SpringerBriefs in Mathematics

The Lower Algebraic K-Theory of Virtually Cyclic Subgroups of the Braid Groups of the Sphere and of ZB4(S2)

Authors: Guaschi, John, Juan-Pineda, Daniel, Millán López, Silvia

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  • Includes many worked examples of K-theory computations for finite groups
  • A useful reference for researchers in K-theory, bringing together a broad array of techniques and references in one place, and mainly self-contained
  • Applies the knowledge of virtually-cyclic subgroups to determine the lower algebraic K-theory for the braid groups of B4(S2)
  • Gives new properties about braid groups of the sphere
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About this book

This volume deals with the K-theoretical aspects of the group rings of braid groups of the 2-sphere. The lower algebraic K-theory of the finite subgroups of these groups up to eleven strings is computed using a wide variety of tools. Many of the techniques extend to the general case, and the results reveal new K-theoretical phenomena with respect to the previous study of other families of groups. The second part of the manuscript focusses on the case of the 4-string braid group of the 2-sphere, which is shown to be hyperbolic in the sense of Gromov. This permits the computation of the infinite maximal virtually cyclic subgroups of this group and their conjugacy classes, and applying the fact that this group satisfies the Fibred Isomorphism Conjecture of Farrell and Jones, leads to an explicit calculation of its lower K-theory. 

Researchers and graduate students working in K-theory and surface braid groups will constitute the primary audience of the manuscript, particularly those interested in the Fibred Isomorphism Conjecture, and the computation of Nil groups and the lower algebraic K-groups of group rings. The manuscript will also provide a useful resource to researchers who wish to learn the techniques needed to calculate lower algebraic K-groups, and the bibliography brings together a large number of references in this respect.

Table of contents (4 chapters)

Table of contents (4 chapters)
  • Introduction

    Pages 1-5

    Guaschi, John (et al.)

  • Lower Algebraic K-Theory of the Finite Subgroups of $$B_{n}(\mathbb S^{2})$$

    Pages 7-42

    Guaschi, John (et al.)

  • The Braid Group $$B_{4}(\mathbb S^{2})$$, and the Conjugacy Classes of Its Maximal Virtually Cyclic Subgroups

    Pages 43-62

    Guaschi, John (et al.)

  • Lower Algebraic K-Theory Groups of the Group Ring $$\mathbb Z[B_4(\mathbb S^{2})]$$

    Pages 63-72

    Guaschi, John (et al.)

Buy this book

eBook $54.99
price for USA in USD
  • ISBN 978-3-319-99489-5
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • Immediate eBook download after purchase and usable on all devices
  • Bulk discounts available
Softcover $69.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
The Lower Algebraic K-Theory of Virtually Cyclic Subgroups of the Braid Groups of the Sphere and of ZB4(S2)
Authors
Series Title
SpringerBriefs in Mathematics
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
The Author(s), under exclusive license to Springer Nature Switzerland AG
eBook ISBN
978-3-319-99489-5
DOI
10.1007/978-3-319-99489-5
Softcover ISBN
978-3-319-99488-8
Series ISSN
2191-8198
Edition Number
1
Number of Pages
X, 80
Number of Illustrations
4 b/w illustrations
Topics