论文标题
均匀的高素质
Uniform hyperfiniteness
论文作者
论文摘要
大约四十年前，康纳斯，费尔德曼和魏斯证明，对于可衡量的等效关系，舒适性和过度丰富性的概念一致。在本文中，我们定义了有界顶点度的可测量图等效关系的均匀版本和过度强度，并证明这两个概念也一致。粗略地说，如果存在$ k \ geq 1 $，则测量图$ \ cg $是统一的高限度，如果存在$ k \ geq 1 $，因此不仅$ \ cg $，而且其所有积极的子级都为$（\ eps，k）$ - hyperfinite。我们还表明，这种情况相当于加权高素质和强大的分数高素质，这是Lovász最近提出的这一概念。 作为推论，我们通过统一的高素质来获得有限生成的组精确性的特征。
Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lovász. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness.
