On p–almost direct products and residual properties of pure braid groups of nonorientable surfaces

Bellingeri, Paolo; Gervais, Sylvain

doi:10.2140/agt.2016.16.547

Cited by 8 publications

(25 citation statements)

References 26 publications

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“…Proof of Theorem 1.4. If M = K, the result follows from Theorem 1.3 (2), and if M is a compact surface without boundary of genus g ≥ 3, the conclusion follows from [4] and from Theorem 6.1. If M = RP 2 , by [18], B n (RP 2 ) is residually nilpotent if n ≤ 2, and if n = 4, B n (RP 2 ) is residually soluble if n < 4.…”

Section: The Case Of Non-orientable Surfaces Of Higher Genusmentioning

confidence: 92%

“…. , x n }) is residually 2-finite by [4]. Therefore P m+2 (RP 2 ) is residually 2-finite Appendix Let M be the Möbius band, and let n ≥ 1.…”

Section: The Case Of Non-orientable Surfaces Of Higher Genusmentioning

confidence: 99%

“…In the case of the projective plane, B n (RP 2 ) is residually nilpotent if and only if n ≤ 2, and if n = 4, B n (RP 2 ) is residually soluble if and only if n < 4. More recently, if M is a non-orientable surface different from RP 2 , Bellingeri and Gervais showed that P n (M ) is residually 2-finite, and so is residually nilpotent [4].…”

Section: Introductionmentioning

confidence: 99%

“…For a non-orientable surface M without boundary of higher genus, we may decide whether B n (M ) is residually nilpotent or residually soluble using results of [4,18]. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Lower central and derived series of semi-direct products, and applications to surface braid groups

Guaschi

Pereiro

2020

Journal of Pure and Applied Algebra

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For an arbitrary semi-direct product, we give a general description of its lower central series and an estimation of its derived series. In the second part of the paper, we study these series for the full braid group B n (M) and pure braid group P n (M) of a compact surface M , orientable or non-orientable, the aim being to determine the values of n for which B n (M) and P n (M) are residually nilpotent or residually soluble. We first solve this problem in the case where M is the 2-torus. We then use the results of the first part of the paper to calculate explicitly the lower central series of P n (K), where K is the Klein bottle. Finally, if M is a non-orientable, compact surface without boundary, we determine the values of n for which B n (M) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

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Section: The Case Of Non-orientable Surfaces Of Higher Genusmentioning

confidence: 92%

“…. , x n }) is residually 2-finite by [4]. Therefore P m+2 (RP 2 ) is residually 2-finite Appendix Let M be the Möbius band, and let n ≥ 1.…”

Section: The Case Of Non-orientable Surfaces Of Higher Genusmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…For a non-orientable surface M without boundary of higher genus, we may decide whether B n (M ) is residually nilpotent or residually soluble using results of [4,18]. Theorem 1.4.…”

Section: Introductionmentioning

confidence: 99%

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Lower central and derived series of semi-direct products, and applications to surface braid groups

Guaschi

Pereiro

2020

Journal of Pure and Applied Algebra

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“…The following theorem summarises some of the known results about the lower central series of braid groups of non-orientable surfaces without boundary [26,30,37], and is the analogue of Theorems 2.3 and 3.1. One may consult [6] for the case of pure braid groups.…”

Section: Remark 42mentioning

confidence: 99%

Lower central series, surface braid groups, surjections and permutations

Bellingeri

Gonçalves

Guaschi

2021

Math. Proc. Camb. Phil. Soc.

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Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n ∈ $\mathbb N$ for which there exists a surjection between the n- and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.

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Alexander invariants and cohomology jump loci in group extensions

Suciu

2021

Preprint

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We study the integral, rational, and modular Alexander invariants, as well as the cohomology jump loci of groups arising as extensions with trivial algebraic monodromy. Our focus is on extensions of the form 1→K→G→Q→1, where Q is an abelian group acting trivially on H1(K;ℤ), with suitable modifications in the rational and mod-p settings. We find a tight relationship between the Alexander invariants, the characteristic varieties, and the resonance varieties of the groups K and G. This leads to an inequality between the respective Chen ranks, which becomes an equality in degrees greater than 1 for split extensions.

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On p–almost direct products and residual properties of pure braid groups of nonorientable surfaces

Cited by 8 publications

References 26 publications

Lower central and derived series of semi-direct products, and applications to surface braid groups

Lower central and derived series of semi-direct products, and applications to surface braid groups

Lower central series, surface braid groups, surjections and permutations

Alexander invariants and cohomology jump loci in group extensions

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