The Ramanujan Property for Simplicial Complexes

Let G be a topological group acting on a simplicial complex X satisfying some mild assumptions. For example, consider a k-regular tree and its automorphism group, or more generally, a regular affine Bruhat-Tits building and its automorphism group. We define and study various types of high-dimensional spectra of quotients of X by subgroups of G. These spectra include the spectrum of many natural operators associated with the quotients, e.g. the high-dimensional Laplacians.
We prove a theorem in the spirit of the Alon-Boppana Theorem, leading to a notion of Ramanujan quotients of X. Ramanujan k-regular graphs and Ramanuajn complexes in the sense of Lubotzky, Samuels and Vishne are Ramanujan in dimension 0 according to our definition (for X, G suitably chosen). We give a criterion for a quotient of X to be Ramanujan which is phrased in terms of representations of G, and use it, together with deep results about automorphic representations, to show that affine buildings of inner forms of GLn over local fields of positive characteristic admit infinitely many quotients which are Ramanujan in all dimensions. The Ramanujan (in dimension 0) complexes constructed by Lubotzky, Samuels and Vishne arise as a special case of our construction. Our construction also gives rise to Ramanujan graphs which are apparently new.
Other applications are also discussed. For example, we show that there are non-isomorphic simiplicial complexes which are isospectral in all dimensions.