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.

我们获得了标准量度空间上强遍历等价关系的谱隙表征。我们使用谱隙标准来证明，由可测的 $ 1 $ 循环产生的一大类偏乘积等价关系具有局部紧凑的阿贝尔群的值，是强烈遍历的。通过类似于Connes在全部因子上的工作，我们介绍了 $ \ text {III} $类型的 强遍历等价关系的Sd和 $ \ unicode [STIX] {x1D70F} $ 不变量。作为我们主要结果的推论，我们证明了对于任何类型的 $ \ text {III} _ {1} $ 遍历等价关系 $ {\ mathcal {R}} $ ，Maharam扩展名 $ \ text {c}（{\ mathcal {R}}）$ 仅当 $ {\ mathcal {R}} $ 是高度遍历且不变的 $ \ unicode [STIX] {x1D70F}（{\ mathcal {R}}）$ 是 $ \ mathbb { R} $ 。我们还获得了近似周期的强遍历等价关系的结构定理，类似于Connes的近似周期全因子的结构定理。最后，我们证明对于双精确群（例如双曲线群）的任意强遍历自由动作，轨道等价关系和相关群的Sd和 $ \ unicode [STIX] {x1D70F} $ 不变量不变，并且度量空间冯·诺伊曼因素重合。