Coboundary and Cosystolic Expansion from Strong Symmetry

Tali Kaufman, Izhar Oppenheim

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Coboundary and cosystolic expansion are notions of expansion that generalize the Cheeger constant or edge expansion of a graph to higher dimensions. The classical Cheeger inequality implies that for graphs edge expansion is equivalent to spectral expansion. In higher dimensions this is not the case: a simplicial complex can be spectrally expanding but not have high dimensional edge-expansion. The phenomenon of high dimensional edge expansion in higher dimensions is much more involved than spectral expansion, and is far from being understood. In particular, prior to this work, the only known bounded degree cosystolic expanders were derived from the theory of buildings that is far from being elementary. In this work we study high dimensional complexes which are strongly symmetric. Namely, there is a group that acts transitively on top dimensional cells of the simplicial complex [e.g., for graphs it corresponds to a group that acts transitively on the edges]. Using the strong symmetry, we develop a new machinery to prove coboundary and cosystolic expansion. It was an open question whether the recent elementary construction of bounded degree spectral high dimensional expanders based on coset complexes give rise to bounded degree cosystolic expanders. In this work we answer this question affirmatively. We show that these complexes give rise to bounded degree cosystolic expanders in dimension two, and that their links are (two-dimensional) coboundary expanders. We do so by exploiting the strong symmetry properties of the links of these complexes using a new machinery developed in this work. Previous works have shown a way to bound the co-boundary expansion using strong symmetry in the special situation of "building like"complexes. Our new machinery shows how to get coboundary expansion for general strongly symmetric coset complexes, which are not necessarily "building like", via studying the (Dehn function of the) presentation of the symmetry group of these complexes.

Original languageEnglish
Title of host publication48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
EditorsNikhil Bansal, Emanuela Merelli, James Worrell
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771955
DOIs
StatePublished - 1 Jul 2021
Event48th International Colloquium on Automata, Languages, and Programming, ICALP 2021 - Virtual, Glasgow, United Kingdom
Duration: 12 Jul 202116 Jul 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume198
ISSN (Print)1868-8969

Conference

Conference48th International Colloquium on Automata, Languages, and Programming, ICALP 2021
Country/TerritoryUnited Kingdom
CityVirtual, Glasgow
Period12/07/2116/07/21

Keywords

  • Coboundary expansion
  • Cosystolic expansion
  • High dimensional expanders

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