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Coboundary expansion of coset complexes

Tali Kaufman, Izhar Oppenheim, Shmuel Weinberger

Abstract

Coboundary expansion is a high dimensional generalization of the Cheeger constant to simplicial complexes. Originally, this notion was motivated by the fact that it implies topological expansion, but nowadays a significant part of the motivation stems from its deep connection to problems in theoretical computer science such as agreement expansion in the low soundness regime. In this paper, we prove coboundary expansion with non-Abelian coefficients for the coset complex construction of Kaufman and Oppenheim. Our proof uses a novel global argument, as opposed to the local-to-global arguments that are used to prove cosystolic expansion.

Coboundary expansion of coset complexes

Abstract

Coboundary expansion is a high dimensional generalization of the Cheeger constant to simplicial complexes. Originally, this notion was motivated by the fact that it implies topological expansion, but nowadays a significant part of the motivation stems from its deep connection to problems in theoretical computer science such as agreement expansion in the low soundness regime. In this paper, we prove coboundary expansion with non-Abelian coefficients for the coset complex construction of Kaufman and Oppenheim. Our proof uses a novel global argument, as opposed to the local-to-global arguments that are used to prove cosystolic expansion.

Paper Structure

This paper contains 17 sections, 28 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Our contribution
  3. Significance of coboundary expansion over the symmetric group within CS
  4. Comparison to similar results
  5. Proof overview
  6. Background
  7. Simplicial complexes
  8. The $A_n$ root system
  9. Coboundary and cosystolic expansion with general group coefficients
  10. Coset complexes and quotients
  11. The Kaufman-Oppenheim coset complexes
  12. Presentation of $\operatorname{SL}_n (\mathbb{F}_p [t])$
  13. Presentation of unipotent groups
  14. Cohomology of quotients of complexes
  15. Chamber and pre-chamber groups
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $n \geq 3$ and $\Lambda$ a finite group. For every prime $p$ that is large enough with respect to $n$ and $\vert \Lambda \vert$, the family $\lbrace X_{n,p}^{(s)} \rbrace_{s > 3n}$ has uniformly bounded degree and (uniformly) $1$-coboundary expansion over $\Lambda$.

Theorems & Definitions (50)

  • Theorem 1.1: Main Theorem - informal, see formal Theorem \ref{['vanishing of coho for X_n,p^s thm']}
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 40 more