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High-dimensional expanders from Kac--Moody--Steinberg groups

Laura Grave de Peralta, Inga Valentiner-Branth

TL;DR

This work develops a general method to construct bounded-degree high-dimensional expanders (HDXs) by forming coset complexes over finite quotients of 2-spherical Kac--Moody--Steinberg groups (KMS groups) of rank $d+1$. By ensuring local injectivity on root-subgroup images and a convenient intersection property, the authors prove that the resulting coset complex is a pure $d$-dimensional HDX with explicit bounds on link expansion determined by the Dynkin diagram and the field size, yielding a spectral HDX with parameter $\gamma' = \gamma/(1-(d-1)\gamma)$ when $\gamma \le 1/d$. The construction recovers known affine Chevalley-group HDX constructions and extends them to root systems such as $\tilde{G}_2$, providing new infinite families of bounded-degree HDXs, including explicit SL$_3$-type cases. The framework also compares favorably to prior coset-complex HDX approaches and suggests a broad generalization potential beyond KMS groups to other complexes of groups with similar expansion properties.

Abstract

High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by the work of Kaufman and Oppenheim, we use coset complexes over quotients of Kac-Moody-Steinberg groups of rank $d+1$, $d$-spherical and purely $d$-spherical. We prove that infinite families of such quotients exist provided that the underlying field is of size at least 4 and the Kac-Moody-Steinberg group is 2-spherical, giving rise to new families of bounded degree high-dimensional expanders. In the case the generalized Cartan matrix we consider is affine, we recover the construction of O'Donnell and Pratt from 2022, (and thus also the one of Kaufman and Oppenheim) by considering Chevalley groups as quotients of affine Kac-Moody-Steinberg groups. Moreover, our construction applies to the case where the root system is of type $\tilde{G}_2$, a case that was not covered in earlier works.

High-dimensional expanders from Kac--Moody--Steinberg groups

TL;DR

This work develops a general method to construct bounded-degree high-dimensional expanders (HDXs) by forming coset complexes over finite quotients of 2-spherical Kac--Moody--Steinberg groups (KMS groups) of rank . By ensuring local injectivity on root-subgroup images and a convenient intersection property, the authors prove that the resulting coset complex is a pure -dimensional HDX with explicit bounds on link expansion determined by the Dynkin diagram and the field size, yielding a spectral HDX with parameter when . The construction recovers known affine Chevalley-group HDX constructions and extends them to root systems such as , providing new infinite families of bounded-degree HDXs, including explicit SL-type cases. The framework also compares favorably to prior coset-complex HDX approaches and suggests a broad generalization potential beyond KMS groups to other complexes of groups with similar expansion properties.

Abstract

High-dimensional expanders are a generalization of the notion of expander graphs to simplicial complexes and give rise to a variety of applications in computer science and other fields. We provide a general tool to construct families of bounded degree high-dimensional spectral expanders. Inspired by the work of Kaufman and Oppenheim, we use coset complexes over quotients of Kac-Moody-Steinberg groups of rank , -spherical and purely -spherical. We prove that infinite families of such quotients exist provided that the underlying field is of size at least 4 and the Kac-Moody-Steinberg group is 2-spherical, giving rise to new families of bounded degree high-dimensional expanders. In the case the generalized Cartan matrix we consider is affine, we recover the construction of O'Donnell and Pratt from 2022, (and thus also the one of Kaufman and Oppenheim) by considering Chevalley groups as quotients of affine Kac-Moody-Steinberg groups. Moreover, our construction applies to the case where the root system is of type , a case that was not covered in earlier works.
Paper Structure (21 sections, 30 theorems, 73 equations)

This paper contains 21 sections, 30 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. High-dimensional spectral expanders (HDX)
  4. Coset complexes
  5. Chamber systems
  6. Kac--Moody--Steinberg groups
  7. Construction and properties
  8. Action on (twin) buildings and its consequences
  9. Finite quotients of Kac--Moody--Steinberg groups
  10. Main theorem
  11. Connectivity of the links
  12. Bounded degree
  13. Link expansion
  14. Symmetries of the construction
  15. Quotients in Chevalley groups
  16. ...and 6 more sections

Key Result

Theorem 1

Let $G$ be a finite group, $\phi$ a surjective morphism from a rank $d+1$, $d$-spherical, purely $d$-spherical Kac--Moody--Steinberg group $\mathcal{U}_A(k)$ to $G$ and let $H_i$ be the images of the $U_{\hat{\imath}}$ in $G$. We consider the coset complex $X$ of $G$ with respect to the subgroups $\ If the map $\phi: \mathcal{U}_A(k) \to G$ satisfies then $X$ is a pure $d$-dimensional simplicial

Theorems & Definitions (71)

  • Theorem 1: \ref{['thm: main thm']}
  • Corollary 2: \ref{['corol: main corollary']}
  • Definition 2.1: oppenheim2018local
  • Definition 2.2
  • Theorem 2.3: oppenheim2018local
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Remark 2.7
  • Definition 2.8
  • ...and 61 more