Coboundary expansion inside Chevalley coset complex HDXs

Abstract

Recent major results in property testing~\cite{BLM24,DDL24} and PCPs~\cite{BMV24} were unlocked by moving to high-dimensional expanders (HDXs) constructed from ${\tilde{C}}_d$-type buildings, rather than the long-known ${\tilde{A}}_d$-type ones. At the same time, these building quotient HDXs are not as easy to understand as the more elementary (and more symmetric/explicit) \emph{coset complex} HDXs constructed by Kaufman--Oppenheim~\cite{KO18} (of $A_d$-type) and O'Donnell--Pratt~\cite{OP22} (of $B_d$-, $C_d$-, $D_d$-type). Motivated by these considerations, we study the $B_3$-type generalization of a recent work of Kaufman--Oppenheim~\cite{KO21}, which showed that the $A_3$-type coset complex HDXs have good $1$-coboundary expansion in their links, and thus yield $2$-dimensional topological expanders. The crux of Kaufman--Oppenheim's proof of $1$-coboundary expansion was: (1)~identifying a group-theoretic result by Biss and Dasgupta~\cite{BD01} on small presentations for the $A_3$-unipotent group over~${\mathbb{F}}_q$; (2)~``lifting'' it to an analogous result for an $A_3$-unipotent group over polynomial extensions~${\mathbb{F}}_q[x]$. For our $B_3$-type generalization, the analogue of~(1) appears to not hold. We manage to circumvent this with a significantly more involved strategy: (1)~getting a computer-assisted proof of vanishing $1$-cohomology of $B_3$-type unipotent groups over~${\mathbb{F}}_5$; (2)~developing significant new ``lifting'' technology to deduce the required quantitative $1$-cohomology results in $B_3$-type unipotent groups over ${\mathbb{F}}_{5^k}[x]$.

130 pages