2024/11/08 by Ryan O'Donnell, Ryan O’Donnell, O'Donnell, Ryan +2 · 1 voice
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Black Holes and Theoretical Physics #cs.DM #math.GR
paper · pdf · doi:10.48550/arxiv.2411.05916
openalex publication_date 2024/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
Recent major results in property testing~\citeBLM24,DDL24 and PCPs~\citeBMV24 were unlocked by moving to high-dimensional expanders (HDXs) constructed from \widetildeCd-type buildings, rather than the long-known \widetildeAd-type ones. At the same time, these building quotient HDXs are not as easy to understand as the more elementary (and more symmetric/explicit) coset complex HDXs constructed by Kaufman--Oppenheim~\citeKO18 (of Ad-type) and O'Donnell--Pratt~\citeOP22 (of Bd-, Cd-, Dd-type). Motivated by these considerations, we study the B3-type generalization of a recent work of Kaufman--Oppenheim~\citeKO21, which showed that the A3-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~\citeBD01 on small presentations for the A3-unipotent group over~\mathbbFq; (2)~``lifting'' it to an analogous result for an A3-unipotent group over polynomial extensions~\mathbbFq[x]. For our B3-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 B3-type unipotent groups over~\mathbbF5; (2)~developing significant new ``lifting'' technology to deduce the required quantitative 1-cohomology results in B3-type unipotent groups over \mathbbF5k[x].