Non-Abelian expansion of congruence KMS complexes

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.