Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-12-01T00:11:43.933Z Has data issue: false hasContentIssue false
  • English
  • Français

Reduction of cocycles with hyperbolic targets

Published online by Cambridge University Press:  14 October 2010

Scot Adams
Scot Adams
Affiliation:
School of Mathematics, Vincent Hall, 206 Church St. SE, University of Minnesota, Minneapolis, MN 55455, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that any cocycle from an ergodic, finite measure preserving action of a higher-rank group to a closed subgroup of the isometry group of a proper, geodesic hyperbolic, ‘at most exponential’ metric space is necessarily cohomologous to a cocycle with values in a compact subgroup. Philosophically, this says that higher-rank dynamics is incompatible with hyperbolic dynamics; since hyperbolicity is, in some sense, generic among finitely presented groups, this places strong restrictions on the possible dynamics of higher-rank groups. We derive some applications of the main result. When the target group of the cocycle has no small subgroups, we show that the main result holds for a wider class of domain groups.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 6 , December 1996 , pp. 1111 - 1145
Copyright
Copyright © Cambridge University Press 1996

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

[Adi] Adams, S.. Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups. Topology 33 (1994), 765–783.CrossRefGoogle Scholar
[Ad2] Adams, S.. Indecomposability of equivalence relations generated by word hyperbolic groups. Topology 33 (1994), 785–798.CrossRefGoogle Scholar
[Ad3] Adams, S.. An equivalence relation that is not freely generated. Proc. Amer. Math. Soc. 102 (1988), 565–566.CrossRefGoogle Scholar
[Ad4] Adams, S.. Reduction of cocycles with hyperbolic targets (unabridged version). Preprint.Google Scholar
[AEG1] Adams, S., Elliott, G. and Giordano, T.. Amenable actions of groups. Trans. Amer. Math. Soc. 344 (1994), 803–822.CrossRefGoogle Scholar
[AS1] Adams, S., and Spatzier, R.. Kazhdan groups, cocycles and trees. Amer. J. Math. 112 (1990), 271–287.CrossRefGoogle Scholar
[BD1] Bröcker, T., and Dieck, T. torn. Representations of Compact Lie Groups. Springer, New York, 1985.CrossRefGoogle Scholar
[Bui] Burgess, J.. A selection theorem for group actions. Pac. J. Math. 80 (1979), 333–336.CrossRefGoogle Scholar
[CFW1] Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1 (1981), 431–450.CrossRefGoogle Scholar
[FHM1] Feldman, J., Hahn, P. and Moore, C.. Orbit structure and countable sections for actions of continuous groups. Adv. Math. 28 (1978), 186–230.CrossRefGoogle Scholar
[FM1] Feldman, J., and Moore, C.. Ergodic equivalence relations, cohomology and von Neumann algebras I. Trans. Amer. Math. Soc. 234 (1977), 289–324.CrossRefGoogle Scholar
[GH1] Ghys, E., and Harpe, P. de la. Sur les Groupes Hyperboliques d'après Mikhael Gromov. Birkhäuser, Boston, 1990.CrossRefGoogle Scholar
[Grl] Gromov, M.. Rigid transformation groups. Géometrie Differentielle. Eds Bernard, D. and Choquet-Bruhat, Y.. Hermann, Paris, 1988.Google Scholar
[Iwl] Iwasawa, K.. On some types of topological groups. Ann. Math. 50 (1949), 507–558.CrossRefGoogle Scholar
[Kel] Kechris, A.. Countable sections for locally compact group actions. Ergod. Th. & Dynam. Sys. 12 (1992), 283–295.CrossRefGoogle Scholar
[Ral] Raghunathan, M.. Discrete Subgroups of Lie Groups. Springer, New York, 1972.CrossRefGoogle Scholar
[Rul] Rudin, W.Functional Analysis. McGraw-Hill, New York, 1973.Google Scholar
[SZ1] Spatzier, R. and Zimmer, R.. Fundamental groups of negatively curved manifolds and actions of semisimple groups. Topology 30 (1991), 591–601.CrossRefGoogle Scholar
[Wai] Watatani, Y.. Property (T) of Kazhdan implies property (FA) of Serre. Math. Jap. 27 (1982), 97–103.Google Scholar
[Zil] Zimmer, R.. Ergodic Theory and Semisimple Groups. Birkhäuser, Boston, 1984.CrossRefGoogle Scholar
[Zi2] Zimmer, R.. Kazhdan groups acting on compact manifolds. Invent, math. 75 (1984), 425–436.CrossRefGoogle Scholar
[Zi3] Zimmer, R.. Groups generating transversals to semisimple Lie group actions. Israel J. Math. 73 (1991), 151–159.CrossRefGoogle Scholar
[Zi4] Zimmer, R.. Ergodic theory, semisimple Lie groups and foliations by manifolds of negative curvature. Publ. Math. IHES 55 (1982), 37–62.CrossRefGoogle Scholar
[Zi5] Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Fund. Anal. 27 (1978), 350–372.CrossRefGoogle Scholar
[Zi6] Zimmer, R.. Ergodic actions of semisimple groups and product relations. Ann. Math. 118 (1983), 9–19.CrossRefGoogle Scholar