The first Ramanujan graphs were obtained as quotients of the regular tree, which is the Bruhat-Tits building of type Ã1. In a similar manner, a quotient of the building of type Ãd - 1 is Ramanujan if the spectrum of the colored Hecke operators on the quotient is contained in the spectrum on the building. I will describe a construction of Ramanujan complexes which are quotient by arithmetic lattices of inner form of PGLd(F), F a local field of prime characteristic. It turns out that if d is a prime, then every such quotient is Ramanujan, while if d is composite then there are infinitely many Ramanujan and non-Ramanujan quotients. For outer forms, there are non-Ramanujan constructions for arbitrary d>3. The proof makes use of the strong approximation, local representation theory of GLd, the Jacquet-Langlands correspondence, and Lafforgue's recent results on temperedness for spherical cuspidal automorphic representations.
Related techniques provide pairs of iso-spectral (non-isomorphic) Riemannian surfaces, and pairs of iso-spectral (non-isomorphic) regular graphs.
This is a joint work with A.Lubotzky and B.Samuels.