Random graphs, weak coarse embeddings, and higher index theory

R. Willett
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引用次数: 7

Abstract

This paper studies higher index theory for a random sequence of bounded degree, finite graphs with diameter tending to infinity. We show that in a natural model for such random sequences the following hold almost surely: the coarse Baum-Connes assembly map is injective; the coarse Baum-Connes assembly map is not surjective; the maximal coarse Baum-Connes assembly map is an isomorphism. These results are closely tied to issues of expansion in graphs: in particular, we also show that such random sequences almost surely do not have geometric property (T), a strong form of expansion. The key geometric ingredients in the proof are due to Mendel and Naor: in our context, their results imply that a random sequence of graphs almost surely admits a weak form of coarse embedding into Hilbert space.
随机图,弱粗嵌入,高指标理论
研究了一类有界度随机序列的高指标理论。我们证明了在这种随机序列的自然模型中,下列条件几乎肯定成立:粗糙的Baum-Connes集合映射是内射的;粗糙的Baum-Connes集合映射不是满射的;最大粗Baum-Connes装配映射是一个同构映射。这些结果与图中的展开问题密切相关:特别是,我们还证明了这种随机序列几乎肯定不具有几何性质(T),这是一种强展开形式。证明中的关键几何成分要归功于孟德尔和诺尔:在我们的背景下,他们的结果表明,一个随机的图序列几乎肯定承认在希尔伯特空间中存在弱形式的粗嵌入。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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