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arXiv · 1606.01098

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

Uriya A. First
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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.

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Uriya A. First. 2016-06-03. Highlights from "The Ramanujan Property for Simplicial Complexes" [arXiv:1605.02664]. https://arxiv.org/abs/1606.01098

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