Strongly ergodic equivalence relations: spectral gap and type III invariants

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.

On full groups of non-ergodic probability-measure-preserving equivalence relations

This article generalizes our previous results [Le Maître. The number of topological generators for full groups of ergodic equivalence relations. Invent. Math. 198 (2014), 261–268] to the non-ergodic case by giving a formula relating the topological rank of the full group of an aperiodic probability-measure-preserving (pmp) equivalence relation to the cost of its ergodic components. Furthermore, we obtain examples of full groups that have a dense free subgroup whose rank is equal to the topological rank of the full group, using a Baire category argument. We then study the automatic continuity property for full groups of aperiodic equivalence relations, and find a connected metric for which they have the automatic continuity property. This allows us to provide an algebraic characterization of aperiodicity for pmp equivalence relations, namely the non-existence of homomorphisms from their full groups into totally disconnected separable groups. A simple proof of the extreme amenability of full groups of hyperfinite pmp equivalence relations is also given, generalizing a result of Giordano and Pestov to the non-ergodic case [Giordano and Pestov. Some extremely amenable groups related to operator algebras and ergodic theory. J. Inst. Math. Jussieu6(2) (2007), 279–315, Theorem 5.7].

Borel complexity of isomorphism between quotient Boolean algebras

In response to a question of Farah, “How many Boolean algebras are there?” [Far04], one of us (Oliver) proved that there are continuum-many nonisomorphic Boolean algebras of the form with I a Borel ideal on the natural numbers, and in fact that this result could be improved simultaneously in two directions:(i) “Borel ideal” may be improved to “analytic P-ideal”(ii) “continuum-many” may be improved to “E0-many”; that is, E0 is Borel reducible to the isomorphism relation on quotients by analytic P-ideals.See [Oli04].In [AdKechOO], Adams and Kechris showed that the relation of equality on Borel sets (and therefore, any Borel equivalence relation whatsoever) is Borel reducible to the equivalence relation of Borel bireducibility. (In somewhat finer terms, they showed that the partial order of inclusion on Borel sets is Borel reducible to the quasi-order of Borel reducibility.) Their technique was to find a collection of, in some sense, strongly mutually ergodic equivalence relations, indexed by reals, and then assign to each Borel set B a sort of “direct sum” of the equivalence relations corresponding to the reals in B. Then if B1, ⊆ B2 it was easy to see that the equivalence relation thus induced by B1 was Borel reducible to the one induced by B2, whereas in the opposite case, taking x to be some element of B / B2, it was possible to show that the equivalence relation corresponding to x, which was part of the equivalence relation induced by B1, was not Borel reducible to the equivalence relation corresponding to B2.

Tail-fields of products of random variables and ergodic equivalence relations

We prove the following result. Let $G$ be a countable discrete group with finite conjugacy classes, and let $(X_n, n\in\mathbb Z)$ be a two-sided, strictly stationary sequence of $G$-valued random variables. Then $\mathscr T_\infty =\mathscr T_\infty ^*$, where $\mathscr T_\infty$ is the two-sided tail-sigma-field $\bigcap_{M\ge1}\sigma (X_m:|m|\ge M)$ of $(X_n)$ and $T_\infty ^*$ the tail-sigma-field $\bigcap_{M\ge0}\sigma (Y_{m,n}:m,n\ge M)$ of the random variables $(Y_{m,n}, m,n\ge0)$ defined as the products $Y_{m,n}=X_n\dots X_{-m}$. This statement generalises a number of results in the literature concerning tail triviality of two-sided random walks on certain discrete groups.