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Strongly ergodic equivalence relations: spectral gap and type III invariants

Published online by Cambridge University Press:  27 November 2017

CYRIL HOUDAYER ,
AMINE MARRAKCHI  and
PETER VERRAEDT
CYRIL HOUDAYER
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email cyril.houdayer@math.u-psud.fr, amine.marrakchi@math.u-psud.fr, peter.verraedt@math.u-psud.fr
AMINE MARRAKCHI
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email cyril.houdayer@math.u-psud.fr, amine.marrakchi@math.u-psud.fr, peter.verraedt@math.u-psud.fr
PETER VERRAEDT
Affiliation:
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France email cyril.houdayer@math.u-psud.fr, amine.marrakchi@math.u-psud.fr, peter.verraedt@math.u-psud.fr
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Abstract

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 7 , July 2019 , pp. 1904 - 1935
Copyright
© Cambridge University Press, 2017 

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