Non-Abelian expansion of congruence KMS complexes

TL;DR

This paper demonstrates that certain classical Chevalley groups' congruence subgroups produce new examples of non-Abelian coboundary expansion in high-dimensional complexes, expanding the known instances of this phenomenon.

Contribution

It introduces four new sources of non-Abelian coboundary expansion examples using classical Chevalley groups, enriching the understanding of high-dimensional expansion.

Findings

Non-Abelian coboundary expansion occurs in complexes from classical Chevalley groups.

Four new examples of such complexes are identified.

This significantly broadens the known landscape of high-dimensional expansion phenomena.

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.