Dimension expanders

Dimension expanders via quiver representations

We relate the notion of dimension expanders to quiver representations and their general subrepresentations, and use this relation to establish sharp existence results.

Ramanujan Complexes and bounded degree topological expanders

Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to simplicial complexes, among them stand out coboundary expansion and topological expanders. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov, is whether bounded degree high dimensional expanders, according to these definitions, exist for d >= 2. We …

High-Dimensional Expanders from Chevalley Groups

Let $Φ$ be an irreducible root system (other than $G_2$) of rank at least $2$, let $\mathbb{F}$ be a finite field with $p = \operatorname{char} \mathbb{F} > 3$, and let $\mathrm{G}(Φ,\mathbb{F})$ be the corresponding Chevalley group. We describe a strongly explicit high-dimensional expander (HDX) family of dimension $\mathrm{rank}(Φ)$, where $\mathrm{G}(Φ,\mathbb{F})$ acts simply transitively on the top-dimensional faces; these are $λ$-spectral HDXs with $λ\to 0$ as $p \to \infty$. This generalizes a construction of Kaufman and Oppenheim (STOC 2018), which corresponds to the case $Φ= A_d$. Our…

来自KMS群的新型余上同调高维展开图

New cosystolic high-dimensional expanders from KMS groups

本文利用Kac-Moody-Steinberg群的有限商群构造KMS复形，首次给出任意维数下余上同调展开图的陪集复形构造，并证明其链结为余边界展开图，从而获得无限族有界度余上同调展开图。

Bounded Degree Cosystolic Expanders of Every Dimension

In this work we present a new local to global criterion for proving a form of high dimensional expansion, which we term cosystolic expansion. Applying this criterion on Ramanujan complexes, yields for every dimension, an infinite family of bounded degree complexes with the topological overlapping property. This answer affirmatively an open question raised by Gromov.

超正则图与高维膨胀图

Hyper-regular graphs and high dimensional expanders

该研究定义了n维超正则图（HRG）与新型图乘积，构造出团联合邻域连通的无限族n维HRG，还依托已有成果得到具有良好膨胀性质的任意大n的无限族HRG，解答了Dinur的相关存在性问题。