High-Dimensional Expanders from Chevalley Groups

03/07/2022

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by   Ryan O'Donnell, et al.

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Let Φ be an irreducible root system (other than G_2) of rank at least 2, let 𝔽 be a finite field with p = char𝔽 > 3, and let G(Φ,𝔽) be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension rank(Φ), where G(Φ,𝔽) acts simply transitively on the top-dimensional faces; these are λ-spectral HDXs with λ→ 0 as p →∞. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case Φ = A_d. Our work gives three new families of spectral HDXs of any dimension ≥ 2, and four exceptional constructions of dimension 4, 6, 7, and 8.