From Grassmannian to Simplicial High-Dimensional Expanders

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 an abelian group, 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 of our Cayley simplicial complexes. 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 the squared logarithm of N. No prior constructions in the literature have been shown to achieve as large a 1-homology group.