Powering of high-dimensional expanders

Wed, 03/04/2019 - 10:30

Powering the adjacency matrix of an expander graph results in a better expander of higher degree. High dimensional expanders are simplicial complexes which generalize the notion of expanders. In these settings, we look for an analogue of the powering operation. We show that the naive approach to powering does not yield high dimensional expanders in general, but that for quotients of Bruhat Tits buildings a powering operation arises from so-called "geodesic walks". The analysis of the expansion in the power-complex boils down to intricate combinatorial relations between special flags in a free module over the ring Z/(p^r). Based on joint work with Tali Kaufman.