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14
Random walks on finite groups
 In Probability on Discrete Structures, Encyclopedia of Mathematical Sciences
, 2004
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Two Kazhdan constants and mixing of random walks
 Int. Math. Res. Not. 2002
, 2001
"... Let G be a group with Kazhdan’s property (T), and let S be a transitive generating set (there exists a group H ⊂ Aut(G) which acts transitively on S.) In this paper we relate two definitions of the Kazhdan constant and the eigenvalue gap in this case. Applications to various random walks on groups, ..."
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Cited by 11 (1 self)
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Let G be a group with Kazhdan’s property (T), and let S be a transitive generating set (there exists a group H ⊂ Aut(G) which acts transitively on S.) In this paper we relate two definitions of the Kazhdan constant and the eigenvalue gap in this case. Applications to various random walks on groups, and the product replacement random algorithm, are also presented. 1
Connectivity of the Product Replacement Algorithm Graph of PSL(2, q
, 2007
"... Abstract. The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph Γk(G) whose vertices are the generating ktuples of G (for a fixed k). We show that if G = PSL(2, q) or PGL(2, q), where q is a prime ..."
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Abstract. The product replacement algorithm is a practical algorithm to construct random elements of a finite group G. It can be described as a random walk on a graph Γk(G) whose vertices are the generating ktuples of G (for a fixed k). We show that if G = PSL(2, q) or PGL(2, q), where q is a prime power, then Γk(G) is connected for any k ≥ 4. This generalizes former results obtained by Gilman and Evans. 1.
Commutator maps, measure preservation, and Tsystems
"... Let G be a finite simple group. We show that the commutator map α: G × G → G is almost equidistributed as G  → ∞. This somewhat surprising result has many applications. It shows that a for a subset X ⊆ G we have α −1 (X)/G  2 = X/G  + o(1), namely α is almost measure preserving. From this w ..."
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Cited by 4 (2 self)
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Let G be a finite simple group. We show that the commutator map α: G × G → G is almost equidistributed as G  → ∞. This somewhat surprising result has many applications. It shows that a for a subset X ⊆ G we have α −1 (X)/G  2 = X/G  + o(1), namely α is almost measure preserving. From this we deduce that almost all elements g ∈ G can be expressed as commutators g = [x, y] where x, y generate G. This enables us to solve some open problems regarding Tsystems and the Product Replacement Algorithm (PRA) graph. We show that the number of Tsystems in G with two generators tends to infinity as G  → ∞. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators. Some of our results apply for more general finite groups, and more general word maps. Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function ζG (s) = ∑ χ∈Irr(G) χ(1)−s plays a key role in the proofs.
CONNECTIVITY OF THE PRODUCT REPLACEMENT GRAPH OF SIMPLE GROUPS OF BOUNDED LIE RANK
, 2008
"... The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating ktuples of the group (for a fixed k). We show that there is a function c(r) such that for any f ..."
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Cited by 3 (2 self)
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The Product Replacement Algorithm is a practical algorithm for generating random elements of a finite group. The algorithm can be described as a random walk on a graph whose vertices are the generating ktuples of the group (for a fixed k). We show that there is a function c(r) such that for any finite simple group of Lie type, with Lie rank r, the Product Replacement Graph of the generating ktuples is connected for any k ≥ c(r). The proof uses results of Larsen and Pink [17] and does not rely on the classification of finite simple groups.
A polynomialtime theory of matrix groups and black box groups
 in these Proceedings
"... We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic proble ..."
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We consider matrix groups, specified by a list of generators, over finite fields. The two most basic questions about such groups are membership in and the order of the group. Even in the case of abelian groups it is not known how to answer these questions without solving hard number theoretic problems (factoring and discrete log); in fact, constructive membership testing in the case of 1 × 1 matrices is precisely the discrete log problem. So the reasonable question is whether these problems are solvable in randomized polynomial time using number theory oracles. Building on 25 years of work, including remarkable recent developments by several groups of authors, we are now able to determine the order of a matrix group over a finite field of odd characteristic, and to perform constructive membership testing in such groups, in randomized polynomial time, using oracles for factoring and discrete log. One of the new ingredients of this result is the following. A group is called semisimple if it has no abelian normal subgroups. For matrix groups over finite fields, we show that the order of the largest semisimple quotient can be determined in randomized polynomial time (no number theory oracles required and no restriction on parity). As a byproduct, we obtain a natural problem that belongs to BPP and is not known to belong either to RP or to coRP. No such problem outside the area of matrix groups appears to be known. The problem is the decision version of the above: Given a list A of nonsingular d × d matrices over a finite field and an integer N, does the group generated by A have a semisimple quotient of order ≥ N? We also make progress in the area of constructive recognition of simple groups, with the corollary that for a large class of matrix groups, our algorithms become Las Vegas.
Out (Fn) and the Spectral Gap Conjecture
"... For n>2, given ϕ1,...,ϕn randomly chosen isometries of S2, it is well known that the group Γ generated by ϕ1,...,ϕn acts ergodically on S2. In 1999, Gamburd, Jakobson, and Sarnak conjectured that for almost every choice of ϕ1,...,ϕn, this action is strongly ergodic. This is equivalent to the spec ..."
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For n>2, given ϕ1,...,ϕn randomly chosen isometries of S2, it is well known that the group Γ generated by ϕ1,...,ϕn acts ergodically on S2. In 1999, Gamburd, Jakobson, and Sarnak conjectured that for almost every choice of ϕ1,...,ϕn, this action is strongly ergodic. This is equivalent to the spectrum of ϕ1 + ϕ−1 1 + ·· · + ϕn + ϕ−1 n as an operator on L2 (S2) having a spectral gap, that is, all eigenvalues but the largest one being bounded above by some λ1 <2n. (The largest eigenvalue λ0, corresponding to constant functions, is 2n.) In this paper, we show that if n>2, then either the conjecture is true or almost every ntuple fails to have a gap. In fact, the same result holds for any ntuple ϕ1,...,ϕn in any compact group K that is an almost direct product of SU(2) factors with L 2 (S 2) replaced by L2 (X), where X is any homogeneous K space. A weaker result is proven for n = 2 and some conditional results for similar actions of Fn on homogeneous spaces for more general compact groups. 1
On Kazhdan Constants and Mixing of Random Walks
"... Abstract Let G be a group with Kazhdan's property (T), and let S be a transitive generating set (there exists a group H ae Aut(G) which acts transitively on S.) In this paper we relate two definitions of the Kazhdan constant and the eigenvalue gap in this case. Applications to various random wa ..."
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Abstract Let G be a group with Kazhdan's property (T), and let S be a transitive generating set (there exists a group H ae Aut(G) which acts transitively on S.) In this paper we relate two definitions of the Kazhdan constant and the eigenvalue gap in this case. Applications to various random walks on groups, and the product replacement random algorithm, are also presented.