New cosystolic high-dimensional expanders from KMS groups

TL;DR

The paper constructs new cosystolic high-dimensional expanders of bounded degree in arbitrary dimension by leveraging finite quotients of KMS groups to form coset complexes. It introduces a novel iterative cone-function technique that extends cone structures by adding vertex-sets, connecting local link expansion (via opposition complexes in spherical buildings) to global cosystolic expansion through established local-to-global results. The main results prove that in the $n$-classical KMS setting over large fields, all proper links are coboundary expanders and the (n-1)-skeletons are cosystolic expanders, yielding topological overlapping via DKW. This work generalizes prior 2-dimensional cosystolic constructions from Chevalley quotients and affine buildings to higher dimensions and arbitrary finitely generated Abelian coefficients, offering a symmetric, elementary framework grounded in coset complexes and spherical building oppositions. The approach significantly broadens the catalog of bounded-degree HDX constructions and deepens the connection between algebraic group actions, building theory, and high-dimensional expansion phenomena.

Abstract

Cosystolic expansion is a high-dimensional generalization of the Cheeger constant for simplicial complexes. Originally, this notion was motivated by the fact that it implies the topological overlapping property, but more recently it was shown to be connected to problems in theoretical computer science such as list agreement expansion and agreement expansion in the low soundness regime. There are only a few constructions of high-dimensional cosystolic expanders and, in dimension larger than $2$, the only known constructions prior to our work were (co-dimension 1)-skeletons of quotients of affine buildings. In this paper, we give the first coset complex construction of cosystolic expanders for an arbitrary dimension. Our construction is more symmetric and arguably more elementary than the previous constructions relying on quotients of affine buildings. The coset complexes we consider arise from finite quotients of Kac--Moody--Steinberg (KMS) groups and are known as KMS complexes. KMS complexes were introduced in recent work by Grave de Peralta and Valentiner-Branth where it was shown that they are local-spectral expanders. Our result is that KMS complexes, satisfying some minor condition, give rise to infinite families of bounded degree cosystolic expanders of arbitrary dimension and for any finitely generated Abelian coefficient group. This result is achieved by observing that proper links of KMS complexes are joins of opposition complexes in spherical buildings. In order to show that these opposition complexes are coboundary expanders, we develop a new method for constructing cone functions by iteratively adding sets of vertices. Hence we show that the links of KMS complexes are coboundary expanders. Using the prior local-to-global results, we obtain cosystolic expansion for the (co-dimension 1)-skeletons of the KMS complexes.

Figures (1)

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

Theorems & Definitions (82)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Definition 2.2: Cone function
• Definition 2.3: Radius of a cone
• Remark 2.4
• Example 2.5
• Proposition 2.6
• proof
• Theorem 2.8