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Powering the adjacency matrix of an expander graph results in a betterexpander of higher degree. In this paper we seek an analogue operation forhigh-dimensional expanders. We show that the naive approach to powering doesnot preserve high-dimensional expansion, and define a new power operation,using geodesic walks on quotients of Bruhat-Tits buildings. Applying thisoperation results in high-dimensional expanders of higher degrees. The crux ofthe proof is a combinatorial study of flags of free modules over finite localrings. Their geometry describes links in the power complex, and showing thatthey are excellent expanders implies high dimensional expansion for thepower-complex by Garland's local-to-global technique. As an application, we useour power operation to obtain new efficient double samplers.

Free flags over local rings and powering of high dimensional expanders