In this paper, we present a new construction of simplicial complexes of subpolynomial degree with arbitrarily good local spectral expansion. Previously, the only known high-dimensional expanders (HDXs) with arbitrarily good expansion and less than polynomial degree were based on one of two constructions, namely Ramanujan complexes and coset complexes. In contrast, our construction is a Cayley complex over the group $\mathbb{F}_2^k$, with Cayley generating set given by a Grassmannian HDX. Our construction is in part motivated by a coding-theoretic interpretation of Grassmannian HDXs that we present, which provides a formal connection between Grassmannian HDXs, simplicial HDXs, and LDPC codes. We apply this interpretation to prove a general characterization of the 1-homology groups over $\mathbb{F}_2$ of Cayley simplicial complexes over $\mathbb{F}_2^k$. Using this result, we construct simplicial complexes on $N$ vertices with arbitrarily good local expansion for which the dimension of the 1-homology group grows as $\Omega(\log^2N)$. No prior constructions in the literature have been shown to achieve as large a 1-homology group.

翻译：本文提出了一种新的单纯复形构造方法，其度数为次多项式级，且具有任意良好的局部谱扩展性质。此前，唯一已知的拥有任意良好扩展性且度数低于多项式的高维扩展器（HDXs）基于两种构造之一：Ramanujan复形和陪集复形。与之相对，我们的构造是基于群$\mathbb{F}_2^k$的Cayley复形，其Cayley生成集由格拉斯曼HDX给出。这一构造部分受我们提出的格拉斯曼HDX的编码理论解释所启发，该解释建立了格拉斯曼HDX、单纯HDX与LDPC码之间的形式联系。我们应用该解释证明了$\mathbb{F}_2^k$上Cayley单纯复形在$\mathbb{F}_2$上1-同调群的一般性刻画。利用这一结果，我们构造了顶点数为$N$、具有任意良好局部扩展性的单纯复形，其1-同调群的维数可达$\Omega(\log^2N)$。现有文献中尚未有构造能够达到如此大的1-同调群规模。