From Grassmannian to Simplicial High-Dimensional Expanders
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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 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 𝔽_2 of Cayley simplicial complexes over 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 Ω(log^2N). No prior constructions in the literature have been shown to achieve as large a 1-homology group.
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