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We construct explicit generating sets Sn and S̃n of the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), Sn) and C(Sym(n), S̃n) into a family of bounded degree expanders for all n. This answers affirmatively an old question which has been asked many times in the literature. These expanders have many applications in the theory of… (More)
On the Automorphism Tower of Free Nilpotent Groups Martin Dimitrov Kassabov 2003 In this thesis I study the automorphism tower of free nilpo-tent groups. Our main tool in studying the automorphism tower is to embed every group as a lattice in some Lie group. Using known rigidity results the automorphism group of the discrete group can be embedded into the… (More)
We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal lattices Gd,k = SLd(Z[x1, . . . , xk]) with respect to several generating sets. As an application of the above result we show that the Cayley graphs of the finite groups SL3k(Fp) can be made expanders… (More)
We prove the following three closely related results: (1) Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. (2) If G is a finite simple group, F a field and M is an FG-module, then dim H.G;M/ .17:5/ dim M . (3) If G is a finite group, F a field and M is an irreducible faithful FG-module, then dim H.G;M/… (More)
We prove that if a Cartesian product of alternating groups is topologically finitely generated, then it is the profinite completion of a finitely generated residually finite group. The same holds for Cartesian producs of other simple groups under some natural restrictions.
Guba and Sapir asked, in their joint paper , if the simultaneous conjugacy problem was solvable in Diagram Groups or, at least, for Thompson’s group F . We give an elementary proof for the solution of the latter question. This relies purely on the description of F as the group of piecewise linear orientationpreserving homeomorphisms of the unit interval… (More)
Article history: Received 29 July 2008 Available online 17 June 2009 Communicated by Peter Webb Dedicated to the memory of Karl Gruenberg
We prove that there exist k in and 0 < epsilon in such that every non-abelian finite simple group G, which is not a Suzuki group, has a set of k generators for which the Cayley graph Cay(G; S) is an epsilon-expander.
We construct an explicit generating sets Fn and F̃n of the alternating and the symmetric groups, which make the Cayley graphs C(Alt(n), Fn) and C(Sym(n), F̃n) a family of bounded degree expanders for all sufficiently large n. These expanders have many applications in the theory of random walks on groups and other areas of mathematics. A finite graph Γ is… (More)
In this article we improve the known Kazhdan constant for SLn(Z) with respect to the generating set of the elementary matrices. We prove that the Kazhdan constant is bounded from below by [42 √ n + 860], which gives the exact asymptotic behavior of the Kazhdan constant, as n goes to infinity, since √ 2/n is an upper bound. We can use this bound to improve… (More)