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High-Dimensional Expanders from Chevalley Groups

Authors Ryan O'Donnell, Kevin Pratt

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Ryan O'Donnell
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Kevin Pratt
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA

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Ryan O'Donnell and Kevin Pratt. High-Dimensional Expanders from Chevalley Groups. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 18:1-18:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.18

Abstract

Let Φ be an irreducible root system (other than G₂) of rank at least 2, let 𝔽 be a finite field with p = char 𝔽 > 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • High-dimensional expanders
  • simplicial complexes
  • group theory

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