Expanders and the Affine Building of \(\mathrm{Sp}_{n}\)

Abstract

For \(n \geq 2\) 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.