Abstract | ||
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Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006; Lubotzky, Bull Am Math
Soc, 49:113–162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. 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 (Geom Funct Anal 20(2):416–526, 2010), is whether bounded degree high dimensional expanders exist for \({d \geq 2}\). We present an explicit construction of bounded degree complexes of dimension \({d = 2}\) which are topological expanders, thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on \({\mathbb{F}_2}\) systolic invariants of these complexes, which seem to be the first linear \({\mathbb{F}_2}\) systolic bounds. The expansion results are deduced from these isoperimetric inequalities. |
Year | DOI | Venue |
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2014 | 10.1007/s00039-016-0362-y | Geometric and Functional Analysis |
Keywords | Field | DocType |
Simplicial Complex, Isoperimetric Inequality, Congruence Subgroup, Expander Graph, Spherical Building | Topology,Discrete mathematics,Combinatorics,Expander graph,Open problem,Ramanujan's sum,Invariant (mathematics),Congruence subgroup,Isoperimetric inequality,Conjecture,Mathematics,Bounded function | Journal |
Volume | Issue | ISSN |
abs/1409.1397 | 1 | 1420-8970 |
Citations | PageRank | References |
10 | 1.08 | 8 |
Authors | ||
3 |
Name | Order | Citations | PageRank |
---|---|---|---|
Tali Kaufman | 1 | 499 | 38.33 |
David Kazhdan | 2 | 20 | 1.88 |
Alexander Lubotzky | 3 | 231 | 43.47 |