Ramanujan geometries of type \(\tilde A_{n}\) (Q1402066)

A finite regular graph of degree \(k\) is said to be Ramanujan if, apart from the trivial eigenvalues \(\pm k\), its spectrum is contained in the range \([-2\sqrt{k-1},+2\sqrt{k-1}]\). Extremal spectral properties corresponding to the Ramanujan property have been generalized by Lubotzky to finite graphs with a common universal cover \(X\) and, roughly speaking, the most concentrated spectrum compatible with that constraint. The authors explore the case when \(X\) is the skeleton graph of an \(\tilde{A}_n\) building and present a few conjectural Ramanujan geometries, supported by examples.