Strongly ergodic equivalence relations: spectral gap and type III invariants

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 into locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\tau$ invariants for type $\mathrm{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\mathrm{II}{\mathrm{I}}_1$ 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 $\tau (\mathcal{R})$ is the usual topology on $\mathbf{R}$. 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 $\tau$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

28 pages. To appear in Ergodic Theory Dynam. Systems

References in corpus (2)

• Classification of type III Bernoulli crossed products
• A remark on fullness of some group measure space von Neumann algebras

Cited by in corpus (5)

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• Nonsingular Gaussian actions: beyond the mixing case
• Nonsingular Bernoulli actions of arbitrary Krieger type
• A Markovian and Roe-algebraic approach to asymptotic expansion in measure
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras