  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 expanders. We show that the naive approach to powering does
not preserve high-dimensional expansion, and define a new power operation,
using geodesic walks on quotients of Bruhat-Tits buildings. Applying this
operation results in high-dimensional 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 high dimensional 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.
