## Expansion of Product Replacement Graphs (2001)

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Venue: | Combinatorica |

Citations: | 9 - 1 self |

### BibTeX

@TECHREPORT{Gamburd01expansionof,

author = {Alexander Gamburd and Igor Pak},

title = {Expansion of Product Replacement Graphs},

institution = {Combinatorica},

year = {2001}

}

### Years of Citing Articles

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### Abstract

. We establish a connection between the expansion coefficient of the product replacement graph \Gamma k (G) and the minimal expansion coefficient of a Cayley graph of G with k generators. In particular, we show that the product replacement graphs \Gamma k \Gamma PSL(2; p) \Delta form an expander family, under assumption that all Cayley graphs of PSL(2; p), with at most k generators are expanders. This gives a new explanation of the outstanding performance of the product replacement algorithm and supports the speculation that all product replacement graphs are expanders [LP,P3].

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Citation Context ...onstant of the Cayley graph C(G; S), with jSjsk. This idea is similar in spirit to the paper [DS3], where the eigenvalue gap fi (\Gamma k (G)) was bounded in terms of maximal diameter of C(G; S) (cf. =-=[P3]-=-). For k = \Omega\Gamma059 jGj), the dependence on diameter was later removed in [P4]. Let us say a few words about the proof. The proof is combinatorial in nature and is almost entirely self containe... |

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Citation Context ...o [P1]). While this bound cannot be improved for abelian groups, no better result is known for other classes of groups (cf. [B3]). The first explicit constructions of expanders were found by Margulis =-=[M1]-=-, who used Kazhdan's property (T) from representation theory to prove the expansion. The next breakthrough came in papers [LPS,M2], where the authors used harmonic analysis and number theory to obtain... |

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Citation Context ...der certain conditions, the Cheeger constant of \Gamma k (G) is bounded from below by the minimal Cheeger constant of the Cayley graph C(G; S), with jSjsk. This idea is similar in spirit to the paper =-=[DS3]-=-, where the eigenvalue gap fi (\Gamma k (G)) was bounded in terms of maximal diameter of C(G; S) (cf. [P3]). For k = \Omega\Gamma059 jGj), the dependence on diameter was later removed in [P4]. Let us ... |

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Citation Context ...ain unanswered, such as the connectivity of \Gamma k (G), for general finite groups G. In our running example, it was proved by Gilman that graphs \Gamma k (PSL(2; p)) are connected, for ks3, and ps5 =-=[Gi]-=-. In general, it is known that \Gamma k (G) is connected for all k ? m(G) + d(G) [DS3,P3]. Now, a rigorous study of convergence of random walks on the product replacement graphs \Gamma k (G), for gene... |

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Citation Context ...lity that k independent random elements in G generate the whole group. A major recent result in this direction was completed in a sequence of papers by Dixon [Dx] (see also [B1]), Kantor and Lubotzky =-=[KL]-=-, Liebeck and Shalev [LS1,LS2]. Together, these papers prove that ' 2 (Gn ) ! 1, for any sequence of finite simple groups fGn g, such that jGn j !1. While the overall result is based on classification... |

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Citation Context ...s, provided a known open problem 3) has positive solution: 3) Does group Aut(F k ) have Kazhdan's property (T) ? The problem 3) remains open; an indirect evidence in favor of it is the fact proved in =-=[CV]-=- the it has property (FA) of Serre. It is also known that Aut(F k ) are hyperbolic and thus nonamenable [G1]. There are also some negative indications: Aut(F 2 ) and Aut(F 3 ) are shown not to have (T... |

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Citation Context ...t in computer science, in areas ranging from parallel computation to complexity theory and cryptography; recently they were also used as a key ingredient in connection with the Baum-Connes conjecture =-=[G2]-=- and in computational group theory [LP]. The explicit constructions of expander graphs (by Margulis [M1, M2] and Lubotzky, Phillips, and Sarnak [LPS] ) use deep tools (Kazhdan's property (T), Selberg'... |

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Citation Context ... 8,9.) Therefore,s`(G p ) ! C for infinitely many prime p, where C = C(n) is a fixed constant. In particular, for G(p) = PSL(2; p), we have f(p) = ord(G(p)) = 1 2 p(p \Gamma 1)(p + 1). It is believed =-=[O]-=- that there are infinitely many primes q, such that 6q + 1 and 12q +1 are also primes. Taking p = 12q + 1, this gives f(p) = 12p(6p+1)(12p+ 1), so that `(PSL(2; p))s6 for infinitely many primes p. On ... |

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Citation Context ...of G. Clearly, d(G)sm(G)s`(G)slog 2 jGj: Let ' k (G) denotes the probability that k random group elements generate G. Let ` ffl (G) be the smallest k such that ' k (G) ? 1 \Gamma ffl. It was shown in =-=[P1]-=- that ` ffl (G)s`(G) + C log(1=ffl), for a universal constant C ? 0. Let \Gamma be an (oriented, loops are allowed) graph. Denote by deg(\Gamma) the maximal in and out-degree of a vertex in \Gamma. We... |

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Citation Context ...use deep tools (Kazhdan's property (T), Selberg's Theorem, proved Ramanujan conjectures) to construct families of Cayley graphs of finite groups. The fundamental problem, raised by Lubotzky and Weiss =-=[LW]-=-, is whether being an expander family is a property of the groups alone, independent of the choice of generators (Independence Problem). The product replacement algorithm is a commonly used heuristic ... |

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9 |
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Citation Context ... and details. Let us note that the main theorem is inapplicable to a family of alternating groups fAn g, where ns5. Not all Cayley graphs of An are expanders (see below), and also m(An ) = n \Gamma 2 =-=[W1]-=-, which contradicts the assumptions in Corollary 1. Let us present here an important closely related open problem [B+,Lu,LW]: 4) Is there any sequence of Cayley graphs fC(Sn ; Rn ) \Psi , which is an ... |

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