affine transformations. Preprint.Google Scholar
[Mil]Milnor, J.. On fundamental groups of completely affine flat manifolds. Adv. Math. 25 (1977), 178–187.CrossRefGoogle Scholar
[Neu]Neumann, J. von. Zur allgemeinen theorie des masses. Fund. Math. 13 (1929), 73–116.CrossRefGoogle Scholar
[Nic]Nicholls, P. J.. Ergodic Theory of Discrete Groups. Cambridge University Press, Cambridge, 1989.CrossRefGoogle Scholar
[Pat]Patterson, S. J.. The limit set of a Fuchsian group. Ada Math. 136 (1976), 241–272.Google Scholar
[Ric]Rickert, N.. Amenable groups and groups with the fixed point property. Trans. Amer. Math. Soc. 127 (1967), 221–232.CrossRefGoogle Scholar
[Sul]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. I.H.E.S. 50 (1979), 171–202.CrossRefGoogle Scholar
[Tit]Tits, J.. Free subgroups in linear groups. J. Algebra 20 (1972), 250–270.CrossRefGoogle Scholar
[Tom]Tomanov, G.. The virtual solvability of the fundamental group of a generalized Lorentz space form. J. Diff. Geom. 32 (1990), 539–547.Google Scholar
[Wal]Walters, P.. An Introduction to Ergodic Theory. Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
[Wol]Wolf, J. A.. Growth of finitely generated solvable groups and curvature of riemannian manifolds. J. Diff. Geom. 2 (1968), 421–446.Google Scholar
[Zim]Zimmer, R. J., Ergodic Theory and Semisimple Groups. Birkhäuser, Basel 1984.CrossRefGoogle Scholar