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Actions of amenable equivalence relations on CAT(0) fields

Published online by Cambridge University Press:  30 October 2012

MARTIN ANDEREGG  and
PHILIPPE HENRY
MARTIN ANDEREGG
Affiliation:
University of Geneva, 2-4 rue du Lièvre, Case postale 64, CH-1211 Genève 4, Switzerland (email: martin.anderegg@unige.ch)
PHILIPPE HENRY
Affiliation:
EPFL, SB MATHGEOM EGG, Station 8, CH-1015 Lausanne, Switzerland (email: philippe.henry@epfl.ch)
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Abstract

We present the general notion of Borel fields of metric spaces and show some properties of such fields. Then we make the study specific to the Borel fields of proper CAT(0) spaces and we show that the standard tools we need behave in a Borel way. We also introduce the notion of the action of an equivalence relation on Borel fields of metric spaces and we obtain a rigidity result for the action of an amenable equivalence relation on a Borel field of proper finite dimensional CAT(0) spaces. This main theorem is inspired by the result obtained by Adams and Ballmann regarding the action of an amenable group on a proper CAT(0) space.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 34 , Issue 1 , February 2014 , pp. 21 - 54
Copyright
Copyright © 2012 Cambridge University Press 

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