This paper brings the main definitions and results from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]. No proofs are given. Given a simplicial complex $\mathcal{ X}$ and a group $G$ acting on $\mathcal{ X}$, we define Ramanujan quotients of $\mathcal{ X}$. For $G$ and $\mathcal{ X}$ suitably chosen this recovers Ramanujan $k$-regular graphs and Ramanujan complexes in the sense of Lubotzky, Samuels and Vishne. Deep results in automorphic representations are used to give new examples of Ramanujan quotients when $\mathcal{ X}$ is the affine building of an inner form of $\mathbf{ GL}_n$ over a local field of positive characteristic.
Categories: math.CO math.NT math.RT