Bounded Normal Generation and Invariant Automatic Continuity

Dowerk, Philip A.; Thom, Andreas

doi:10.48550/arxiv.1506.08549

Cited by 3 publications

(4 citation statements)

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“…• Does it have (isometric) representations on a reflexive Banach space? • Does A(q)/F × q have the bounded normal generation property, see [6]? The first question has a positive answer in the case of the hyperfinite II 1 -factors by work of Popa-Takesaki [31], but we were unable to generalize the methods to our setting.…”

Section: Unitary Representabilitymentioning

confidence: 99%

An exotic group as limit of finite special linear groups

Carderi¹,

Thom²

2018

Ann. inst. Fourier

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Section: Unitary Representabilitymentioning

confidence: 99%

An exotic group as limit of finite special linear groups

Carderi¹,

Thom²

2018

Ann. inst. Fourier

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“…Similarly, using results of [29] or [7], if one can approximate a conjugacy class of large cardinality resp. norm, one gets bounded width of the word image.…”

Section: Introductionmentioning

confidence: 99%

Word images in symmetric and classical groups of Lie type are dense

Schneider,

Thom

2018

Preprint

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Let w ∈ F k be a non-trivial word and denote by w(G) ⊆ G the image of the associated word map w : G k → G. Let G be one of the finite groups Sn, GLn(q), Sp 2m (q), GO ± 2m (q), GO2m+1(q), GUn(q) (q a prime power, n ≥ 2, m ≥ 1), or the unitary group Un over C. Let dG be the normalized Hamming distance resp. the normalized rank metric on G when G is a symmetric group resp. one of the other classical groups and write n(G) for the permutation resp. Lie rank of G.For ε > 0, we prove that there exists an integer N (ε, w) such that w(G) is ε-dense in G with respect to the metric dG if n(G) ≥ N (ε, w). This confirms metric versions of a conjectures by Shalev and Larsen. Equivalently, we prove that any non-trivial word map is surjective on a metric ultraproduct of groups G from above such that n(G) → ∞ along the ultrafilter.As a consequence of our methods, we also obtain an alternative proof of the result of Hui-Larsen-Shalev that w1(SUn)w2(SUn) = SUn for non-trivial words w1, w2 ∈ F k and n sufficiently large.

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“…As stated, [3, Theorem 3.9] is not correct and our main result (Theorem 1) should replace it. Note that already in [1], it was pointed out that some of the techniques and results of [3] were flawed. Some corrections on results about bounded generation in the setting of unitary groups on finite-dimensional Hilbert spaces can be found in [1].…”

Section: Introductionmentioning

confidence: 99%

“…Note that already in [1], it was pointed out that some of the techniques and results of [3] were flawed. Some corrections on results about bounded generation in the setting of unitary groups on finite-dimensional Hilbert spaces can be found in [1]. The statement of [3,Theorem 4.20] should be considered as open problem at the moment.…”

Section: Introductionmentioning

confidence: 99%

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

Schneider¹,

Thom²

2017

Preprint

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In [3] it was stated that the lattice of normal subgroups of an ultraproduct of finite simple groups is always linearly ordered. This is false in this form in most cases for classical groups of Lie type. We correct the statement and point out a version of 'relative' bounded generation results for classical quasisimple groups and its implications on the structure of the lattice of normal subgroups of an ultraproduct of quasisimple groups.

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Bounded Normal Generation and Invariant Automatic Continuity

Cited by 3 publications

References 0 publications

An exotic group as limit of finite special linear groups

An exotic group as limit of finite special linear groups

Word images in symmetric and classical groups of Lie type are dense

A note on the normal subgroup lattice of ultraproducts of finite quasisimple groups

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