论文标题
扩展器和右角Artin团体
Expanders and right-angled Artin groups
论文作者
论文摘要
本文的目的是通过右角artin群体对扩展器图的家族进行特征。我们证明了一系列简单图$ \ {γ_i\} _ {i \ in \ mathbb {n}} $构成一个expander图的家族，并且仅当且仅当一个由杯子中的杯子中引起的某个天然微型最大不变时， $ \ {a（γ_i）\} _ {i \ in \ mathbb {n}} $与图形序列的cheeger常数一致，从而使我们能够通过共同体来表征扩展器图。在\ emph {矢量空间扩展器}的更通用框架中证明了这一结果，这是一种新型结构，由配备矢量空间值的矢量空间序列组成，配备了矢量空间值，可满足某些迷你最大条件。这些对象可以被认为是线性代数领域中的扩展器图的类似物，cup乘积在共同体中给出了词典，在这种情况下，词典代表了一种不同的扩展器方法，该方法是Lubotzky-Zelmanov和Bourgain-Yehehudayoff开发的。
The purpose of this article is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs $\{Γ_i\}_{i\in\mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups $\{A(Γ_i)\}_{i\in\mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of \emph{vector space expanders}, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky-Zelmanov and Bourgain-Yehudayoff.
