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Non-Abelian expansion of congruence KMS complexes

Izhar Oppenheim, Inga Valentiner-Branth

TL;DR

This work advances the theory of high-dimensional expanders with non-Abelian coefficients by proving vanishing of the first cohomology and establishing non-Abelian cosystolic expansion for congruence KMS complexes of classical type. It constructs and analyzes congruence KMS complexes $X(\mathring\Phi,K,f)$ over finite fields with irreducible polynomials $f$, showing that, for large $|K|$ and suitable coefficient groups $\Lambda$, the family yields uniform $1$-coboundary expanders. A central methodological achievement is the development of non-Abelian cone functions (NAC) and their robust behavior under joins and vertex-adding operations, enabling a non-Abelian local-to-global expansion theory (via DD-cosys/KMS frameworks) to apply to the opposite complexes in types $A_n$, $C_n$, and $D_n$. The results significantly enrich the catalog of high-dimensional expander constructions by providing four new sources of non-Abelian coboundary expanders and establishing scalable, coefficient-flexible expansion bounds with practical implications for complexity theory and related domains.

Abstract

Coboundary expansion with non-Abelian coefficients is a strong version of high-dimensional expansion for simplicial complexes. One motivation for studying this notion is that it was recently shown to have deep connections to problems in theoretical computer science. However, very few examples of families of simplicial complexes with this type of expansion are known. Namely, prior to our work, the only known examples were quotients of symplectic buildings and a slight variation of the Kaufman-Oppenheim coset complexes construction associated with $\operatorname{SL}_{n} (\mathbb{F}_p [t])$. In this paper, we show that the Grave de Peralta and Valentiner-Branth constructions of KMS complexes have coboundary expansion with non-Abelian coefficients when it is performed with respect to congruence subgroups of Chevalley groups of classical type, i.e., of type $A_n, B_n, C_n$ and $D_n$. This gives four new sources of examples to this expansion phenomenon, thus significantly enriching our list of constructions.

Non-Abelian expansion of congruence KMS complexes

TL;DR

This work advances the theory of high-dimensional expanders with non-Abelian coefficients by proving vanishing of the first cohomology and establishing non-Abelian cosystolic expansion for congruence KMS complexes of classical type. It constructs and analyzes congruence KMS complexes over finite fields with irreducible polynomials , showing that, for large and suitable coefficient groups , the family yields uniform -coboundary expanders. A central methodological achievement is the development of non-Abelian cone functions (NAC) and their robust behavior under joins and vertex-adding operations, enabling a non-Abelian local-to-global expansion theory (via DD-cosys/KMS frameworks) to apply to the opposite complexes in types , , and . The results significantly enrich the catalog of high-dimensional expander constructions by providing four new sources of non-Abelian coboundary expanders and establishing scalable, coefficient-flexible expansion bounds with practical implications for complexity theory and related domains.

Abstract

Coboundary expansion with non-Abelian coefficients is a strong version of high-dimensional expansion for simplicial complexes. One motivation for studying this notion is that it was recently shown to have deep connections to problems in theoretical computer science. However, very few examples of families of simplicial complexes with this type of expansion are known. Namely, prior to our work, the only known examples were quotients of symplectic buildings and a slight variation of the Kaufman-Oppenheim coset complexes construction associated with . In this paper, we show that the Grave de Peralta and Valentiner-Branth constructions of KMS complexes have coboundary expansion with non-Abelian coefficients when it is performed with respect to congruence subgroups of Chevalley groups of classical type, i.e., of type and . This gives four new sources of examples to this expansion phenomenon, thus significantly enriching our list of constructions.
Paper Structure (39 sections, 51 theorems, 111 equations, 1 figure)

This paper contains 39 sections, 51 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Vanishing of the first cohomology
  3. Non-Abelian cosystolic expansion
  4. $A_n$ congruence KMS complexes
  5. Organization
  6. Preliminaries
  7. High-dimensional expanders
  8. Weighted simplicial complexes
  9. Local spectral expansion
  10. Cohomological notations, cosystolic and coboundary expansion
  11. Kac--Moody--Steinberg groups and complexes
  12. Root systems and Chevalley groups
  13. Kac--Moody--Steinberg groups
  14. KMS complexes
  15. Congruence KMS complexes
  16. ...and 24 more sections

Key Result

Theorem 1.1

Let $\Lambda$ be a group, $n \geq 3$ an integer and $\mathring \Phi$ a spherical root system of type $A_n, B_n, C_n$ or $D_n$. Also, let $K$ be a finite field of characteristic $p \neq 2$ such that $\vert K \vert$ is sufficiently large with respect to $n$ and that $\Lambda$ does not have an elemen

Figures (1)

  • Figure 1: The Dynkin diagrams of irreducible spherical root systems

Theorems & Definitions (124)

  • Theorem 1.1: See Corollary \ref{['cor: main result CBE']} for a more general formulation
  • Theorem 1.2: See Corollary \ref{['cor: homology of KMS complex is trivial']}
  • Theorem 1.3: See Theorem \ref{['thm: proper links are CBE']} for a more general formulation
  • Theorem 1.4: \ref{['cor: cosystolic exp of classcial KMS']}
  • Remark 1.5
  • Definition 1.6
  • Theorem 1.7
  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • ...and 114 more