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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
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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.

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

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