Free Flags over Local Rings and Powering of High-dimensional Expanders

Tali Kaufman, Ori Parzanchevski

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)14741-14769
Number of pages29
JournalInternational Mathematics Research Notices
Issue number19
StatePublished - 1 Oct 2022

Bibliographical note

Publisher Copyright:
© The Author(s) 2021. Published by Oxford University Press. All rights reserved.


This work was supported by European Research Council (ERC) and United States-Israel Binational Science Foundation (BSF) grants [to T.K.]; and Israel Science Foundation (ISF) [grant 1031/17 to O.P.]. The authors thank David Kazhdan for helpful discussions, and the anonymous referees for various improvements of the paper.

FundersFunder number
European Research Council
United States-Israel Binational Science Foundation
Israel Science Foundation1031/17


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