Coboundary Expansion Inside Chevalley Coset Complex HDXs

Recent major results in property testing and PCPs were unlocked by moving to high-dimensional expanders (HDXs) constructed from C_d-type buildings, rather than the long-known 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) coset complex HDXs constructed by Kaufman-Oppenheim [KO18] (of A-d-type) and O’Donnell-Pratt [OP22] (of Bd-, Cd-, Dd-type). Motivated by these considerations, we study the B_3-type generalization of a recent work of Kaufman-Oppenheim [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 on small presentations for the A_3-unipotent group over F_q; (2) "lifting" it to an analogous result for an A_3-unipotent group over polynomial extensions 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 F_5; (2) developing significant new "lifting" technology to deduce the required quantitative 1-cohomology results in B_3-type unipotent groups over F_{5^k}[x].

 

Joint work with Noah G. Singer (Carnegie Mellon).