On a question of M. Newman on the number of commutators

Dennis, R. Keith; Vaserstein, L. N.

doi:10.1016/0021-8693(88)90055-5

Cited by 65 publications

(59 citation statements)

References 14 publications

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“…[5], Remark 10 in the commutative case). Its proof in case of matrices over finite commutative ring is slightly more complicated:…”

Section: Proof Of Theoremmentioning

confidence: 97%

Universal lattices and unbounded rank expanders

Kassabov

2007

Invent. math.

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We study the representations of non-commutative universal lattices and use them to compute lower bounds for the τ -constant for the commutative universal latticeswith respect to several generating sets.As an application of the above result we show that the Cayley graphs of the finite groups SL 3k (Fp) can be made expanders using suitable choice of the generators. This provides the first examples of expander families of groups of Lie type where the rank is not bounded and gives a natural (and explicit) counter examples to two conjectures of Alex Lubotzky and Benjamin Weiss.

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“…[5], Remark 10 in the commutative case). Its proof in case of matrices over finite commutative ring is slightly more complicated:…”

Section: Proof Of Theoremmentioning

confidence: 97%

Universal lattices and unbounded rank expanders

Kassabov

2007

Invent. math.

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“…This argument can be generalized to semi-local rings if one is careful about the order of the elementary matrices which appear in the product. See also [6], where a more general result is proved for rings with Bass stable range at most 1.…”

Section: Proof Of Theoremmentioning

confidence: 98%

Universal lattices and property tau

Kassabov

Nikolov

2006

Invent. math.

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We prove that the universal lattices -the groups G = SL d (R) where R = Z[x1, . . . , x k ], have property τ for d ≥ 3. This provides the first example of linear groups with τ which do not come from arithmetic groups. We also give a lower bound for the τ -constant with respect to the natural generating set of G. Our methods are based on bounded elementary generation of the finite congruence images of G, a generalization of a result by Dennis and Stein on K2 of some finite commutative rings and a relative property T of (SL2(R) ⋉ R 2 , R 2 ).

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“…We can take xX = G A n such that the first n by n block in [6] (ii) The proof is similar, using the factorization r) = ^rJA. ',^^ f rorn Lemma 9 of [4]. By decomposing r\ as V^t^V^ where the ^(^z) are upper (lower) triangular matrices, r] 6 e /"_* 0 GL k R, and rearranging these matrices, we obtain 0 = \lr^k9 2 \jf 2 , where i/r, are in B + , k is in B~, and 9 2 € h( n -k) ® O 2k R with k = asr(R) + 1.…”

Section: Introductionmentioning

confidence: 99%

Commutators in pseudo-orthogonal groups

Arlinghaus

Vaserstein²,

You

1995

J Aust Math Soc A

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We study commutators in pseudo-orthogonal groups O 2n R (including unitary, symplectic, and ordinary orthogonal groups) and in the conformal pseudo-orthogonal groups GO^R. We estimate the number of commutators, c(C>2nR) and c(G02 n R), needed to represent every element in the commutator subgroup. We show that c(O 2n R) < 4 if R satisfies the A-stable condition and either n > 3 or n = 2 and 1 is the sum of two units in R, and that c(GO 2n R) < 3 when the involution is trivial and A = R f . We also show that ciO^R) < 3 and c(GO 2n R) < 2 for the ordinary orthogonal group O 2rl R over a commutative ring R of absolute stable rank 1 where either n > 3 or n = 2 and 1 is the sum of two units in R.

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On a question of M. Newman on the number of commutators

Cited by 65 publications

References 14 publications

Universal lattices and unbounded rank expanders

Universal lattices and unbounded rank expanders

Universal lattices and property tau

Commutators in pseudo-orthogonal groups

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