Expanders and the Affine Building of ${\rm Sp}_n$

TL;DR

This paper analyzes the spectral properties of a subgraph related to the affine building of the symplectic group, demonstrating it forms a family of expanders and is non-amenable, thus contributing to the understanding of expander graphs in algebraic structures.

Contribution

It computes the spectral radius of a subgraph of the affine building associated with ${ m Sp}_n(K)$, establishing it as an expander and non-amenable, which is a new insight in the field.

Findings

The subgraph $Y_n$ has a specific spectral radius.

$Y_n$ is an expander graph.

$Y_n$ is non-amenable.

Abstract

For $n \geq 2$ and a local field $K$, let $\Delta_n$ denote the affine building naturally associated to the symplectic group ${\rm Sp}_n(K)$. We compute the spectral radius of the subgraph $Y_n$ of $\Delta_n$ induced by the special vertices in $\Delta_n$, from which it follows that $Y_n$ is an analogue of a family of expanders and is non-amenable.