# Strongly ergodic equivalence relations: spectral gap and type III invariants

## TL;DR

This paper characterizes strongly ergodic equivalence relations using spectral gaps, introduces invariants for type III relations, and explores their properties and implications for group actions and von Neumann algebras.

## Contribution

It provides a spectral gap criterion for strong ergodicity, introduces Sd and τ invariants for type III relations, and establishes their equivalence and structure results.

## Key findings

- Spectral gap characterization of strong ergodicity.
- Invariance of Sd and τ for certain group actions.
- Structure theorem for almost periodic strongly ergodic relations.

## 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 ${\rm III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type ${\rm III_1}$ ergodic equivalence relation $\mathcal R$, the Maharam extension $\mathord{\text {c}}(\mathcal R)$ 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.