Factorization of invertible matrices over rings of stable rank one

Vaserstein, L. N.; Wheland, Ethel R.

doi:10.1017/s1446788700029980

Cited by 16 publications

(8 citation statements)

References 10 publications

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“…Using the same methods a bound the Kazhdan constant for this group can be computed, and it is of the similar to the relative Kazhdan constant. 5 Using a result of L. Vaserstain (see [10]) it can be shown that any element in SLn(Z) can be written as a product of a fixed number of matrices in some H i . The best bound for the number of matrices need to write any element in SLn as such product is around 100 and leads to slightly worse bound for the Kazhdan constant of SLn(Z) than the one obtained in Theorem A.…”

Section: Resultsmentioning

confidence: 99%

“…It is well known (see [2,3,10]) that if n ≥ k + 2, 7 using approximately 2kn elementary operations we can transform any vector system to the canonical vector system U , which contains only standard basis vectors at the first k places and the zero vectors in the other places.…”

Section: Vectors Systems In Z Kmentioning

confidence: 99%

“…The condition n ≥ k + 2, comes from the fact that the ring Z has stable range equal to 2, see[10] for details. Using the fact that SL 3 (Z) is boundedly generated by the elementary matrices it is possible to extend this result to all n ≥ k except n = 2 and k = 1 or k = 2.…”

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confidence: 99%

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KAZHDAN CONSTANTS FOR SL_n(ℤ)

Kassabov

2005

Int. J. Algebra Comput.

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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] −1 , 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 the bounds for the spectral gap of the Cayley graph of SLn(Fp) and for the working time of the product replacement algorithm for abelian groups.

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Section: Resultsmentioning

confidence: 99%

Section: Vectors Systems In Z Kmentioning

confidence: 99%

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KAZHDAN CONSTANTS FOR SL_n(ℤ)

Kassabov

2005

Int. J. Algebra Comput.

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“…We believe that solution of the following two problems is now at hand. Compare the works of Arlinghaus, Leonid Vaserstein, Ethel Wheland and You Hong [43,44,53,3], where this is essentially done for classical groups, over rings subject to sr(R) = 1 or some stronger stability conditions, and the work by Nikolai Gordeev and Jan Saxl [18], where this is essentially done over local rings. Problem 1.…”

Section: Final Remarksmentioning

confidence: 99%

Gauss decomposition for Chevalley groups, revisited

Smolensky,

Sury,

Vavilov

2011

Preprint

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are unipotent radicals of two opposite Borel subgroups B = B(Φ, R) and B − = B − (Φ, R) containing T . It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as sr(R) = 1 or asr(R) = 1. Later the second author noticed that condition sr(R) = 1 is necessary for Gauss decomposition.Here, we show that a slight variation of Tavgen's rank reduction theorem implies that for the elementary group E(Φ, R) condition sr(R) = 1 is also sufficient for Gauss decomposition. In other words, E = HU U − U , where H = H(Φ, R) = T ∩ E. This surprising result shows that stronger conditions on the ground ring, such as being semi-local, asr(R) = 1, sr(R, Λ) = 1, etc., were only needed to guarantee that for simply connected groups G = E, rather than to verify the Gauss decomposition itself.

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“…We can mention results on the width of matrix groups with respect to the set of transvections (see, e.g., [1,4,20]) and the study of the width of verbal subgroups of various free constructions [2,7,8,15,17].…”

Section: Introductionmentioning

confidence: 99%