57 research outputs found

    Ramanujan Complexes and bounded degree topological expanders

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    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.Comment: To appear in FOCS 201

    Coboundary expanders

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    We describe a natural topological generalization of edge expansion for graphs to regular CW complexes and prove that this property holds with high probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main theorem extended to more general random complexe

    Hypergraph expanders from Cayley graphs

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    We present a simple mechanism, which can be randomised, for constructing sparse 33-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over Z2t\mathbb{Z}_2^t and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.Comment: 13 page

    Hypergraph expanders of all uniformities from Cayley graphs

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    Hypergraph expanders are hypergraphs with surprising, non-intuitive expansion properties. In a recent paper, the first author gave a simple construction, which can be randomized, of 33-uniform hypergraph expanders with polylogarithmic degree. We generalize this construction, giving a simple construction of rr-uniform hypergraph expanders for all r≥3r \geq 3.Comment: 32 page
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