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

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