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Revision #1 to TR23-065 | 26th December 2023 03:39
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#### From Grassmannian to Simplicial High-Dimensional Expanders

**Abstract:**
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.

**Changes to previous version:**
Edits to improve exposition, added references

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TR23-065 | 4th May 2023 05:45
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#### From Grassmannian to Simplicial High-Dimensional Expanders

**Abstract:**
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.