The Ramanujan Property for Simplicial Complexes

Abstract: Let $G$ be a topological group acting on a simplicial complex $\mathcal{X}$ 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 $\mathcal{X}$ 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 $\mathcal{X}$. 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 $\mathcal{X}$, $G$ suitably chosen). We give a criterion for a quotient of $\mathcal{X}$ 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 $\mathbf{GL}_n$ 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.