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

Abstract

We study K-theoretical aspects of the braid groups B n pS 2 q on n strings of the 2-sphere, which by results of the second two authors, are known to satisfy the Farrell-Jones fibred isomorphism conjecture [56]. In light of this, in order to determine the algebraic K-theory of the group ring ZrB n pS 2 qs, one should first compute that of its virtually cyclic subgroups, which were classified by D. L. Gonçalves and the first author [47]. We calculate the Whitehead and K´1-groups of the group rings of the finite subgroups (dicyclic and binary polyhedral) of B n pS 2 q for all 4 ď n ď 11. Some new phenomena occur, such as the appearance of torsion for the K´1-groups. We then go on to study the case n " 4 in detail, which is the smallest value of n for which B n pS 2 q is infinite. We show that B 4 pS 2 q is an amalgamated product of two finite groups, from which we are able to determine a universal space for proper actions of the group B 4 pS 2 q. We also calculate the algebraic K-theory of the infinite virtually cyclic subgroups of B 4 pS 2 q, including the Nil groups of the quaternion group of order 8. This enables us to determine the lower algebraic K-theory of ZrB 4 pS 2 qs. arXiv:1209.4791v2 [math.KT]

On the algebraic K-theory of 3-manifold groups

“…J. Guaschi, D. Juan-Pineda and S. Millán performed in [GJPML18], extensive calculations of lower K theory groups of some of the groups that appear as fundamental groups of spherical 3-manifolds, one can manufacture examples with nontrivial Whitehead groups using these calculations.…”

The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient B/[P,P]

“…We note that the classification of virtually cyclic subgroups of surface braid groups was studied for the case of the sphere in [14] and for the projective plane in [15]. The lower algebraic K-theory of the sphere braid groups was computed in [18].…”

“…Besides helping us to understand better the structure of G, in the case G satisfies the Fibered Isomorphism Conjecture of Farrell and Jones (see [11]), the algebraic K-theory of their group rings (over Z) may be computed by means of the algebraic Ktheory groups of their virtually cyclic subgroups, via the so-called "assembly maps". For more details about this topic we refer the reader to [14], [15] and [18] and their references. We note that the classification of virtually cyclic subgroups of surface braid groups was studied for the case of the sphere in [14] and for the projective plane in [15].…”

The Whitehead group of (almost) extra-special p-groups with p odd

“…The computation of the Whitehead group W h(G) of a finite group G is in general very hard, and a compendium on the subject is the book by Bob Oliver ([17]) of 1988. Since then, it seems that not much progress has been made on this subject (see however [16], [15], [22], [21]).…”