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Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree

Authors Yotam Dikstein , Irit Dinur

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Author Details

Yotam Dikstein
  • Institute for Advanced Study, Princeton, NJ, USA
Irit Dinur
  • Weizmann Institute of Science, Rehovot, Israel

Funding

Both authors were supported by Irit Dinur’s ERC grant 772839, and ISF grant 2073/21 when working on this project.
  • Dikstein, Yotam: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1926686.

Acknowledgements

We are deeply grateful to Lewis Bowen for his important feedback on our paper, which greatly improved its clarity and readability.

Cite AsGet BibTex

Yotam Dikstein and Irit Dinur. Coboundary and Cosystolic Expansion Without Dependence on Dimension or Degree. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 62:1-62:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.62

Abstract

We give new bounds on the cosystolic expansion constants of several families of high dimensional expanders, and the known coboundary expansion constants of order complexes of homogeneous geometric lattices, including the spherical building of SL_n(𝔽_q). The improvement applies to the high dimensional expanders constructed by Lubotzky, Samuels and Vishne, and by Kaufman and Oppenheim. Our new expansion constants do not depend on the degree of the complex nor on its dimension, nor on the group of coefficients. This implies improved bounds on Gromov’s topological overlap constant, and on Dinur and Meshulam’s cover stability, which may have applications for agreement testing. In comparison, existing bounds decay exponentially with the ambient dimension (for spherical buildings) and in addition decay linearly with the degree (for all known bounded-degree high dimensional expanders). Our results are based on several new techniques: - We develop a new "color-restriction" technique which enables proving dimension-free expansion by restricting a multi-partite complex to small random subsets of its color classes. - We give a new "spectral" proof for Evra and Kaufman’s local-to-global theorem, deriving better bounds and getting rid of the dependence on the degree. This theorem bounds the cosystolic expansion of a complex using coboundary expansion and spectral expansion of the links. - We derive absolute bounds on the coboundary expansion of the spherical building (and any order complex of a homogeneous geometric lattice) by constructing a novel family of very short cones.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • High Dimensional Expanders
  • HDX
  • Spectral Expansion
  • Coboundary Expansion
  • Cocycle Expansion
  • Cosystolic Expansion

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