Expanders and the Affine Building of ${\mathrm{S}\mathrm{p}}_n$

Abstract

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