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 language | English |
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Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
Publisher | IEEE Computer Society |
Pages | 484-493 |
Number of pages | 10 |
ISBN (Electronic) | 9781479965175 |
DOIs | |
State | Published - 7 Dec 2014 |
Event | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 - Philadelphia, United States Duration: 18 Oct 2014 → 21 Oct 2014 |
Publication series
Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |
Conference
Conference | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 |
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Country/Territory | United States |
City | Philadelphia |
Period | 18/10/14 → 21/10/14 |
Bibliographical note
Publisher Copyright:© 2014 IEEE.
Keywords
- high dimensional expanders
- Ramanujan complexes
- topological expanders
- topological overlapping