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Introduction and Statement of the Main Results

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The Classification of the Virtually Cyclic Subgroups of the Sphere Braid Groups

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Abstract

The main results of this book are presented in this chapter. We start by recalling the basic definitions concerning surface braid groups, as well as some known facts about the braid groups \(B_n(\mathbb S ^2)\) of the sphere \(\mathbb S ^2\), including a presentation, Murasugi’s classification of their torsion elements, and the relation with the mapping class groups \({\mathcal{MCG }(\mathbb{S }^{2},n)}\) of the punctured sphere. In Theorem 5, which is the main result of this book, we classify the isomorphism classes of the infinite virtually cyclic subgroups of \(B_n(\mathbb S ^2)\), up to a finite number of exceptions. As corollaries, in Theorem 7, we obtain the complete classification of the infinite virtually cyclic subgroups of \(B_n(\mathbb S ^2)\) for \(n\geqslant 5\) odd, as well as the corresponding classification for \({\mathcal{MCG }(\mathbb{S }^{2},n)}\) in Theorem 14.

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Authors and Affiliations

  1. Departamento de Matemática - IME, Universidade de São Paulo, Ag. Cidade de São Paulo, São Paulo, 05314-970, Brazil

    Daciberg Lima Gonçalves

  2. Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, UNICAEN, Normandie Université, Caen Cedex, 14032, France

    John Guaschi

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  1. Daciberg Lima Gonçalves
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Lima Gonçalves, D., Guaschi, J. (2013). Introduction and Statement of the Main Results. In: The Classification of the Virtually Cyclic Subgroups of the Sphere Braid Groups. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-00257-6_1

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