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

Abstract

For $n\ge 2$ and a local field $K$, let ${\Delta }_n$ denote the affine building naturally associated to the symplectic group ${\mathrm{Sp}}_n(K)$. 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.

minor corrections made; 11 pages, 2 figures; to appear in Ars Combinatoria