Discrete Groups, Expanding Graphs and Invariant Measures

Lubotzky, Alexander

doi:10.1007/978-3-0346-0332-4

Cited by 582 publications

(623 citation statements)

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“…Therefore if Aut(F k ) (or equivalently A + (F k )) has Kazhdan's property (T), then one can make Cayley graphs on the symmetric (alternating) groups into a sequence of expanders. This would solve positively Open Problem 10.3.4 and negatively Open Problem 10.3.2 in [Lu1].…”

Section: Discussionmentioning

confidence: 99%

“…Hence there exists a common > 0 such that for every Schreier graph X of SL 2 (Z) modulo Γ (m) we have α(X) > . This follows from the celebrated result of Selberg asserting that λ 1 Γ (m) \ H ≥ 3/16, where H is the upper half plane with its hyperbolic metric and λ 1 (M ) means the bottom of the positive spectrum of the Laplacian on the manifold M (see [Lu1], [Lu2] for details).…”

Section: Open Problem 32 Does Aut(f K ) (Or Equivalently a + (F K )mentioning

confidence: 95%

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The product replacement algorithm and Kazhdan’s property (T)

Lubotzky

Pak

2000

J. Amer. Math. Soc.

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The “product replacement algorithm” is a commonly used heuristic to generate random group elements in a finite group G G , by running a random walk on generating k k -tuples of G G . While experiments showed outstanding performance, the theoretical explanation remained mysterious. In this paper we propose a new approach to the study of the algorithm, by using Kazhdan’s property (T) from representation theory of Lie groups.

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Section: Discussionmentioning

confidence: 99%

Section: Open Problem 32 Does Aut(f K ) (Or Equivalently a + (F K )mentioning

confidence: 95%

The product replacement algorithm and Kazhdan’s property (T)

Lubotzky

Pak

2000

J. Amer. Math. Soc.

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“…The tree numbers t 2k have been studied in great detail; see [McK81], and [Lub94]. Good estimates, a recursion, and their generating function are known, but all we need here is a rough estimate.…”

Section: Proof I: Counting Closed Walks In T Dmentioning

confidence: 99%

“…The book by Davidoff, Sarnak, and Valette [DSV03] offers a self-contained description of the beautiful mathematics around it. Lubotzky's book [Lub94] should be consulted as well. We will review in the following sections some recent attempts at solving this problem using combinatorial and probabilistic methods.…”

Section: Ramanujan Graphsmentioning

confidence: 99%

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Expander graphs and their applications

Hoory

Linial

2006

Bull. Amer. Math. Soc.

1,510

1,355

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A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields.But, perhaps, we should start with a few words about graphs in general. They are, of course, one of the prime objects of study in Discrete Mathematics. However, graphs are among the most ubiquitous models of both natural and human-made structures. In the natural and social sciences they model relations among species, societies, companies, etc. In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more. In mathematics, Cayley graphs are useful in Group Theory. Graphs carry a natural metric and are therefore useful in Geometry, and though they are "just" one-dimensional complexes, they are useful in certain parts of Topology, e.g. Knot Theory. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems.The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs. But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected. Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early '70s. The property of being an expander seems significant in many of these mathematical, computational and physical contexts. It is not surprising that expanders are useful in the design and analysis of communication networks. What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness. In mathematics, we will encounter e.g. their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture. Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications. The list of such interesting and fruitful connections goes on and on with so many applications we will not even be able to mention. This universality of expanders is becoming more evident as more connections are discovered. It transpires that expansion is a fundamental mathematical concept, well deserving to be thoroughly investigated on its own.In hindsight, one reason that expanders are so ubiquitous is that their very definition can be given in at least three languages: combinatorial/geometric, probabilistic and algebraic. Combinatorially, expanders are graphs which are highly connected; to disconnect a large part of the graph, one has to sever many edges. Equivalently, using the geometric notion of isoperimetry, every s...

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The action of a few permutations onr-tuples is quickly transitive

Friedman

Joux²,

Roichman

et al. 1998

Random Struct. Alg.

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ABSTRACT:We prove that for every r and dG 2 there is a C such that for most choices of d permutations , , . . . , of S , the following holds: for any two r-tuples of distinct 1 2 d n Ä 4 elements in 1, . . . , n , there is a product of less than C log n of the s which map the first i r-tuple to the second. Although we came across this problem while studying a rather unrelated cryptographic problem, it belongs to a general context of which random Cayley graph quotients of S are good expanders. ᮊ

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Discrete Groups, Expanding Graphs and Invariant Measures

Cited by 582 publications

References 0 publications

The product replacement algorithm and Kazhdan’s property (T)

The product replacement algorithm and Kazhdan’s property (T)

Expander graphs and their applications

The action of a few permutations onr-tuples is quickly transitive

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