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.
| Original language | English |
|---|---|
| Pages (from-to) | 14741-14769 |
| Number of pages | 29 |
| Journal | International Mathematics Research Notices |
| Volume | 2022 |
| Issue number | 19 |
| DOIs | |
| State | Published - 1 Oct 2022 |
Bibliographical note
Publisher Copyright:© The Author(s) 2021. Published by Oxford University Press. All rights reserved.
Funding
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.
| Funders | Funder number |
|---|---|
| European Commission | |
| United States-Israel Binational Science Foundation | |
| Israel Science Foundation | 1031/17 |