Highlights from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]

TL;DR

This paper summarizes key definitions and results related to Ramanujan quotients of simplicial complexes, connecting graph theory, automorphic representations, and algebraic structures, without providing detailed proofs.

Contribution

It introduces the concept of Ramanujan quotients for simplicial complexes and links them to known Ramanujan graphs and complexes, extending the theory using automorphic representations.

Findings

Defines Ramanujan quotients for simplicial complexes

Recovers known Ramanujan graphs and complexes as special cases

Provides new examples via automorphic representations of affine buildings

Abstract

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.