Free flags over local rings and powering of high dimensional expanders

TL;DR

This paper introduces a novel power operation for high-dimensional expanders using geodesic walks on Bruhat-Tits buildings, leading to higher degree expanders and new efficient double samplers.

Contribution

It defines a new powering method for high-dimensional expanders that preserves expansion properties, using geometric and combinatorial techniques involving flags over local rings.

Findings

The new power operation produces high-dimensional expanders of higher degrees.

Application to double samplers yields more efficient constructions.

Geometric analysis of flags over local rings underpins the expansion properties.

Abstract

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.