The Lower Algebraic K-theory Of Virtually Cyclic Subgroups Of The Braid Groups Of The Sphere And Of Zb4(s2) 1st Ed. 2018

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.-- Provided by publisher Preface 6 References 6 Acknowledgements 7 Reference 7 Contents 8 1 Introduction 10 References 13 2 Lower Algebraic K-Theory of the Finite Subgroups of Bn() 15 2.1 Classification of the Virtually Cyclic Subgroups of Bn() 15 2.2 Conjugacy Classes of Binary Polyhedral Groups 17 2.3 Whitehead Groups of the Finite Subgroups of Bn() 21 2.4 widetildeK0(mathbbZ[G]) for the Finite Subgroups of Bn() 22 2.5 K-1(mathbbZ[G]) for the Finite Subgroups of Bn() 26 2.5.1 Torsion of K-1(mathbbZ[G]) for Finite Subgroups of Bn() 26 2.5.2 The Rank of K-1(mathbbZ[G]) for the Finite Subgroups of Bn() 32 2.6 The Lower Algebraic K-Theory of the Finite Subgroups of Bn() for 4leqnleq11 47 References 49 3 The Braid Group B4(), and the Conjugacy Classes of Its Maximal Virtually Cyclic Subgroups 51 3.1 Generalities about B4() 51 3.2 Maximal Virtually Cyclic Subgroups of B4() 55 3.2.1 Proof of Parts (a) and (b) of Theorem 41 56 3.2.2 Proof of Parts (c) and (d) of Theorem 41 58 3.2.3 Proof of the Existence of Maximal Subgroups timesmathbbZ in Part (c) of Theorem 41 63 3.3 Conjugacy Classes of Maximal Infinite Virtually Cyclic Subgroups in B4() 68 References 70 4 Lower Algebraic K-Theory Groups of the Group Ring mathbbZ[B4()] 71 4.1 The Lower K-Theory of Infinite Virtually Cyclic Groups 72 4.2 Preliminary K-Theoretical Calculations for mathbbZ[B4()] 73 4.3 Nil Group Computations 74 References 79 A The Fibred Isomorphism Conjecture 81 The Setup 81 The Conjecture 81 Appendix B Braid Groups 83 References 87 Front Matter ....Pages i-x Introduction (John Guaschi, Daniel Juan-Pineda, Silvia Millán López)....Pages 1-5 Lower Algebraic K-Theory of the Finite Subgroups of \(B_{n}(\mathbb S^{2})\) (John Guaschi, Daniel Juan-Pineda, Silvia Millán López)....Pages 7-42 The Braid Group \(B_{4}(\mathbb S^{2})\), and the Conjugacy Classes of Its Maximal Virtually Cyclic Subgroups (John Guaschi, Daniel Juan-Pineda, Silvia Millán López)....Pages 43-62 Lower Algebraic K-Theory Groups of the Group Ring \(\mathbb Z[B_4(\mathbb S^{2})]\) (John Guaschi, Daniel Juan-Pineda, Silvia Millán López)....Pages 63-72 Back Matter ....Pages 73-80