Strongly ergodic equivalence relations: spectral gap and type III invariants

摘要

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 tau invariants for type III strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type III1 ergodic equivalence relation R, the Maharam extension c(R) is strongly ergodic if and only if R is strongly ergodic and the invariant tau(R) is the usual topology on 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.