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Lower central series, surface braid groups, surjections and permutations

Part of: Special aspects of infinite or finite groups Permutation groups

Published online by Cambridge University Press:  05 April 2021

PAOLO BELLINGERI ,
DACIBERG LIMA GONÇALVES  and
JOHN GUASCHI
PAOLO BELLINGERI
Affiliation:
Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14000 Caen, France. e-mail: paolo.bellingeri@unicaen.fr
DACIBERG LIMA GONÇALVES
Affiliation:
Departamento de Matemática - IME-USP, Rua do Matão 1010 CEP: 05508-090 - São Paulo - SP - Brazil. e-mail: dlgoncal@ime.usp.br
JOHN GUASCHI
Affiliation:
Normandie Univ., UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, 14000 Caen, France. e-mail: john.guaschi@unicaen.fr
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Abstract

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.

MSC classification

Primary: 20F36: Braid groups; Artin groups
Secondary: 20F14: Derived series, central series, and generalizations 20B15: Primitive groups
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 373 - 399
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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