Ramanujan complexes and bounded degree topological expanders

Tali Kaufman, David Kazhdan, Alexander Lubotzky

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

21 Scopus citations

Abstract

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 present an explicit construction of bounded degree complexes of dimension d = 2 which are high dimensional expanders. More precisely, our main result says that the 2-skeletons of the 3-dimensional Ramanujan complexes are topological expanders. Assuming a conjecture of Serre on the congruence subgroup property, infinitely many of them are also coboundary expanders.

Original languageEnglish
Title of host publicationProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
PublisherIEEE Computer Society
Pages484-493
Number of pages10
ISBN (Electronic)9781479965175
DOIs
StatePublished - 7 Dec 2014
Event55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 - Philadelphia, United States
Duration: 18 Oct 201421 Oct 2014

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
ISSN (Print)0272-5428

Conference

Conference55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014
Country/TerritoryUnited States
CityPhiladelphia
Period18/10/1421/10/14

Bibliographical note

Publisher Copyright:
© 2014 IEEE.

Keywords

  • Ramanujan complexes
  • high dimensional expanders
  • topological expanders
  • topological overlapping

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