Title: 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 ${\rm III}$ 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.