Lp expander Complexes

TL;DR

This paper introduces a new class of high-dimensional expanders called Lp-expander complexes, generalizing expander and Ramanujan graphs to higher-dimensional structures, with spectral and zeta function analyses.

Contribution

It defines Lp-expander complexes via two equivalent combinatorial approaches, computes spectral gaps, and links Ramanujan properties to the Riemann hypothesis for associated zeta functions.

Findings

Explicit spectral gaps calculated for Lp-expander complexes

Established equivalence of two definitions for affine buildings

Connected Ramanujan complexes to the Riemann hypothesis of zeta functions

Abstract

We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of an Lp-expander complex. We calculate explicit spectral gaps on many combinatorical operators, on any Lp-expander complex. We associate with any complex a natural "zeta function", generalizing the Ihara-Hashimoto zeta function of a finite graph. We generalize a well known theorem of Hashimoto, showing that a complex is Ramanujan if and only if the zeta function satisfies the Riemann hypothesis.