Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps

Abstract

We introduce the notion of local spectral expansion of a simplicial complex as a possible analogue of spectral expansion defined for graphs. We then show that the condition of local spectral expansion for a complex yields various spectral gaps in both the links of the complex and the global Laplacians of the complex.

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Acknowledgements

The author would like to thank Matthew Kahle for many useful discussions and Alex Lubotzky for the inspiration to pursue this subject.

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Correspondence to Izhar Oppenheim.

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Editor in Charge: János Pach

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Oppenheim, I. Local Spectral Expansion Approach to High Dimensional Expanders Part I: Descent of Spectral Gaps. Discrete Comput Geom 59, 293–330 (2018). https://doi.org/10.1007/s00454-017-9948-x

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Keywords

  • High dimensional expanders
  • Graph Laplacian
  • Simplicial complexes
  • Spectral gap

Mathematics Subject Classification

  • Primary 05E45
  • Secondary 05A20
  • 05C81