Free Flags Over Local Rings and Powering of High-dimensional Expanders
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 powering operation, using geodesic walks on quotients of Bruhat-Tits buildings. When applying our geodesic powering operation to these complexes (and in particular to Ramanujan complexes), the resulting complexes are high-dimensional expanders of higher degrees. The crux of the proof is a combinatorial study of the spherical geometry of flags of free modules over finite local rings. This geometry, which is interesting in its own right, arises from the study of links of cells in the power complex. We show that these links are excellent expanders, thus establishing high dimensional expansion for the power-complex by the Garland technique. 
As an application, we use our powering operation to obtain new efficient double samplers, which were recently defined by Dinur and Kaufman. We achieve this by first showing that geodesic walks on Ramanujan complexes sample the vertices well, and then use the power-complex to construct a "large triangle walk", which samples well the geodesics.