Free Flags Over Local Rings and Powering of High-dimensional Expanders. (11th June 2021)

• Record Type: Journal Article
• Authors: Kaufman, Tali; Parzanchevski, Ori
• Is Part Of: International mathematics research notices. Volume 2022:Number 19(2022)
• Page Start: 14741
• Page End: 14769
• Publication Date: 2021-06-11
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab126
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: Powering the adjacency matrix of an expander graph results in a better expander of higher degree. In this paper we seek an analogue operation for high-dimensional (HD) expanders. We show that the naive approach to powering does not preserve HD expansion and define a new power operation, using geodesic walks on quotients of Bruhat–Tits buildings. Applying this operation results in HD expanders of higher degrees. The crux of the proof is a combinatorial study of flags of free modules over finite local rings. Their geometry describes links in the power complex, and showing that they are excellent expanders implies HD expansion for the power complex by Garland's local-to-global technique. As an application, we use our power operation to obtain new efficient double samplers.