John Guaschi
email: john.guaschi@unicaen.fr

Daniel Juan-Pineda
email: daniel@matmor.unam.mx

Silvia Millán López

The lower algebraic K-theory of virtually cyclic subgroups of the braid groups of the sphere and of ZrB 4 pS 2 qs

. 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.

We wish to thank the following colleagues for helpful and fruitful discussions: Bruno Anglès, for help with the Galois theory of Section 2.5.2; Eric Jespers, Gerardo Raggi and Ángel del Río (and the GAP package Wedderga [14]), for aiding us with the Wedderburn decomposition of dicyclic and binary polyhedral groups; Jean-François Lafont, Ivonne Ortiz and Stratos Prassidis, for conversations on K-theoretical aspects of our work during the early stages of the writing of this paper; and Chuck Weibel for valuable comments at various points. We would also like to thank the referees for their comments on the

Chapter 1 Introduction

Given a group G, the K-theoretic fibred isomorphism conjecture of F. T. Farrell and L. E. Jones asserts that the algebraic K-theory of its integral group ring ZrGs may be computed from the knowledge of the algebraic K-theory groups of its virtually cyclic subgroups (see [START_REF] Farrell | Isomorphism conjectures in algebraic K-theory[END_REF] or Appendix A for the statement). This conjecture has been verified for a number of classes of groups, such as discrete cocompact subgroups of virtually connected Lie groups [START_REF] Farrell | Isomorphism conjectures in algebraic K-theory[END_REF], finitely-generated Fuchsian groups [START_REF] Berkove | A geometric approach to the lower algebraic K-theory of Fuchsian groups[END_REF], Bianchi groups [START_REF] Berkove | The Farrell-Jones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic K-theory of Bianchi groups[END_REF], pure braid groups of aspherical surfaces [START_REF] Aravinda | Algebraic K-theory of pure braid groups[END_REF], braid groups of aspherical surfaces [START_REF] Farrell | The Whitehead groups of braid groups vanish[END_REF] and for some classes of mapping class groups [START_REF] Berkove | Algebraic K-theory of mapping class groups[END_REF]. In [START_REF] Lafont | Lower algebraic K-theory of hyperbolic 3-simplex reflection groups[END_REF], Lafont and Ortiz presented explicit computations of the lower algebraic K-theory of hyperbolic 3-simplex reflection groups, and then together with Magurn, for that of certain reflection groups [START_REF] Lafont | Lower algebraic K-theory of certain reflection groups[END_REF]. Similar calculations were performed for virtually free groups in [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF].

Let n P N, let M be a surface, and let B n pMq (resp. P n pMq) denote the n-string braid group (resp. n-string pure braid group) of M [START_REF] Birman | Braids, links and mapping class groups[END_REF][START_REF] Hansen | Braids and coverings: Selected topics[END_REF]. Some basic information and facts about surface braid groups are given in Appendix B. The braid groups of the 2-sphere S 2 were first studied by Zariski, and then later by Fadell and Van Buskirk during the 1960's [START_REF] Fadell | The braid groups of E 2 and S 2[END_REF][START_REF] Zariski | On the Poincaré group of rational plane curves[END_REF]. If M either is the 2-sphere S 2 or the projective plane RP 2 , the results of [START_REF] Aravinda | Algebraic K-theory of pure braid groups[END_REF][START_REF] Farrell | The Whitehead groups of braid groups vanish[END_REF] do not apply to its braid groups, the principal reason being that these groups possess torsion [START_REF] Fadell | The braid groups of E 2 and S 2[END_REF][START_REF] Van Buskirk | Braid groups of compact 2-manifolds with elements of finite order[END_REF]. The second two authors of this book proved that the conjecture of Farrell and Jones holds also for the braid groups of these two surfaces, which using the method prescribed by the conjecture, enabled them to carry out complete computations of the lower algebraic K-groups for P n pS 2 q and P n pRP 2 q [START_REF] Juan-Pineda | The Whitehead group and the lower algebraic K-theory of braid groups on S 2 and RP 2[END_REF]. One necessary ingredient in this process is the knowledge of the virtually cyclic subgroups of P n pMq. For n ě 4, P n pS 2 q has only one non-trivial finite subgroup, generated by the 'full twist' braid, which is central and of order 2, and from this, it is straightforward to see that P n pS 2 q has very few isomorphism classes of virtually cyclic subgroups. The classification of the isomorphism classes of the virtually cyclic subgroups of P n pRP 2 q, which was established in [START_REF] Gonçalves | Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane[END_REF] and used subsequently in [START_REF] Juan-Pineda | The Whitehead group and the lower algebraic K-theory of braid groups on S 2 and RP 2[END_REF] to compute the K-theory groups of ZrP n pRP 2 qs, is rather more involved.

Our aim in this manuscript is to implement similar K-theoretical computations for the group ring ZrB n pS 2 qs of the full braid groups of S 2 . In order to do so, one must determine initially the virtually cyclic subgroups (finite, and then infinite) of B n pS 2 q, and then compute the K-groups of these subgroups. If n ď 3 then B n pS 2 q is finite, and so we shall assume in much of this manuscript that n ě 4. The torsion of B n pS 2 q was determined in [START_REF] Gillette | The word problem and consequences for the braid groups and mapping class groups of the 2-sphere[END_REF], and its fi-nite order elements were classified in [START_REF] Murasugi | Seifert fibre spaces and braid groups[END_REF]. It was shown by D. L. Gonçalves in collaboration with the first author that up to isomorphism, the finite subgroups of B n pS 2 q are cyclic, dicyclic or binary polyhedral (see [START_REF] Gonçalves | The classification and the conjugacy classes of the finite subgroups of the sphere braid groups[END_REF] or Theorem 2). As for the corresponding pure braid groups, one must then determine the infinite virtually cyclic subgroups of B n pS 2 q with the aid of the characterisation due to Epstein and Wall of infinite virtually cyclic groups [START_REF] Epstein | Ends[END_REF][START_REF] Juan-Pineda | On classifying spaces for the family of virtually cyclic subgroups, in Recent developments in algebraic topology[END_REF][START_REF] Wall | Poincaré complexes I[END_REF]. Up to isomorphism and with a few exceptions in the case that n is a small even number, this was achieved in [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]. A taste of the results is given in Theorem 4 when n is odd and in Theorem 39 when n " 4.

In the ensuing quest to compute the lower algebraic K-theory of the group ring ZrB n pS 2 qs, we encountered a number of difficulties, among them: (a) the family of virtually cyclic subgroups of B n pS 2 q is relatively large, and depends on n, contrasting sharply with the case of the pure braid groups analysed in [START_REF] Juan-Pineda | The Whitehead group and the lower algebraic K-theory of braid groups on S 2 and RP 2[END_REF]. (b) the lower algebraic K-theory of even the finite subgroups of B n pS 2 q is poorly understood, and the investigation of the K-groups of dicyclic and binary polyhedral groups presents additional technical obstacles compared to that of the dihedral and polyhedral groups that appear in [START_REF] Lafont | Lower algebraic K-theory of certain reflection groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of hyperbolic 3-simplex reflection groups[END_REF] for example. (c) in order to apply the method of calculation suggested by the fibred isomorphism conjecture, one needs not only to compute the various Nil groups, but also to discover a suitable universal space for the family of virtually cyclic subgroups of B n pS 2 q. In spite of the rich topological and geometric structures of the braid groups and their associated configuration spaces, this space has thus far proved to be elusive for n ě 5.

Chapter 2 is devoted to the second point, that of the computation of the lower K-theory groups of many of the finite subgroups of B n pS 2 q. In Section 2.1, we recall the classification up to isomorphism of the finite subgroups of B n pS 2 q, and of its virtually cyclic subgroups when n is odd or n " 4. In Section 2.2, we compute the number of different types of conjugacy classes of the binary polyhedral groups in Proposition 7. These results are used later in the chapter, when we determine the lower algebraic K-theory of the group rings of these groups. In Sections 2.3 and 2.5, we calculate the Whitehead and the K ´1-groups respectively of the integral group rings of many of the finite subgroups of B n pS 2 q. To our knowledge, these sections contain a number of original results, as well as some new phenomena, such as the existence of torsion for some K ´1-groups, that did not appear in previous work [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of certain reflection groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of hyperbolic 3-simplex reflection groups[END_REF]. This necessitates alternative techniques, notably the application of results of Yamada to determine local Schur indices [START_REF] Yamada | On the group algebras of metacyclic groups over algebraic number fields[END_REF][START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF], which enables us to calculate the torsion of our K ´1-groups. We believe that the methods that we use to calculate these K-groups for dicyclic groups of certain orders may be extended to dicyclic groups of other orders. The Whitehead groups are given in Proposition 10. The main results concerning the K ´1-groups are Theorem 25 for dicyclic groups of order 4m, where m is an odd prime, Proposition 27 for the generalised quaternions (the dicyclic groups of order a power of 2), Proposition 28 for the binary polyhedral groups. We also compute the K ´1-groups of the dicyclic groups of order 24, 36 and 40 in Proposition 29, and of cyclic groups of order 2p q , 12 and 20, where p is prime and q P N in Proposition 30. In Section 2.4, we recall briefly the work of Swan pertaining to the calculation of the r K 0 -groups of the group rings of the binary polyhedral groups, and of the dicyclic groups of order 4m for m ď 11 [START_REF] Swan | Projective modules over binary polyhedral groups[END_REF], and in Theorem 14, we compute r K 0 pZrGsq where G is a cyclic group of order 18, 20 or 22. For dicyclic groups of higher order, the situation is complicated, and little seems to be known about the corresponding r K 0 -groups. In Section 2.6, we sum up the results of many of our computations in Table 2.1, which lists the lower K-theory groups of the finite subgroups of B n pS 2 q for all 4 ď n ď 11.

From this table, we may also obtain the lower K-theory groups of B n pS 2 q in the cases where B n pS 2 q is finite, namely for n P t1, 2, 3u. The aim of the remaining two chapters is to determine the lower algebraic K-theory of ZrB 4 pS 2 qs. We study the case of B 4 pS 2 q in detail and show how the algebraic and geometric features of this group interact, thus allowing us to compute its lower K-groups. In preparation for the explicit computations in Chapter 4, in Chapter 3, we describe the ingredients for the corresponding computations in the case of infinite virtually cyclic subgroups. We start by recalling some basic facts about the group B 4 pS 2 q in Section 3.1. One striking property, which was proved in [START_REF] Gonçalves | The lower central and derived series of the braid groups of the sphere[END_REF]Theorem 1.3(3)], is that it possesses a finite normal subgroup isomorphic to the quaternion group of order 8. This enables us to show in Proposition 37 that B 4 pS 2 q is an amalgamated product of the generalised quaternion group of order 16 and the binary tetrahedral group, the amalgamation being along this normal subgroup, from which we deduce in Remark 38 that it is hyperbolic in the sense of Gromov. In Section 3.2, we determine the isomorphism classes of the maximal virtually cyclic subgroups of B 4 pS 2 q in Theorem 41, and in Section 3.3, we show that there are an infinite number of conjugacy classes for each of the isomorphism classes of the infinite maximal virtually cyclic subgroups. These properties aid greatly, not just in the computation of the K-groups of the virtually cyclic subgroups of B 4 pS 2 q and of the corresponding Nil groups, but also to exhibit an appropriate universal space referred to in (c) above.

In Chapter 4, we bring together the results of the previous chapters to compute the lower K-groups of B 4 pS 2 q. In Section 4.1, we recall some facts and results about the lower K-theory of infinite virtually cyclic groups. In Section 4.2, we determine the lower K-groups of B 4 pS 2 q up to the computations of the associated Nil groups, and in Section 4.3, we determine these Nil groups. One result that is interesting is its own right is Proposition 52 where we calculate the Bass Nil groups NK i pZrQ 8 sq for i " 0, 1 of the quaternion group Q 8 of order 8. Our calculations show that both WhpB 4 pS 2 qq and r K 0 pZrB 4 pS 2 qsq are infinitely-generated Abelian groups, and contain infinite direct sums of Abelian 2-groups. In contrast, we shall see that K ´1pZrB 4 pS 2 qsq -Z ' Z 2 . Compared with the families of groups considered in [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of certain reflection groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of hyperbolic 3-simplex reflection groups[END_REF], the existence of torsion here is once more a new phenomenon. We summarise these results as follows.

Theorem 1. The group B 4 pS 2 q has the following lower algebraic K-groups: WhpB 4 pS 2 qq -Z ' Nil 1 , r K 0 pZrB 4 pS 2 qsq -Z 2 ' Nil 0 , and K ´1pZrB 4 pS 2 qsq -Z 2 ' Z, K ´ipZrB 4 pS 2 qsq " 0 for all i ě 2, where for i " 0, 1, the groups Nil i are isomorphic to a countably-infinite direct sum of Z 2 , Z 4 or Z 2 ' Z 4 .

For n ě 5, we cannot expect the group B n pS 2 q to enjoy properties, such as hyperbolicity, similar to those of B 4 pS 2 q. Furthermore, we have not been able as yet to determine an appropriate model for the universal space for the family of virtually cyclic subgroups of B n pS 2 q. There are some candidates suggested by the theory of Brunnian braids, but the corresponding subgroups are of large index, and do not seem to be terribly useful from a practical viewpoint. On the positive side, for small odd values of n, the family of virtually cyclic subgroups of B n pS 2 q is relatively small, and our techniques enable us to determine the corresponding K-groups of these subgroups. If we are able to find an appropriate universal space, we hope to be able to determine the K-groups of B n pS 2 q for other values of n.

Chapter 2

Lower algebraic K-theory of the finite subgroups of B n pS 2 q

Classification of the virtually cyclic subgroups of B n pS 2 q

If G is a group that satisfies the Farrell-Jones fibred isomorphism conjecture, the lower algebraic K-theory of the group ring ZrGs may be calculated in principle if one knows the lower algebraic K-theory of the group rings of the virtually cyclic subgroups of G (see Appendix A). Recall that a group is said to be virtually cyclic if it possesses a cyclic subgroup of finite index. Clearly any finite group is virtually cyclic. By results of Epstein and Wall [START_REF] Epstein | Ends[END_REF][START_REF] Wall | Poincaré complexes I[END_REF], an infinite group is virtually cyclic if and only if it has two ends. This allows us to show that any infinite virtually cyclic group G is isomorphic either to F ¸Z or to G 1 ˚F G 2 , where F is a finite normal subgroup of G, and in the second case, F is of index 2 in both G 1 and G 2 . Consequently, in order to determine the virtually cyclic subgroups of G, one must first discover its finite subgroups. Let G " B n pS 2 q, and if m ě 2, let Dic 4m denote the dicyclic group of order 4m, with presentation: Dic 4m " @ x, y x m " y 2 , yxy ´1 " x ´1D .

(2.1)

If m is a power of 2, then we shall also say that Dic 4m is a generalised quaternion group, and denote it by Q 4m . Using a presentation of B n pS 2 q, such as that given in Theorem 34, if n ď 3, B n pS 2 q may be seen to be finite. The group B 1 pS 2 q is trivial, B 2 pS 2 q is isomorphic to Z 2 , and B 3 pS 2 q is isomorphic to Dic 12 , and its subgroups may be obtained easily. So in most of what follows, we shall assume that n ě 4, in which case B n pS 2 q is infinite. The finite subgroups of B n pS 2 q were classified up to isomorphism in [START_REF] Gonçalves | The classification and the conjugacy classes of the finite subgroups of the sphere braid groups[END_REF] as follows. More information on T ˚, O ˚and I ˚, to which we refer collectively as the binary polyhedral groups, may be found in [START_REF] Adem | Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Coxeter | Regular complex polytopes[END_REF][START_REF] Coxeter | Generators and relations for discrete groups[END_REF][START_REF] Wolf | Spaces of constant curvature[END_REF]. It is well known that the subgroups of dicyclic and binary polyhedral groups are cyclic, dicyclic or binary polyhedral (see [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Proposition 85] for the binary polyhedral case). We recall from [43, page 759] that the finite subgroups of B n pS 2 q are periodic of period 1, 2 or 4. Further, by [START_REF] Gonçalves | The classification and the conjugacy classes of the finite subgroups of the sphere braid groups[END_REF]Proposition 1.5], any two finite subgroups of B n pS 2 q that are isomorphic are also conjugate, with the exception of those subgroups that are isomorphic to Z 4 and Dic 4r if n is even and r divides n{2 or pn ´2q{2, in which case are two conjugacy classes in each isomorphism class. Consequently, any such subgroup H of B n pS 2 q satisfies the following three conditions (see [START_REF] Adem | Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften[END_REF] or [74, page 20]): (a) the p 2 -condition: for any prime divisor p of |H| (|H| denotes the order of H), H contains no subgroup isomorphic to Z p ˆZp . (b) the 2p-condition: for any prime divisor p of |H|, any subgroup of H of order 2p is cyclic. (c) the Milnor condition: if H has an element of order 2, this element is unique (and so is central in H).

Remarks 3.

(i) The p 2 -condition implies that the Sylow p-subgroups of H are either cyclic or generalised quaternion, the latter case occurring only if p " 2.

(ii) If G is a dicyclic or binary polyhedral group, the centre ZpGq is generated by the unique element of order 2.

The second step in the process is to classify the infinite virtually cyclic subgroups of B n pS 2 q. Up to isomorphism, and with a finite number of exceptions, this was achieved in [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]. The statement of the main result of [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF] is somewhat long to explain here, but to give a flavour of the results, we state the classification when n is odd, in which case the classification is complete for all values of n. We shall also recall the case n " 4 later in Theorem 39.

Theorem 4 [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Theorem 7]). Let n ě 3 be odd. Then up to isomorphism, the virtually cyclic subgroups of B n pS 2 q are as follows. (I) The isomorphism classes of the finite virtually cyclic subgroups of B n pS 2 q are: (i) Dic 4m , where m ě 3 divides n or n ´2. (ii) Z m , where m P N divides 2n, 2pn ´1q or 2pn ´2q. (II) If in addition n ě 5, then the following groups are the isomorphism classes of the infinite virtually cyclic subgroups of B n pS 2 q. (i) Z m ¸θ Z, where θp1q P tId, ´Idu, m is a strict divisor of 2pn ´iq, for i P t0, 2u, and m ‰ n ´i. (ii) Z m ˆZ, where m is a strict divisor of 2pn ´1q. (iii) Dic 4m ˆZ, where m ě 3 is a strict divisor of n ´i for i P t0, 2u. (iv) Z 4q ˚Z2q Z 4q , where q divides pn ´1q{2. (v) Dic 4q ˚Z2q Dic 4q , where q ě 2 is a strict divisor of n ´i, and i P t0, 2u.

The aim of the rest of this chapter is to compute the lower algebraic K-theory of the group rings of many of the finite groups of B n pS 2 q. In Section 2.2, we start by determining the number of different types of conjugacy classes in the binary polyhedral groups. The main result of that section, Proposition 7, will be used in the rest of the chapter to determine the lower algebraic K-theory of the group rings of T ˚, O ˚and I ˚. In Sections 2.3, 2.4 and 2.5, we calculate respectively the Whitehead, r K 0 -and K ´1-groups of the group rings of many groups that appear in the statement of Theorem 2. This allows us in Section 2.6 to determine the lower algebraic K-theory of the group rings of the isomorphism classes of the finite groups of B n pS 2 q for all 4 ď n ď 11, the results being summarised in Table 2.1.

Conjugacy classes of binary polyhedral groups

In this section, we compute the number of certain types of conjugacy classes of elements of the binary polyhedral groups. Some of these numbers will be used in the calculations of the lower algebraic K-theory of the group rings of these groups. Recall first that O ˚is generated by the elements X, P, Q and R, subject to the following relations [83, page 198]:

# X 3 " 1, P 2 " Q 2 " R 2 , PQP ´1 " Q ´1, XPX ´1 " Q, XQX ´1 " PQ RXR ´1 " X ´1, RPR ´1 " QP, RQR ´1 " Q ´1. (2.
2)

It follows that O ˚contains T ˚as an index 2 subgroup generated by X, P and Q that are subject to the relations given in the first line of (2.2). The subgroup xP, Qy is isomorphic to Q 8 , and X is of order 3 and acts by conjugation on xP, Qy by permuting P, Q and PQ cyclically, so that T ˚-Q 8 ¸Z3 . Further, O ˚z T ˚is comprised of 12 elements of order 4 and twelve of order 8. We recall also that |I ˚| " 120, that I ˚is comprised of the trivial element, one element of order 2, thirty elements of order 4, twenty elements of order 3 and twenty of order 6, twenty-four elements of order 5 and twenty-four elements of order 10. The group I ˚also contains subgroups isomorphic to T ˚. The following lemma will be useful in some of our computations.

Lemma 5.

Let G be a dicyclic or binary polyhedral group, and let g P G be an element of order greater than or equal to 3. Then the centraliser C G pgq of g in G is cyclic.

Proof. Let g P G be of order at least 3. Then g P ZpC G pgqq, so |ZpC G pgqq| ě 3. The subgroups of G are cyclic, dicyclic or binary polyhedral (see [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Proposition 85] for the binary polyhedral case). Then ZpC G pgqq is cyclic because the centre of a dicyclic or binary polyhedral group is isomorphic to Z 2 by Remarks 3(ii).

If G is a finite group and d is a divisor of |G|, let νpdq be the number of elements of order d in G, let r 0 pdq be the number of conjugacy classes of elements of order d in G, let r 1 pdq be the number of conjugacy classes of unordered pairs g, g ´1( of elements of order d in G, and let r 2 pdq be the number of conjugacy classes of cyclic subgroups of G of order d in G. If g, g 1 are elements of G of the same order d such that g, g ´1( is conjugate to g 1 , g 1´1 ( , then there exists h P G such that either hgh ´1 " g 1 or hgh ´1 " g 1´1 , so h xgy h ´1 " xg 1 y " @ g 1´1 D , and thus r 1 pdq ě r 2 pdq. It thus follows that: νpdq ě r 0 pdq ě r 1 pdq ě r 2 pdq.

(2.3)

For small d, the inequality r 1 pdq ě r 2 pdq is an equality.

Lemma 6. Let G be a finite group. Then r 1 pdq " r 2 pdq for all d P t1, 2, 3, 4, 6u.

Proof of Lemma 6. If d P t1, 2u and if g P G is of order d then the pair g, g ´1( reduces to tgu and then clearly r 1 pdq " r 2 pdq. So assume that d P t3, 4, 6u. From above, it suffices to show that r 1 pdq ď r 2 pdq. Note that if g P G is of order d then the elements of xgy of order d are precisely g and g ´1. If g 1 P G is also of order d and xgy and xg 1 y are conjugate then g is conjugate to g 1 or g 1´1 , so g, g ´1( is conjugate to g 1 , g 1´1 ( , which completes the proof of the lemma.

The following proposition summarises the values of νpdq, r 0 pdq, r 1 pdq and r 2 pdq for each of the three binary polyhedral groups. It will be used in the calculations of Whitehead and K ´1-groups in Proposition 10 and Proposition 28 respectively.

Proposition 7.

(a) If G " T ˚, νpdq, r 0 pdq, r 1 pdq and r 2 pdq are given by:

d 1 2 3 4 6 νpdq 1 1 8 6 8 r 0 pdq 1 1 2 1 2 r 1 pdq 1 1 1 1 1 r 2 pdq 1 1 1 1 1 If d P t3,
r 0 pdq 1 1 1 1 2 1 2 r 1 pdq 1 1 1 1 2 1 2 r 2 pdq 1 1 1 1 1 1 1
If d " 5 (resp. d " 10), and g P T ˚is of order d, g and g 2 (resp. g and g 3 ) are representatives of the two conjugacy classes of elements of order d.

Proof. Since the binary polyhedral groups have exactly one element of order 1 and of order 2, the elements of the columns for d P t1, 2u are all equal to 1. So we suppose that d ě 3.

(a) Let G " T ˚. We make use of the presentation of whose relations are given by the first line of (2.2). Let d " 3. The subgroups of T ˚of order 3 are the Sylow 3-subgroups of T ˚, so they are pairwise conjugate. Thus r 2 p3q " 1, and r 1 p3q " 1 by Lemma 6. We now compute r 0 p3q. Let g P T ˚be of order 3. Its centraliser C T ˚pXq contains g and the central element P 2 of T ˚of order 2, so C T ˚pgq contains the cyclic subgroup @ gP 2 D of order 6. Now T ˚contains no element of order greater than 6, and C T ˚pgq is cyclic by Lemma 5. It follows that C T ˚pgq " @ gP 2 D . By the orbit-stabiliser theorem, the conjugacy class of g contains 4 elements, and since T ˚possesses 8 elements of order 4, we deduce that r 0 p3q " 2. The fact that r 1 p3q " 1 implies that g and g ´1 belong to different conjugacy classes, so g and g ´1 are representatives of the two conjugacy classes of elements of order 3. By adjoining P 2 to g, we see that r i p6q " r i p3q for all i P t0, 1, 2u. Now let d " 4. The six elements of T ˚of order 4 are contained in the subgroup xP, Qy isomorphic to Q 8 . We have QPQ ´1 " P ´1, so P is conjugate to P ´1. Further, conjugation by X permutes P, Q and PQ, so T ˚contains a single conjugacy class of elements of order 4, r 0 p4q " 1, and thus r 1 p4q " r 2 p4q " 1 by (2.3). (b) Let G " O ˚. First let d " 3. All of the elements of O ˚of order 3 are contained in its subgroup T ˚, and so r 0 p3q ď 2. From (2.2), we have the relation RXR ´1 " X ´1, where X is of order 3. Since X and X ´1 are representatives of the two conjugacy classes of elements of order 3 in T ˚, it follows that there is a single conjugacy class of elements of order 3 in O ˚, so r 0 p3q " 1, and r 1 p3q " r 2 p3q " 1 by (2.3). Once more, the values for d " 6 are obtained by adjoining P 2 to X. Now let d " 4. From the case G " T ˚, the six elements of order 4 that belong to the subgroup T ˚of O ˚are pairwise conjugate, and since T ˚is normal in O ˚, they form a complete conjugacy class of elements of order 4. Now let U denote the set of twelve elements of order 4 that belong to O ˚z T ˚, and let g P U . Since C O ˚pgq is cyclic by Lemma 5 and contains xgy, it follows that |C O ˚pgq| P t4, 8u. Suppose that |C O ˚pgq| " 8. Then C O ˚pgq -Z 8 , and there exists h P C O ˚pgq of order 8 such that g " h 2 . But since T ˚is of index 2 in O ˚, it follows that h 2 P T ˚, which contradicts the fact that g R T ˚. So |C O ˚pgq| " 4, and C O ˚pgq " xgy. The orbit-stabiliser theorem implies that the conjugacy class of g contains twelve elements, which must be the elements of U . We thus conclude that there are two conjugacy classes in O ˚of elements of order 4, so r 0 p4q " 2. it also follows that if g 1 and g 2 are elements of O ˚of order 4 such that g 1 P T ˚and g 2 R T ˚then g 1 and g 2 are representatives of these two conjugacy classes. If g P O ˚is of order 4, then either it belongs to T ˚, and then g ´1 P T ˚, or it belongs to O ˚z T ˚, and then g ´1 P O ˚z T ˚. In both cases, it follows that g and g ´1 are conjugate in O ˚. Thus r 1 p4q " 2, and hence r 2 p4q " 2 by Lemma 6.

Finally, let d " 8, let g P O ˚be of order 8, and let H be a Sylow 2-subgroup that contains g. Then |H| " 16, and since O ˚has no element of order 16, it follows from Remark 3(i) that H -Q 16 . The group Q 16 contains a unique cyclic subgroup of order 8, and hence H is the only Sylow 2-subgroup that contains g. Since the Sylow 2-subgroups are pairwise conjugate, it follows that the three cyclic subgroups of order 8 are pairwise conjugate, and hence r 2 p8q " 1. The centraliser C O ˚pgq contains xgy, and is cyclic by Lemma 5, and the fact that O ˚has no element of order greater than 8 implies that C O ˚pgq " xgy. The orbitstabiliser theorem implies the conjugacy class of g contains 6 elements, and since O ˚contains 12 elements of order 8, we conclude that r 0 p8q " 2. Using the presentation (2.1) of Q 16 , we see that ygy ´1 " g ´1 for all y P Hz xgy, so g and g ´1 are conjugate, and it follows that r 1 p8q " 2 also. We also deduce that g and g 3 are representatives of the two conjugacy classes of elements of order 8. (c) Let G " I ˚. First suppose that d P t3, 5u, and let g P I ˚be an element of order d. Then C O ˚pgq contains xgy and the unique element ω of I ˚of order 2, and since C O ˚pgq is cyclic by Lemma 5, we see as in the previous cases that C O ˚pgq " xωgy, and |C O ˚pgq| " 2d. The orbit- stabiliser theorem then implies that the conjugacy class of g contains 60{d elements. If d " 3, this conjugacy class is the set of elements of order 3, so r 0 p3q " 1, and hence r 1 p3q " r 2 p3q " 1 by (2.3). If d " 5, the conjugacy class of g contains 12 elements, from which we conclude that there are two conjugacy classes C 1 and C 2 of elements of order 5, so r 0 p5q " 2. The subgroups of I ˚of order 5 are its Sylow 5-subgroups, which are pairwise conjugate, so r 2 p5q " 1. This also implies that each such subgroup contributes two elements to each of C 1 and C 2 . So if g P O ˚is of order 5, it is conjugate to exactly one element h of xgy z tgu. Note that h ‰ g 2 (resp. h ‰ g ´2) for otherwise g would be conjugate to g 2 (resp. g ´2), then g 2 would be conjugate to g ´1 (resp. to g), and the conjugacy class of g would contain at least three elements of xgy, which is not possible. Hence g is conjugate to g ´1 for every element g P I ˚of order 5, but is not conjugate to g 2 or to g ´2. This proves that the conjugacy class of g, g ´1( is equal to that of g, and so r 1 p5q " 2, and that g and g 2 are representatives of the two conjugacy classes of elements of order 5. Once more, if d P t3, 5u, then r i p2dq " r i pdq for all i P t0, 1, 2u.

It remains to study the case d " 4. Let g P I ˚be an element of order 4. Using once more the fact that C O ˚pgq contains xgy and is cyclic, we see that C O ˚pgq " xgy, and then that the conjugacy class of g contains thirty elements, which is the number of elements of I ˚of order 4. So r 0 p4q " 1, and r 1 p3q " r 2 p3q " 1 by (2.3).

Whitehead groups of the finite subgroups of B n pS 2 q

If G is a finite group, recall that its Whitehead group WhpGq is a finitely-generated Abelian group, and so may be written in the form:

WhpGq " Z r ' SK 1 pZrGsq, (2.4)
where SK 1 pZrGsq is isomorphic to the torsion subgroup of WhpGq [START_REF] Wall | Norms of units in group rings[END_REF]. The following proposition implies that to determine WhpGq, where G is a finite subgroup of B n pS 2 q, it suffices to compute r. Proposition 8. Let n P N, and let G be a finite subgroup of B n pS 2 q. Then SK 1 pZrGsq is trivial.

Proof. As we mentioned in Section 2.1, any finite subgroup G of B n pS 2 q is cyclic, dicyclic or binary polyhedral. If G is cyclic, dicyclic of order 8m, m P N, or binary polyhedral the result follows from [75, Theorem A, parts (1), (3), ( 5), ( 6) and ( 7 Consequently, if G is a finite subgroup of B n pS 2 q, then by equation (2.4), WhpGq is a free Abelian group, and it remains to calculate its rank. This is achieved in the following proposition.

Proposition 10. Let n P N, let G be a finite subgroup of B n pS 2 q, and if q P N, let δpqq denote the number of divisors of q. Then WhpGq -Z r , where:

r " $ ' ' ' ' ' ' & ' ' ' ' ' ' % X m 2 \ `1 ´δpmq if G -Z m , m P N m `1 ´δp2mq if G -Dic 4m , m ě 2 0 if G -T 1 if G -O 2 if G -I ˚.
Proof. Let G be isomorphic to a finite subgroup of B n pS 2 q. We recall once more that G is cyclic, dicyclic or binary polyhedral. Let r 1 denote the number of conjugacy classes of unordered pairs g, g

´1(
in G, where g P G, and let r 2 be the number of conjugacy classes of cyclic subgroups of G. By [59, page 39], the rank r of WhpGq is equal to r 1 ´r2 , and so:

r " ÿ d |G| `r1 pdq ´r2 pdq ˘. (2.5)
We treat the possibilities for G separately.

(a) G -Z m , where m P N. Since G is Abelian, r 1 is just the number of unordered pairs g, g

´1(
in G, where g runs over the elements of G, and r 2 is the number of cyclic subgroups of G. Since g " g ´1 if and only if |xgy| P t1, 2u, we have that r 1 " m´1 2 `1 if m is odd, and

r 1 " m´2 2 `2 if m is even. So r 1 " X m 2 \ `1.
Since the subgroups of Z m are in bijection with the divisors of m, we have r 2 " δpmq, so r " r 1 ´r2 " X m 2 \ `1 ´δpmq as required. (b) G -Dic 4m , where m ě 2. Let G " xxy š xxy y be given by equation (2.1). Since the elements of xxy y are of order 4, it follows from Lemma 6 and equation (2.5) that they do not contribute to r. So we just need to consider the contributions of the elements of xxy to r 1 and r 2 . Using equation (2.1), the conjugacy classes of the elements of xxy in G are x i , x ´i( , where 0 ď i ď m. Since xxy is of order 2m, as in the cyclic case, its elements contribute m `1 to the r 1 -term, and δp2mq to the r 2 -term, and thus r " m `1 ´δp2mq. (c) If G is binary polyhedral, the rank of WhpGq may be easily deduced using (2.5) and the tables of Proposition 7.

r

K 0 pZrGsq for the finite subgroups of B n pS 2 q

Let G be a finite group. The calculation of r K 0 pZrGsq is a difficult problem, even when the order of G is small. It is known that r K 0 pZrGsq is isomorphic to the ideal class group ClpZrGsq of ZrGs [START_REF] Curtis | with applications to finite groups and orders[END_REF]Section 49.11]. The following theorems summarise some results about r K 0 pZrGsq for certain finite groups.

Theorem 11 [START_REF] Curtis | with applications to finite groups and orders[END_REF]Corollary 50.17], [START_REF] Endô | Finite groups with trivial class groups[END_REF]). If G is Abelian then r K 0 pZrGsq is trivial if and only if G is either cyclic of order n, n P t1, 2, . . . , 11, 13, 14, 17, 19u, or is isomorphic to Z 2 ' Z 2 . If G is non Abelian and r K 0 pZrGsq " 1 then G is isomorphic to one of Dih 2q , q ě 3, A 4 , S 4 or A 5 , Dih 2q being the dihedral group of order 2q.

Theorem 12 ([73, Theorems III and IV, Corollary 10.12]).

(a) r K 0 pZrDic 4m sq - $ ' & ' % Z 2 if m P t2, 3, 4, 5, 7, 8, 11u Z 2 2 if m " 9 Z 3 2 if m P t6, 10u. (b) r K 0 pZrT ˚sq -Z 2 , r K 0 pZrO ˚sq -Z 2 2 and r K 0 pZrI ˚sq -Z 3 2 .
In Section 2.6, we will determine the lower algebraic K-theory of the finite subgroups of B n pS 2 q for all 4 ď n ď 11. With this in mind, we now compute r K 0 pZrGsq for some other finite cyclic groups. Before proving our results, we state the following result concerning the Bass cyclic units of the group ring ZrZ n s.

Theorem 13 ([7, p. 403]). Let G denote the cyclic group of order n. Let n, k P N, let g P Z n be an element of order n, and let m be a multiple of φpnq. Then k m " 1 mod n. Further, the Bass cyclic units are defined by: u k,m pgq " `1 `g `¨¨¨`g k´1 ˘m `1 ´km n p1 `g `¨¨¨`g n´1 q,

where k and n are relatively prime, and they generate all the units of infinite order in ZrZ n s.

Theorem 14. Let G " Z n , where n P t18, 20, 22u. Then r K 0 pZrGsq -Z 3 if n P t18, 22u, and

r K 0 pZrGsq -Z 5 2 if n " 20.
Proof. Let G " Z n , where n P t18, 20, 22u. We make use of an appropriate Cartesian square and the associated Mayer-Vietoris sequence as follows. We begin with the Rim square associated to Z 2 :

ZrZ 2 s ÝÝÝÑ Z § § đ § § đ Z ÝÝÝÑ F 2 .
(2.6) By (2.6), we obtain the following Cartesian square:

ZrGs ÝÝÝÑ ZrZ n{2 s § § đ § § đ ZrZ n{2 s ÝÝÝÑ F 2 rZ n{2 s.
(2.7) Proposition 8 and Theorem 11 imply that r K 0 pZrZ n{2 sq " 0 and SK 1 pZrZ n{2 sq " 0, hence the Mayer-Vietoris sequence associated with (2.7) becomes:

¨¨¨ÝÑ UpZrZ n{2 sq ' UpZrZ n{2 sq ÝÑ UpF 2 rZ n{2 sq ÝÑ r K 0 pZrGsq ÝÑ 0.
In the rest of this proof, UpRq will denote the group of Bass cyclic units of an integral group ring R. We therefore need to understand the following homomorphism:

UpZrZ n{2 sq ' UpZrZ n{2 sq ÝÑ UpF 2 rZ n{2 sq (2.8)
that is induced by reduction modulo 2. If n " 18 (resp. n " 22), the ring F 2 rZ n{2 s is semisimple, and is isomorphic to F 2 ' F 2 pξ 3 q ' F 2 pξq (resp. to F 2 ' F 2 pξq), where ξ is a primitive pn{2q th root of unity, and is a root of the polynomial x 6 `x3 `1 (resp. of x 10 `x9 `¨¨¨x 2 `x `1). Both of these polynomials are irreducible in F 2 rxs. Recall that F 2 pξq is a field with 64 (resp. 1024) elements [START_REF] Cox | Galois theory[END_REF], and its group of units is cyclic of order 63 (resp. of order 1023 " 31. 11. 3). Suppose that n " 18. As we mentioned, we are taking UpZrZ 9 sq to be generated by the Bass cyclic units that are of infinite order in ZrZ 9 s. These cyclic units are described by Theorem 13, and are u 2,6 , u 4,6 , u 5,6 , u 7,6 and u 8,6 . The image of u 7,6 in F 2 pξq is: p1 `ξ `¨¨¨`ξ 6 q 6 " pξ 7 `ξ8 q 6 " ξ 6 p1 `ξq 6 " ξ 6 p1 `ξ2 q 3 " ξ 6 p1 `ξ2 `ξ4 `ξ6 q " ξ 6 `ξ8 `ξ `ξ3 " 1 `ξ `ξ8 .

So the image of u 2 7,6 in F 2 pξq is 1 `ξ2 `ξ7 , the image of u 6 7,6 in F 2 pξq is the image of pu 2 7,6 q 2 u 2 7,6 in F 2 pξq, which is equal to:

p1 `ξ4 `ξ5 qp1 `ξ2 `ξ7 q " 1 `ξ2 `ξ7 `ξ4 `ξ6 `ξ2 `ξ5 `ξ7 `ξ3 " ξ 4 `ξ5 .

Thus the image of u 7 7,6 in F 2 pξq is the image of u 6 7,6 u 7,6 in F 2 pξq, which is equal to: pξ 4 `ξ5 qp1 `ξ `ξ8 q " ξ 4 `ξ5 `ξ3 `ξ5 `ξ6 `ξ4 " 1.

Hence the image of u 7,6 in F 2 pξq is of order 7, and the image of u 7,6 u 8,6 in F 2 pξq is of order 21. We now show that the three other cyclic units are each of order 21.

(i) u 2,6 : its image in F 2 pξq is p1 `ξq 6 " p1 `ξ2 q 3 " 1 `ξ2 `ξ4 `ξ6 . So the image of u 3 2,6 in F 2 pξq is equal to the image of u 2 2,6 u 2,6 in F 2 pξq, which in turn is equal to: p1 `ξ4 `ξ8 `ξ3 qp1 `ξ2 `ξ4 `ξ6 q " 1 `ξ2 `ξ4 `ξ6 `ξ4 `ξ6 `ξ8 `ξξ 8 `ξ `ξ3 `ξ5 `ξ3 `ξ5 `ξ7 `1 " ξ 2 `ξ7 .

Hence the image of u 6 2,6 in F 2 pξq is equal to the image of pu 3 2,6 q 2 in F 2 pξq, which is equal to ξ 4 `ξ5 , and the image of u 7 2,6 in F 2 pξq is equal to the image of pu 6 2,6 qu 2,6 in F 2 pξq, which in turn is equal to: p1 `ξ2 `ξ4 `ξ6 qpξ 4 `ξ5 q " ξ 4 `ξ6 `ξ8 `ξ `ξ5 `ξ7 `1 `ξ2 " ξ 3 , which is of order 3. It follows that the image of u 2,6 in F 2 pξq is of order 21.

(ii) The image of u 4,6 in F 2 pξq is: p1 `ξ `ξ2 `ξ3 q 6 " p1 `ξ2 `ξ4 `ξ6 q 3 " p1 `ξ2 `ξ4 `ξ6 q 2 p1 `ξ2 `ξ4 `ξ6 q " p1 `ξ4 `ξ8 `ξ3 qp1 `ξ2 `ξ4 `ξ6 q " pξ 2 `ξ3 `ξ4 q 2 " ξ 4 `ξ6 `ξ8 " ξ 4 p1 `ξ2 `ξ4 q.

Then the image of u 2 4,6 in F 2 pξq is ξ 8 p1 `ξ4 `ξ8 q " ξ 8 `ξ3 `ξ7 " ξ 3 p1 `ξ4 `ξ5 q, and the image of u 3 4,6 " u 2 4,6 u 4,6 in F 2 pξq is: ξ 7 p1 `ξ2 `ξ4 qp1 `ξ4 `ξ5 q " ξ 7 p1 `ξ4 `ξ5 `ξ2 `ξ6 `ξ7 `ξ4 `ξ8 `1q " ξ 7 pξ 5 `ξ2 `ξ6 `ξ7 `ξ8 q " ξ 7 pξ 6 `ξ7 q " ξ 4 p1 `ξq.

Thus the image of u 6 4,6 " pu 3 4,6 q 2 is ξp1 `ξ7 q, and the image of u 7 4,6 " u 6 4,6 u 4,6 is: ξ 5 p1 `ξ2 `ξ4 qp1 `ξ7 q " ξ 5 p1 `ξ2 `ξ4 `ξ7 `1 `ξ2 q " ξ 6 .

It follows that the image of u 4,6 in F 2 pξq is of order 21.

(iii) The image of u 5,6 in F 2 pξq is:

p1 `ξ `¨¨¨`ξ 4 q 6 " p1 `ξ2 `ξ4 `ξ6 `ξ8 q 3 " pξ 3 `ξ4 `ξ5 q 3 " p1 `ξ `ξ2 q 3 " p1 `ξ `ξ2 q 2 p1 `ξ `ξ2 q " p1 `ξ2 `ξ4 qp1 `ξ `ξ2 q " p1 `ξ `ξ2 `ξ2 `ξ3 `ξ4 `ξ4 `ξ5 `ξ6 q " ξp1 `ξ4 q.

So the image of u 3 5,6 in F 2 pξq is ξ 3 p1 `ξ4 `ξ8 `ξ3 q " ξ 3 `x7 `ξ2 `ξ6 " 1 `ξ2 `ξ7 , and the image of u 6 5,6 " pu 3 5,6 q 2 is 1 `ξ4 `ξ5 . Hence the image of u 7 5,6 " u 6 5,6 u 5,6 is:

p1 `ξ4 `ξ5 qξp1 `ξ4 q " ξp1 `ξ4 `ξ5 `ξ4 `ξ8 `1q " ξpξ 5 `ξ8 q " ξ 3 .

It follows that the image of u 5,6 in F 2 pξq is of order 21.

We conclude that the image in F 2 pξq of the subgroup generated by the cyclic units is of order 21. Hence the cokernel in (2.8) is of order 3, and from that equation we see that r K 0 pZrZ 18 sq -Z 3 , thus proving the result in the case n " 18. The proofs in the cases n " 20 and n " 22 are similar. First suppose that n " 22. As above, we see that the Bass cyclic units are of the form u k,10 for k " 2, 3, 4, . . . , 10, and making use of a Mathematica [67] routine written by José Hernandez (CCM-UNAM), to whom we are grateful, one may check that the image in F 2 pξq of the subgroup generated by the cyclic units is of order 341, and that the cokernel of (2.8) is of order 3, so once more, r K 0 pZrZ 22 sq -Z 3 . Finally, suppose that n " 20. In this case, the ring F 2 rZ 10 s is not semi-simple. It is isomorphic to F 2 rxs{px 5 ´1q 2 , and its group of units is isomorphic to the direct product UpF 2 rZ 10 sq " pZ 2 q 5 ˆZ3 ˆZ5 . Applying the Mathematica routine once more to the cyclic units u 3,4 , u 7,4 and u 9,4 , we see that the image of the group generated by these units in (2.8) is of order 15, and hence the cokernel of (2.8) is isomorphic to Z 5 2 . This completes the proof of the theorem.

K ´1pZrGsq for the finite subgroups of B n pS 2 q

Let G be a finite subgroup of B n pS 2 q. In order to determine K ´1pZrGsq, we shall use the following special case of a result of Carter. Similar results have recently been obtained independently by B. Magurn in [START_REF] Magurn | Negative K-theory of generalized quaternion groups and binary polyhedral groups[END_REF] for generalised quaternion and binary polyhedral groups.

First we recall that a simple Artinian ring A is isomorphic to M n pDq for some positive integer n and some skew field D. Further, D is finite dimensional over its centre E, the dimension being a square rD : Es, and the Schur index of A is equal to a rD : Es [22, Section 27].

Theorem 15 ([15, Theorem 1]

). Let G be a finite group of order q. Then K ´1pZrGsq -Z r ' Z s 2 , (2.9)

where r is given by r " 1 ´rQ `ÿ p |G| `rQ p ´rF p ˘, (2.10)

r Q (resp. r Q p , r F p ) denotes the number of isomorphism classes of irreducible Q-(resp. Q p -, F p -)
representations of G, and s is equal to the number of simple components of QrGs that have even Schur index m but have odd local Schur indices m Q at every finite prime Q of the centre which divides q.

So to calculate K ´1pZrGsq, we must determine the quantities r F for the various fields appearing in equation (2.10), as well as the number s. For the finite subgroups G of B n pS 2 q, we divide this calculation into two parts. In Section 2.5.1, we determine r, which yields the torsion of K ´1pZrGsq. In Section 2.5.2, we compute s, which is the rank of K ´1pZrGsq. We then obtain K ´1pZrGsq from (2.9).

Torsion of K ´1pZrGsq for finite subgroups of B n pS 2 q

Let G be a finite subgroup of B n pS 2 q, and let s be as defined in equation (2.9). As remarked in [15, page 1928], a consequence of Theorem 15 is that K ´1pZrGsq is torsion free if G is Abelian. In particular, if G is cyclic, then s " 0. If G is non cyclic, then as we shall see, K ´1pZrGsq may have torsion. Although equation (2.9) clearly allows for this possibility, this appears to be a new phenomenon, and contrasts with the calculations given in [START_REF] Lafont | Lower algebraic K-theory of certain reflection groups[END_REF][START_REF] Lafont | Lower algebraic K-theory of hyperbolic 3-simplex reflection groups[END_REF] for example. We thus require new techniques to calculate the torsion of K ´1pZrGsq. If G is dicyclic, we make use of results due to Yamada concerning the computation of the (local) Schur indices of the simple components of QrGs [START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF]. If G is binary polyhedral, then one may apply induction/restriction techniques and the Mackey formula.

Assume first that G -Dic 4m is dicyclic, where m ě 3 is odd. If m is an odd prime then we determine K ´1pZrDic 4m sq. In principle, our method should apply to any odd value of m, not just for prime values. If m is odd, the Wedderburn decomposition over Q of the algebra QrDic 4m s is given in [START_REF] Curtis | I, with applications to finite groups and orders[END_REF]Example 7.40]:

QrDic 4m s -QrDih 2m s ' Qpiq ' ˜à d 0 |m, d 0 ą1 H 2d 0 - Q 2 ' ˜à d|m, dą2 M 2 ´Q ´ζd `ζ´1 d ¯¯¸' Qpiq ' ˜à d 0 |m, d 0 ą1 H 2d 0 ¸, (2.11)
where ζ d is a primitive d th root of unity, and

H d " E d ' E d i ' E d j ' E d k (2.12)
is the quaternion skew field with centre

E d " Q ´ζd `ζ´1 d ¯. In particular, if m " µ is prime then QrDic 4µ s -Q 2 ' M 2 `Eµ ˘' Qpiq ' H 2µ . (2.13)
Note that the number of components in equation (2.13) is equal to the number of conjugacy classes of cyclic subgroups of Dic 4µ , and that the components are in one-to-one correspondence with the irreducible Q-representations of Dic 4µ . The first four components of equation (2.13) are matrix rings over fields, and so their Schur index is equal to one. By equation (2.9), the torsion of K ´1pZrDic 4µ sq is then either trivial or equal to Z 2 depending on the Schur and local Schur indices of the remaining component H 2µ . We now determine precisely this torsion using results of Yamada [START_REF] Yamada | On the group algebras of metacyclic groups over algebraic number fields[END_REF][START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF].

Proposition 16. If µ be an odd prime, the torsion of K ´1pZrDic 4µ sq is trivial if µ " 3 mod 4, and is equal to Z 2 if µ " 1 mod 4.

Proof. We apply the results of [START_REF] Yamada | On the group algebras of metacyclic groups over algebraic number fields[END_REF][START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF], and refer the reader to these papers for the notation used in this proof. If n P N and w P Z is coprime with n, then w mod ˆn will denote w as an element of the multiplicative group of integers modulo n. With the notation of [85, Proposition 4], we have m " 2µ, r " 2µ ´1, s " 2, h " µ and u is the order of 2µ ´1 mod ˆ2µ, so u " s " 2. 2µ gcd p2µ,1q " 2µ and v 1 " 2µ gcd p2µ,µq " 2. Let p be a finite prime of the centre E µ of H 2µ that divides 4µ. Then p divides 2µ, and since µ is odd, p divides p, where p P t2, µu. We distinguish these two possibilities, the notation being that of [START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF]Proposition 9]. (a) Suppose that p divides 2. Then we have p " 2, b " z " 1, a " 1 and t 1 is the order of 2µ ´1 mod ˆµ, so t 1 " 2. Thus e p " 1, and hence

c p " Λ p " 1. (b) Suppose that p divides µ. Then p " µ, b " 0, z " 2, a " 1, @ 2µ ´1 mod ˆ2D " @ µ mod ˆ2D " t1u, f " r f " t 1 " 1, q " µ, e p " 2, c p " gcd p2, µ ´1q " 2 and Λ p " 2 gcd ´2, µ´1 2 ¯" # 1 if µ " 1 mod 4 2 if µ " 3 mod 4 by [85, Proposition 9(II)].
Thus if µ " 1 mod 4, the simple component H 2µ of QrGs whose Schur index is equal to two satisfies the property that its local Schur indices at every finite prime of the centre are odd. Hence the integer s of equation (2.9) is equal to one, so the torsion of K ´1pZrGsq is Z 2 . If µ " 3 mod 4 then Λ p " 2 for any finite prime p that divides µ, so s " 0, and hence K ´1pZrGsq is torsion free.

As another example, we calculate the torsion of K ´1pZrDic 4m sq in the case where m is a power of 2 (so Dic 4m is a generalised quaternion group).

Proposition 17. The torsion of K

´1pZrQ 2 k sq is trivial if k " 3, and is equal to Z 2 if k ě 4. Proof. Let k ě 3. Then Dic 2 k " Q 2 k . From [22, Example 7.40, case 1], QrQ 2 k s -QrDih 2 k´1 s ' H 2 k´1 , (2.14)
where H 2 k´1 is the quaternion skew field defined by equation (2.12). Using [START_REF] Curtis | I, with applications to finite groups and orders[END_REF]Example 7.39], each simple component of QrDih 2 k´2 s is a matrix ring over a field, and so its Schur index is equal to one. As in the proof of Proposition 16, one may show that the Schur index of the remaining simple component H 2 k´1 of equation (2.14) is equal to two, and that this component corresponds to the irreducible representation U p2q 1,0 . To study the local Schur index Λ p of each finite prime p dividing the centre E 2 k´1 of the simple component H 2 k´1 , we again apply [84, Proposition 9]. With the same notation, we have m " 2 k´1 , r " 2 k´1 ´1, u " s " 2, h " 2 k´2 , d 1 " 2 k´1 and v 1 " 2. If p does not divide 2 k´1 then Λ p " 1 by [84, Proposition 9(I)]. So suppose that p divides 2 k´1 . With the notation of [84, Proposition 9(II)], b " z " 1 and p " 2. If k " 3 then we are in the exceptional case of [84, Proposition 9(II)], so Λ p " 2. Thus there exists a finite prime of the centre E 2 k´1 of H 2 k´1 dividing 2 k with even local Schur index, and it follows from Theorem 15 that the torsion of K ´1pZrQ 8 sq is trivial. Assume then that k ě 4. So f " r f " t 1 " 1 and e p " q " 2, thus c p " Λ p " 1. Then the simple component H 2 k´1 of QrQ 2 k s whose Schur index is equal to two satisfies the property that its local Schur indices at every finite prime of the centre dividing 2 k are odd. Hence the integer s of equation (2.9) is equal to one, and thus the torsion of K ´1pZrQ 2 k sq is equal to Z 2 as required. Now let G be a binary polyhedral group. We recall that a group is said to be 2-hyperelementary if it is a semi-direct product of a cyclic normal subgroup of odd order and a 2group. Since G is not itself 2-hyper-elementary, induction/restriction techniques may be used to calculate the torsion of K ´1pZrGsq.

Proposition 18. The torsion of K

´1pZrGsq is trivial if G -T ˚, and is equal to Z 2 if G -O ˚or G -I ˚.
Proof. Let G be a binary polyhedral group. Applying [15, Theorem 3(iii) and page 1936], we have the composition

' H K ´1pZrHsq ind ÝÑ K ´1pZrGsq res ÝÑ ' H K ´1pZrHsq, (2.15)
where ind and res are the usual induction and restriction maps that are surjective and injective respectively when restricted to the corresponding torsion subgroups, and H runs over the conjugacy classes of the

(resp. L " Dic 20 ) if G " O ˚(resp. G " I ˚), the torsion of K ´1pZrLsq is Z 2 .
The injectivity of res in equation (2.15) implies that the torsion of K ´1pZrT ˚sq is trivial, which gives the result in this case. So let G " O ˚or I ˚, and let L be as defined above. Now G possesses a single conjugacy class of subgroups isomorphic to L [73, Lemma 14.3], and since L is the only subgroup of G for which the torsion of K ´1pZrLsq is non trivial, we need only to consider the restriction of equation (2.15) to the factor H " L:

K ´1pZrLsq ind ÝÑ K ´1pZrGsq res ÝÑ K ´1pZrLsq.
(2.16)

It thus suffices to show that the restriction of (2.16) to the corresponding torsion subgroups is the identity. Now K ´1p¨q is a Mackey functor [70, Theorem 11.2], so we may apply Mackey's formula that describes the composition (2.16) as the sum of the maps:

K ´1pZrLsq res ÝÑ K ´1pZrx ´1 i Lx i X Lsq c x i ÝÑ K ´1pZrLsq, (2.17)
where G " š Lx i L is a double coset decomposition of G, and the map

c x i is induced by the homomorphism x ´1 i Lx i X L ÝÑ L defined by y Þ ÝÑ x i yx ´1 i [70, Section 11a]. Let N G pLq denote the normaliser of L in G. If x i R N G pLq then the torsion of K ´1pZrx ´1
i Lx i X Lsq is trivial, and the corresponding map (2.17) contributes zero to the restriction of (2.16) to the torsion subgroups. If on the other hand, x i P N G pLq, the corresponding map (2.17) is an isomorphism. Now L Ă N G pLq Ă G, and since L is not normal in G and G has no proper subgroup that strictly contains L [47, Proposition 85], it follows that N G pLq " L. So there is only one double coset representative x i that belongs to N G pLq, and for this x i , it follows that the restriction of (2.16) to the torsion subgroups is equal to the restriction of the isomorphism (2.17) to the torsion subgroups. Since the torsion of K ´1pZrLsq is Z 2 , the same conclusion holds for K ´1pZrGsq. , ψ G pZrGsq is a maximal order Γ G that is completely described in [73, page 79], from which one may prove that Impψ G q is equal to H d , where d " 8 (resp. d " 5) if G " O ˚(resp. G " I ˚), in other words, H d appears as a factor in the Wedderburn decomposition of QrGs. On the other hand, from equation (2.14) (resp. equation (2.13)), we know that H d also appears in the Wedderburn decomposition of QrQ 16 s (resp. QrDic 20 s), and from the proof of Proposition 17 (resp. Proposition 16), that it contributes a Z 2 -term to the torsion of K ´1pZrQ 16 sq (resp. K ´1pZrDic 20 sq). It follows then from [START_REF] Carter | Lower K-theory of finite groups[END_REF]Theorem 1] that K ´1pZrGsq has non-trivial torsion.

(c) Using the GAP package Wedderga [START_REF] Broche Cristo | Wedderga -Wedderburn Decomposition of Group Algebras[END_REF], one may obtain the complete Wedderburn decomposition for the binary polyhedral groups:

QrT ˚s -Q ' Qpζ 3 q ' M 3 pQq ' H 4 ' HpQpζ 3 qq (2.18) QrO ˚s -Q 2 ' M 2 pQq
QrI ˚s -Q ' M 4 pQq ' H 5 ' M 2 p p Hq ' M 5 pQq ' M 3 pHpQqq 'M 3 pQp ? 5qq, (2.20)
where p H is the quaternion algebra p´1, ´3q{Q. This algebra admits a basis t1, i, j, ku as a Q-vector space, and the algebra multiplication satisfies ij " ´ji " k, i 2 " ´1 and j 2 " ´3. Somewhat surprisingly, we were not able to find the decompositions (2.18)-(2.20) in the literature.

In order to prove Theorem 31 and to obtain Table 2.1 given on page 34, we will need to calculate K ´1pZrGsq for some other dicyclic groups, namely G " Dic 4µ , where µ P t6, 9, 10u. We now compute the torsion of K ´1pZrDic 4µ sq for µ " 9, as well as the case where µ " 2τ, where τ is an odd prime, which includes the cases µ " 6 and µ " 10.

Proposition 20.

(a) If µ " 2τ, where τ is an odd prime, then the torsion of K ´1pZrDic 4µ sq is Z 2 . (b) The group K ´1pZrDic 36 sq is torsion free.

Proof. (a) Let µ " 2τ, where τ is an odd prime. From [START_REF] Curtis | I, with applications to finite groups and orders[END_REF]Example 7.40] or [73, pp. 75-76], and using (2.11) and the notation of Section 2.5.1, we have:

QrDic 4µ s -QrDic 8τ s -Q 4 ' ˜à dPtτ,2τu M 2 ´Q ´ζd `ζ´1 d ¯¯¸' H 4 ' H 4τ . (2.21)
The first three factors of equation (2.21) are matrix rings over fields, so their Schur index is equal to one. Further, the factor H 4 appears in the Wedderburn decomposition of the Qalgebra QrT ˚s, and since K ´1pZrT ˚sq is torsion free by Proposition 18, H 4 does not contribute to the torsion of K ´1pZrDic 4µ sq. It remains to determine the Schur and local Schur indices of the remaining factor H 4τ . Once more, we follow the proof of Proposition 16, and we use the results and notation of [START_REF] Yamada | On the group algebras of metacyclic groups over algebraic number fields[END_REF][START_REF] Yamada | On the group algebras of metabelian groups over algebraic number fields I[END_REF], taking m " 4τ, r " 4τ ´1, h " 2τ, and u " s " 2. Using [85, Proposition 5], the representation U p2q 1,0 gives rise to an irreducible representation of QrDic 4µ s, and the last part of [85, Example 3, Section 6] implies that its Schur index is equal to two. Since the Schur index of each of the first four components of equation (2.21) is equal to one, it follows that the simple component H 4τ of QrDic 4µ s corresponds to U p2q 1,0 . With the notation of [85, Proposition 9], we have d 1 " 4τ and v 1 " 2. Let p be a finite prime of the centre of H 4τ that divides 4τ. Then p divides 2 or τ. If p 2, then p " 2. We are not in the exceptional case of [85, Proposition 9(II)] since the order of r in Z τ is equal to 2. Further, a " 2, and t 1 is equal to the order of 4τ ´1 in Z τ, so t 1 " 2. If p τ, then p " τ, a " 1, and t 1 is equal to the order of 4τ ´1 in Z 4 , so t 1 " 2 also. So in both cases e p " s{t 1 " 1, hence c p " Λ p " 1. Thus the simple component H 4τ of QrDic 4µ s whose Schur index is equal to two satisfies the property that its local Schur indices at every finite prime of the centre are odd. Hence the integer s of equation (2.9) is equal to one, and therefore the torsion of K ´1pZrDic 4µ sq is Z 2 . (b) By (2.11), the Wedderburn decomposition of the Q-algebra QrDic 36 s is given by:

QrDic 36 s -Q 2 ' M 2 pE 3 q ' M 2 pE 9 q ' Qpiq ' H 6 ' H 18 .
(2.22)

The first four factors of equation (2.22) are matrix rings over fields, so their Schur index is equal to one. Further, the factor H 6 also appears in the Wedderburn decomposition of the Q-algebra QrDic

The rank of K ´1pZrGsq for the finite subgroups of B n pS 2 q

Let G be a finite subgroup of B n pS 2 q. To calculate the rank of K ´1pZrGsq, we shall apply equation (2.10). In each case, we will thus need to calculate the number r F of distinct irreducible FrGs-modules, where F is equal respectively to Q, Q p and F p . Before doing so, we recall the requisite theory (see [22, pages 492 and 508] or [70, pages 25-26]).

Let F be a field of characteristic p ě 0, where p is prime if p ą 0. If G is a finite group of exponent m, let:

p m " # m if p " 0 m{p a if p ą 0,
where a is the largest power of p that divides m.

Let Fpζ p m q be a field extension of F by a primitive p m th root of unity, which we denote by ζ p m . Then Fpζ p m q is a Galois extension of F, whose Galois group, denoted by GalpFpζ p m q{Fq, is given by: GalpFpζ p m q{Fq " tφ : Fpζ p m q ÝÑ Fpζ p m q | φ is an automorphism and φpzq " z for all z P F u. Each automorphism σ P GalpFpζ p m q{Fq is uniquely determined by its action on ζ p m , and is given by σpζ p m q " ζ t p m , where t is an integer that is uniquely defined modulo p m. Hence t corresponds to an element of the multiplicative group of units Z p m , and there is an injective group homomorphism:

# φ : GalpFpζ p m q{Fq ÝÑ Z p m σ Þ ÝÑ t, (2.23)
defined by φpσq " t. We now recall the definition of F-conjugacy class.

(a) If f , g are elements of G, we say that they are F-conjugate if there exists t P Impφq and α P G such that f t " αgα ´1. The F-conjugacy relation is an equivalence relation on G, and the F-equivalence class of f in G will be denoted by r f s F . (b) Let G 1 p " tg P G | gcd pp, opgqq " 1u , be the set of p-regular elements of G, where opgq denotes the order of g P G. An F-conjugacy class of G is said to be p-regular if it is contained in G 1 p . If f P G then we denote its usual conjugacy class by r f s. By the Witt-Berman Theorem, we have the following result that will be used to compute r F for our groups. We also need the following results concerning the structure of the Galois groups.

Theorem 23 ([72]). Suppose that n is odd or divisible by 4. Then Q p pζ n q{Q p is a Galois extension of Q p , and its Galois group G is as follows. (a) If p does not divide n then G is cyclic, and there exists an element σ P G, the Frobenius element of the extension satisfying σpζ n q " ζ p n that generates G. Further, the order of σ is the order of p considered as an element of Z n. (b) If n " p m , m ě 1, then G is of order p m´1 pp ´1q, and we have a group isomorphism G -Z pm . Hence G is cyclic if p is odd or if p " m " 2, and is isomorphic to the direct product Z 2 (generated by the class of ´1) and Z p m´2 (generated by the class of 5) if p " 2 and m ě 3. (c) Suppose that n " p m n 1 , where n 1 ě 2 and p does not divide n 1 . Let ζ 1 be a primitive n 1 th root of unity, and let ρ be a primitive p mth root of unity. Then G -GalpQ p pζ 1 q{Q p q ˆGalpQ p pρq{Q p q.

Theorem 24 ([19]). Let k denote the order of p considered as an element of Z n. Then the group GalpF p pζ n q{F p q is isomorphic to Z k .

We suppose in what follows that G is dicyclic of order 4m. We first apply the above results in order to determine the rank of K ´1pZrGsq where m is an odd prime. We then go on to to study the case where m is a power of 2.

Theorem 25. Let m be an odd prime, and let λ be the number of Q 2 -conjugacy classes (or equivalently F 2 -conjugacy classes) of the elements of Dic 4m of order m. Then

K ´1pZrDic 4m sq - # Z λ ' Z 2 if m " 1 mod 4 Z λ if m " 3 mod 4.
Proof. Let G " Dic 4m be given by the presentation (2.1). By Proposition 16 and equation (2.9), it suffices to show that the rank of K ´1pZrDic 4m sq is equal to λ. The group G has one element each (e and x m respectively) of order 1 and 2, pm ´1q elements of order 2m, of the form x i , i odd, 1 ď i ď 2m ´1, and i ‰ m, pm ´1q elements of order m, of the form x i , i even, 2 ď i ď 2m ´2, and 2m elements of order 4, of the form y, xy, . . . , x 2m´1 y. The elements of order 1 and 2 each form a single (usual) conjugacy class, those of order 4 form 2 conjugacy classes,

x i y | 0 ď i ď 2m ´2, i even ( and x i y | 1 ď i ď 2m ´1, i odd (
, while those of order m and 2m form pm ´1q conjugacy classes of the form x i , x ´i( for i " 1, . . . , m ´1. Since r Q is equal to the number of simple components in the Wedderburn decomposition of QrDic 4m s, it follows from equation (2.13) that r Q " 5. This may also be obtained by observing that the subgroups of Dic 4m of order 4 are its Sylow 2-subgroups, and so Dic 4m possesses a single conjugacy class of subgroups of order 4.

We must thus calculate r Q p and r F p for p P t2, mu, which we do using Theorem 22. Since there is a unique conjugacy class of elements of order 1 and 2, these elements contribute 1 to each of r Q p and r F p , except in the case of r F 2 , where the element of order 2 is not 2-regular, so contributes zero. We thus focus on the elements of order 4, m and 2m. According to [70, page 26], it suffices to analyse the F-conjugacy classes of the elements of order 4, m and 2m adjoining an n th root of unity to F for n " 4, m, 2m, where F " Q 2 or Q m .

• Q 2 -conjugacy classes of the order 4 elements: by Theorem 23(b), the monomorphism φ : GalpQ 2 pζ 4 q{Q 2 q ÝÑ Z 4 is an isomorphism and Impφq " t1, 3u. By equation (2.24), rys Q 2 " rys Y ry 3 s " rys Y rx m ys as y 3 " y 2 . y " x m y, and so there is a single Q 2 -class of order 4 elements because m is odd.

• Q 2 -conjugacy classes of the elements of order m and 2m: by hypothesis, the number of Q 2 -conjugacy classes of the elements of order m is equal to λ. Theorem 23(c) implies that the number of Q 2 -conjugacy classes of the elements of order 2m is also equal to λ.

We conclude that r Q 2 " 2λ `3.

• Q m -conjugacy classes of the order 4 elements: by Theorem 23(a), we have a monomorphism φ : GalpQ m pζ 4 q{Q m q ÝÑ Z 4 , GalpQ m pζ 4 q{Q m q is cyclic, and its order is equal to that of m considered as an element of Z 4 . If m " 3 mod 4 then φ is an isomorphism and Impφq " t1, 3u. By equation (2.24), rys Q m " rys Y ry 3 s " rys Y rx m ys as y 3 " y 2 . y " x m y, and so there is a single Q m -class of order 4 elements since m is odd. If m " 1 mod 4 then GalpQ m pζ 4 q{Q m q is trivial and Impφq " t1u. In this case, the Q m -conjugacy classes coincide with the usual conjugacy classes, so there are two Q m -conjugacy classes of elements of order 4.

• Q m -conjugacy classes of the elements of order m: by Theorem 23(b), the monomorphism φ : GalpQ m pζ m q{Q m q ÝÑ Z m is an isomorphism and Impφq " t1, . . . , m ´1u. By equa- tion (2.24), rx 2 s Q m " m´1 ď i"1 rx 2i s, so there is a single Q m -class of order m elements.

• Q m -conjugacy classes of the elements of order 2m: as m is an odd prime, we have that Q m pζ 2m q " Q m pζ m q, so φ : GalpQ m pζ m q{Q m q ÝÑ Z m -Z m´1 is an isomorphism, and we conclude that there is a single Q m -class of order 2m elements.

It thus follows that r Q m " 6 if m " 1 mod 4, and r Q m " 5 if m " 3 mod 4.

• 2-regular F 2 -conjugacy classes: we have G 1 2 " e, x 2 , x 4 , . . . , x 2m´2 ( , which splits as the disjoint union of pm `1q{2 (usual) conjugacy classes in Dic 4m , comprised of teu, and x 2i , x 2pm´iq ( for i " 1, . . . , pm ´1q{2. We thus need to study the F 2 -conjugacy classes of the elements of order m. By Theorem 24, we have φ : GalpF 2 pζ m q{F 2 q ÝÑ Z m, where GalpF 2 pζ m q{F 2 q is cyclic, of order that of 2 considered as an element of Z m, and Impφq " x2y.

We return for a moment to the Q 2 -conjugacy classes of the elements of order m. Replacing φ by φ 1 to distinguish it from the monomorphism φ of the previous paragraph, by Theorem 23(a), we have φ 1 : GalpQ 2 pζ m q{Q 2 q ÝÑ Z m, and GalpQ 2 pζ m q{Q 2 q is cyclic, of order that of 2 considered as an element of Z m. Thus Impφ 1 q " x2y also. In particular, the F 2 -conjugacy class of an element of Dic 4m of order m is equal to its Q 2 -conjugacy class, and thus the number of F 2 -conjugacy classes of elements of order m is equal to λ. We deduce that r F 2 " λ `1.

• m-regular F m -conjugacy classes: we have

G 1
m " e, x m , y, xy, x 2 y, . . . , x 2m´2 y, x 2m´1 y ( .

The four (usual) conjugacy classes in Dic 4m are:

teu, tx m u, y, x 2 y, . . . , x 2m´2 y ( and xy, x 3 y, . . . , x 2m´1 y ( .

It is thus necessary to study the F m -conjugacy classes of the latter two classes, which are those of the elements of Dic 4m of order 4. By Theorem 24, we have the monomorphism φ : GalpF m pζ 4 q{F m q ÝÑ Z 4 , and GalpF m pζ 4 q{F m q is cyclic, of order that of m considered as an element of Z 4 . As in the case of the Q m -conjugacy classes of the order 4 elements, if m " 3 mod 4, there is a single F m -class of order 4 elements, while if m " 1 mod 4, the F m -conjugacy classes coincide with the usual conjugacy classes, and so there are two F mconjugacy classes of order 4 elements. Hence r F m " 4 if m " 1 mod 4 and r F m " 3 if m " 3 mod 4.

So by equation (2.10), the rank r of K ´1pZrDic 4m sq is given by:

r " 1 ´rQ `pr Q 2 ´rF 2 q `pr Q m ´rF m q " # 1 ´5 `p2λ `3q ´pλ `1q `p6 ´4q if m " 1 mod 4 1 ´5 `p2λ `3q ´pλ `1q `p5 ´3q if m " 3 mod 4 " λ.
If m is an odd prime, the proof of Theorem 25 indicates that the number λ of Q 2 -conjugacy classes of the elements of Dic 4m of order m is related to the order of the subgroup x2y in Z m. The question of when 2 generates Z m is open and constitutes a special case of Artin's primitive root conjecture. The following proposition shows that it is also interesting for us to know whether ´1 belongs to x2y, and enables us to determine the rank of K ´1pZrDic 4m sq solely in terms of |x2y|.

L 2 |x2y| if ´1 R x2y.
Examples.

(a) Suppose that m is a Fermat number, of the form 2 2 s `1, where s P N. Then |x2y| " 2 s`1 and ´1 P x2y, so the rank of K ´1pZrDic 4m sq is equal to λ " 2 2 s ´s´1 . For example, if m " 257 then λ " 16 and K ´1pZrDic 1 028 sq -Z 2 ' Z 16 . (b) Suppose that m is a Mersenne prime, of the form 2 p ´1, where p is prime. Then |x2y| " p and ´1 R x2y, so the rank of K ´1pZrDic 4m sq is equal to λ " 2 p ´2 2p " 2 p´1 ´1 p

. For example, if m " 127 then λ " 9 and K ´1pZrDic 508 sq -Z 9 , and if m " 8 191 then λ " 315 and K ´1pZrDic 32 728 sq -Z 315 .

Proof of Proposition 26. Using equation (2.1), the elements of Dic 4m of order m are of the form x 2i , 1 ď i ď m ´1, and rx 2i s " x 2i , x ´2i ( , in particular, they form pm ´1q{2 distinct (usual) conjugacy classes in Dic 4m . Let 1 ď i ď m ´1. Since m is prime, there exist τ, µ P Z such that τi `µm " 1. One may check easily that the maps

rx 2 s Q 2 ÝÑ rx 2i s Q 2 and rx 2i s Q 2 ÝÑ rx 2 s Q 2 ,
defined respectively by w Þ ÝÑ w i and z Þ ÝÑ z τ , are mutual inverses, and hence rx 2i s Q 2 has the same number of elements as rx 2 s Q 2 . Thus the number of Q 2 -conjugacy classes of the elements of order m, which is equal to λ, is just pm ´1q divided by the cardinal of rx 2 s Q 2 . Theorem 23(a) and equation (2.24) imply that:

rx 2 s Q 2 " ď tPx2y rx 2t s " x 2i ˇˇi P x2y ( Y x ´2i ˇˇi P x2y ( " x 2i ˇˇi P x2y ( Y x 2i ˇˇi P ´x2y
( . Now ´x2y is the x2y-coset of ´1 in Z m, so x 2i i P x2y ( and x 2i i P ´x2y ( have the same cardinality |x2y|, and are either equal or disjoint. Since ´1 P ´x2y, they are equal if and only if ´1 P x2y. This being the case, the cardinality of rx 2 s Q 2 is equal to |x2y|, and λ " pm ´1q{ |x2y|. If ´1 R x2y, the two cosets x2y and ´x2y are disjoint, thus the cardinality of rx 2 s Q 2 is equal to 2 |x2y|, and λ " pm ´1q{2 |x2y| as required.

The methods used above allow us in theory to calculate K ´1pZrDic 4m sq for any m ě 2, not just for m an odd prime. As another example, consider the case where m is a power of 2, so G -Q 2 k is the generalised quaternion group of order 2 k , where m " 2 k´2 . Proposition 27. K ´1pZrQ 2 k sq is trivial if k " 3, and is isomorphic to Z 2 if k ě 4.

Proof. By Theorem 15 and Proposition 17, it suffices to show that for all k ě 3, the rank of K ´1pZrQ 2 k sq is zero, which we do using Theorem 22. We must calculate r Q , r Q 2 and r F 2 . Using the presentation (2.1) of Q 2 k , we see that Q 2 k " xxy š xxy y, and that the elements of xxy y are all of order 4. So G 1 2 consists of the identity element, whence r F 2 " 1. We now determine the number r Q of Q-conjugacy classes, which by Remark 21(b), is equal to the number of conjugacy classes of cyclic subgroups in Q 2 k . The elements of Q 2 k are of order 2 l , 0 ď l ď k ´1, and if l ‰ 2 then the elements of order 2 l are contained entirely within xxy. Thus there is just one subgroup of order 2 l for each such l, and so these subgroups contribute k ´1 to r Q . Suppose then that l " 2. Using the relations ypx i yqy ´1 " x ´iy, xpx i yqx ´1 " x i`2 y and px i yq ´1 " x i`2 k´2 y (2.25)

in Q 2 k , we see that there are at most three conjugacy classes of cyclic subgroups of order 4, represented by the subgroups @ x 2 k´3 D , xyy and xxyy. Since @ x 2 k´3 D is contained in the normal subgroup xxy of Q 2 k , it cannot be conjugate to the two other subgroups, and using relations (2.25), we see that xyy and xxyy are non conjugate. We thus conclude that r Q " k `2. This number may also be obtained by counting the number of simple components in the Wedderburn decomposition (2.11) of QrQ 2 k s.

Finally we calculate r Q 2 . Consider the elements of Q 2 k of order 2 l , where 0 ď l ď k ´1. If l P t0, 1u then there is just one element of order 2 l , and so the contribution to r Q 2 is one in each case. If l " 2 then GalpQ 2 pζ 2 2 q{Q 2 q -Z 22 " t1, 3u by Theorem 23(b). Hence for every element z of Q 2 k of order 4, rzs Q 2 " rzs Y rz 3 s " rzs Y rz ´1s " rzs since in Q 2 k , every element is conjugate to its inverse. Thus the elements of Q 2 k of order 4 contribute 3 to r Q 2 . Suppose then that l ě 3. The elements of order 2 l are contained in xxy, are elements of the subgroup @ x 2 k´l´1 D of the form x 2 pk´l´1q r , where gcd pr, 2 l q " 1, and so are of the form x 2 pk´l´1q r , where r P 1, 3, . . . , 2 l ´1( . On the other hand, applying Theorem 23(b), we see that GalpQ 2 pζ 2 l q{Q 2 q -Z 2l . Now Z 2l " 1, 3, . . . , 2 l ´1( , and thus

" x 2 k´l´1 ‰ Q 2 " " x 2 k´l´1 ‰ Y " x 3p2 k´l´1 q ‰ Y ¨¨¨Y " x p2 l ´1qp2 k´l´1 q ‰ .
From above, this is precisely the set of all elements of order 2 l , and hence for each 3 ď l ď k ´1, the elements of order 2 l contribute one to r Q 2 . Summing over all possible values of l yields r Q 2 " k `2, and applying equation (2.10), we obtain r " 1 ´rQ `rQ 2 ´rF 2 " 0, which proves the proposition.

We now turn to the calculation of K ´1pZrGsq, where G is a binary polyhedral group.

Proposition 28. K ´1pZrT ˚sq -Z, K ´1pZrO ˚sq -Z 2 ' Z and K ´1pZrI ˚sq -Z 2 ' Z 2 .
Proof. Let G be a binary polyhedral group. By Proposition 18, it suffices to calculate the rank of K ´1pZrGsq, which we do using (2. (a) We first calculate the rank of K ´1pZrT ˚sq, where a presentation of T ˚" xP, Q, Xy is given by the first line of equation (2.2).

• The set G 1 3 consists of the union of the elements of T ˚of order 1, 2 and 4. By Proposition 7(a), if m P t1, 2, 4u, the elements of order m form a single conjugacy class, and thus form a single Q p -conjugacy class for p P t2, 3u, whence r F 3 " 3.

• The set G 1 2 consists of the identity and the 8 elements of T ˚of order 3, and by Proposition 7(a), there are two conjugacy classes of the elements of order 3, of which X and X ´1 are representatives. By Theorem 24, we have an isomorphism φ : GalpF 2 pζ 3 q{F 2 q ÝÑ Z 3 , and Impφq " t1, 2u. Thus rXs F 2 " rXs Y rX 2 s " rXs Y rX ´1s. It follows that there is a single F 2 -conjugacy class of elements of order 3, and so r F 2 " 2.

• Since there is a single conjugacy class of elements of order d, where d P t1, 2, 4u, it remains to determine the number of Q p -conjugacy classes, p P t2, 3u, of the elements of T ˚of order 3 and 6. We first calculate the number of Q p -conjugacy classes of the elements of order 3. By Theorem 23(a) and (b), φ : GalpQ p pζ 3 q{Q p q ÝÑ Z 3 is an isomorphism, Impφq " t1, 2u, and rXs Q p " rXs Y rX 2 s " rXs Y rX ´1s, which is the union of the two (usual) conjugacy classes of elements of order 3. We have the same result for the elements of order 6 of T ˚, since they are obtained from those of order 3 by adjoining the central element of T ˚of order 2. So for all d P t1, 2, 3, 4, 6u and p P t2, 3u, there is a single Q p -conjugacy class of the elements of order d, and hence r Q 2 " r Q 3 " 5.

• By equation (2.10), the rank r of K ´1pZrT ˚sq is equal to r " 1 ´rQ `rQ 2 ´rF 2 `rQ 3 ´rF 3 " 1, and thus K ´1pZrT ˚sq -Z. (b) We now calculate the rank of K ´1pZrO ˚sq.

• Recall first that T ˚is a subgroup of O ˚of index 2, and that O ˚z T ˚consists of twelve elements of order 4 and of order 8. So G 1 3 is contained in T ˚, and as in case (a) we obtain r F 2 " 2. Further, the elements of O ˚of order 1, 2, 3 and 6 each give rise to a single Q pconjugacy class of O ˚for p P t2, 3u. It remains to calculate the number of Q p -conjugacy classes of the elements of order 4 and 8, as well as r F 3 .

• To calculate the number of Q p -conjugacy classes of the elements of order 8, recall from Proposition 7(b) that there are two conjugacy classes of elements of order 8, for which representatives are g and g 3 , where g is any element of O ˚of order 8. By Theorem 23(a) and (b), φ : GalpQ p pζ 8 q{Q p q ÝÑ Z 8 satisfies Impφq Ą t1, 3u, hence rgs Q p Ą rgs Y rg 3 s, and there is a single Q p -conjugacy class of elements of order 8.

• To calculate the number of Q p -conjugacy classes of the elements of order 4, recall from Proposition 7(b) that there are two conjugacy classes of elements of order 4, C 1 and C 2 , where C 1 (resp. C 2 ) is the intersection of the set of elements of O ˚of order 4 with T ˚(resp. with O ˚z T ˚). In particular, if g P C 1 Y C 2 then g and g ´1 are conjugate. By Theorem 23(a) and (b), φ : GalpQ p pζ 4 q{Q p q ÝÑ Z 4 is an isomorphism, Impφq " t1, 3u, and rgs Q p " rgs Y rg ´1s " rgs for all g P O ˚of order 4. So the number of Q p -conjugacy classes of elements of order 4 is equal to 2.

• From the above computations, if d P t1, 2, 3, 6, 8u and p P t2, 3u, there is a single Q pconjugacy class of elements of order d, and there are two Q p -conjugacy class of elements of order 4, whence r Q p " 7.

• To calculate r F 3 , first note that G 1 3 consists of the union of the elements of O ˚of order 1, 2, 4 and 8, and that there is a single conjugacy class of elements of order 1 and 2. Let m P t4, 8u. By Theorem 24, φ : GalpF 3 pζ m q{F 3 q ÝÑ Z m satisfies Impφq " t1, 3u, and we see that the number of F 3 -conjugacy classes of elements of order m is just the number of Q pconjugacy classes of these elements, i.e. there are two F 3 -conjugacy classes of elements of order 4, and one F 3 -conjugacy class of elements of order 8. We conclude that r F 3 " 5.

• By equation (2.10), the rank r of K ´1pZrO ˚sq is equal to r " 1 ´rQ `rQ 2 ´rF 2 `rQ 3 ´rF 3 " 1, and thus K ´1pZrO ˚sq -Z 2 ' Z. (c) Finally, we determine the rank of K ´1pZrI ˚sq.

• By Proposition 7(c), r 2 plq " 1 for all l P t1, 2, 3, 4, 6u, so there is a single conjugacy class of elements of order l, and there are two conjugacy classes of elements of order 5 and 10. Hence it suffices to study the various F-conjugacy classes for the elements of order 5 and 10.

• We compute the number of the elements Q p -conjugacy classes of elements of order 5 for p P t2, 3, 5u. By Proposition 7(c), if g P I ˚is of order 5, g and g 2 are representatives of the two conjugacy classes of elements of order 5. Using Theorem 23(a) and (b), the homomorphism φ : GalpQ p pζ 5 q{Q p q ÝÑ Z 5 is an isomorphism, and so there is a single Q p -conjugacy class of elements of order 5 in I ˚for all p P t2, 3, 5u. By adjoining the central element of I ˚of order 2 to g, it follows that the same is true for the elements of order 10, from which it follows that r Q p " 7 for all p P t2, 3, 5u.

• To compute the number of F 2 -and F 3 -conjugacy classes of I ˚, note that G 1 2 is the union of the elements of I ˚of order 1, 3 and 5, and G 1 3 is the union of the elements of I ˚of order 1, 2, 4, 5 and 10. By Proposition 7(c), there is a single conjugacy class in I ˚of elements of order 1, 2, 3 and 4. By Theorem 24, for p P t2, 3u, the homomorphism GalpF p pζ 5 q{F p q ÝÑ Z 5 is an isomorphism, and Impφq " t1, 2, 3, 4u. Thus there is a single F p -conjugacy class of the 5-regular elements of order 5. By adjoining the central element of I ˚of order 2 to g, it follows that the same is true for the elements of order 10 in the case p " 3. We conclude that r F 2 " 3 and r F 3 " 5.

• To compute the number of F 5 -conjugacy classes of I ˚, the set G 1 5 is the union of the elements of I ˚of order 1, 2, 3, 4 and 6. Since there is a single conjugacy class in I ˚of elements of each of these orders, it follows that r F 5 " 5.

• By equation (2.10), the rank r of K ´1pZrI ˚sq is equal to r " 1 ´rQ `rQ 2 ´rF 2 `rQ 3 ´rF 3 rQ 5 ´rF 5 " 2, and thus K ´1pZrI ˚sq -Z 2 ' Z 2 . As we mentioned in Section 2.5.2, in order to prove Theorem 31 and to obtain Table 2.1, we need to compute K ´1pZrDic 4µ sq for µ P t6, 9, 10u. The torsion of these groups was already determined in Proposition 20. To end this section, we calculate their rank. Proposition 29. If µ P t6, 9, 10u, the rank of K ´1pZrDic 4µ sq is equal to 2.

Proof. (a)

We first consider the cases where µ P t6, 10u, so µ{2 is an odd prime. Making use of the presentation of the form (2.1) of Dic 4µ , the following table summarises the elements of each order of Dic 4µ . order d elements of order d number of elements 1 e 1 2

x µ 1 4

x µ{2 , x 3µ{2 , x i y, i P t0, 1, . . . , 2µ ´1u 2µ `2 µ{2

x 4 , x 8 , . . . , x 2pµ´2q pµ ´2q{2 µ x 2 , x 6 , . . . , x µ´4 , x µ`4 , x µ`8 , . . . , x 2pµ´2q pµ ´2q{2 2µ

x i , i P t1, 3, . . . , 2µ ´1u z tµ{2, 3µ{2u µ ´2

We compute the number of F-conjugacy classes for each of the fields F that appear in (2.10).

• From the above table, Dic 4µ possesses a single cyclic subgroup of order r for all r P t1, 2, µ{2, µ, 2µu, and using (2.1), it has three conjugacy classes of elements of order 4, namely x µ{2 , x 3µ{2 ( , x i y i P t1, 3, . . . , 2µ ´1u ( and x i y i P t0, 2, . . . , 2µ ´2u ( . So Dic 4µ has three conjugacy classes of (cyclic) subgroups of order 4. We conclude that Dic 4µ has eight conjugacy classes of cyclic subgroups, hence r Q " 8.

• The set of 2-regular elements of Dic 4µ consists of e and the pµ ´2q{2 elements of order µ{2. Since the order of 2 in Z μ{2 is equal to pµ ´2q{2, which is the order of Z μ{2 , the injective homomorphism φ : GalpF 2 pζ µ{2 q{F 2 q ÝÑ Z μ{2 is an isomorphism. Using (2.24), it follows that the F 2 -conjugacy class of x 4 is equal to x 4 , x 8 , . . . , x 2pm´2q ( , and thus r F 2 " 2. • The set of µ{2-regular elements of Dic 4µ consists of e, x µ , which is of order 2, and the 2µ `2 elements of order 4. The image of the injective homomorphism φ : GalpF µ{2 pζ 4 q{F µ{2 q ÝÑ Z 4 is contained in t1, 3u, and so is equal to t1u or t1, 3u. But px µ{2 q 3 " x 3µ{2 , and for all i P t0, 1, . . . , 2µ ´1u, px i yq 3 " px i yq ´1 " y ´1x ´i " y ´1x ´iy. y 2 . y " x i`µ y. It follows that if z P Dic 4µ is of order 4, rzs " rz 3 s, and by (2.24), we have rzs Ă rzs F µ{2 Ă rzs Ă rz 3 s " rzs, so rzs F µ{2 " rzs. Thus the F 2 -conjugacy classes of the µ{2-regular elements of Dic 4µ of order 4 coincide with the usual conjugacy classes, whence r F µ{2 " 5.

We now compute r Q 2 . To do so, we need to determine the number of Q 2 -conjugacy classes of the elements of order 4, µ{2, µ and 2µ.

• We calculate the number of Q 2 -conjugacy classes of the elements of order 4. By Theorem 23(b), the injective homomorphism φ : GalpQ 2 pζ 4 q{Q 2 q ÝÑ Z 4 is an isomorphism, and Impφq " 1, 3 ( . As in the analysis of the µ{2-regular elements of order 4, it follows that rzs Q 2 " rzs for every element z P Dic 4µ of order 4, and so the Q 2 -conjugacy classes of the elements of order 4 coincide with the usual conjugacy classes, and hence there are three Q 2 -conjugacy classes of elements of order 4.

• We determine the number of Q 2 -conjugacy classes of the elements of order µ{2, µ and 2µ. Let j P t0, 1, 2u. Then the injective homomorphism φ : GalpQ 2 pζ 2 j µ{2 q{Q 2 q ÝÑ Z 2j µ{2 is an isomorphism using Theorem 23(b) and (c) because GalpQ 2 pζ 2 {Q 2 q is trivial and the group GalpQ 2 pζ 4 {Q 2 q is of order 2. Thus there is a single Q 2 -conjugacy class of elements of order 2 j µ{2 for all j P t0, 1, 2u.

• It follows from these calculations that there is a single Q 2 -conjugacy class of elements of order r for all r P t1, 2, µ{2, µ, 2µu, and three Q 2 -conjugacy classes of elements of order 4, so r Q 2 " 8.

We now compute r Q µ{2 . To do so, we need to determine the number of Q µ{2 -conjugacy classes of the elements of order 4, µ{2, µ and 2µ.

• Let us determine the number of Q µ{2 -conjugacy classes of the elements of order µ{2 and µ. If j P t0, 1u, using Theorem 23(b) and (c), we see that the injective homomorphism φ : GalpQ µ{2 pζ 2 j µ{2 q{Q µ{2 q ÝÑ Z 2j µ{2 is an isomorphism because GalpQ µ{2 pζ 2 {Q 2 q is trivial. Thus there is a single Q µ{2 -conjugacy class of elements of order 2 j µ{2 for all j P t0, 1u.

• To calculate the number of Q µ{2 -conjugacy classes of the elements of order 2µ, consider the injective homomorphism φ : GalpQ µ{2 pζ 2µ q{Q µ{2 q ÝÑ Z 2µ . By Theorem 23(c), the group GalpQ µ{2 pζ 2µ q{Q µ{2 q is isomorphic to the direct product GalpQ µ{2 pζ µ{2 q{Q µ{2 q ˆGalpQ µ{2 pζ 4 q{Q µ{2 q, which by Theorem 23(b) is isomorphic to Z μ{2 ˆZ2 (resp. Z μ{2 ) if µ{2 " 3 mod 4 (resp. if µ{2 " 1 mod 4). We now distinguish the two cases µ " 6 and µ " 10.

-If µ " 6, φ is an isomorphism, and there is a single Q 3 -conjugacy class of elements of order 12.

-If µ " 10, GalpQ 5 pζ 20 q{Q 5 q is isomorphic to Z 4 by Theorem 23(b) and (c), so the image of φ is a subgroup of Z 20 . Now Z 20 is isomorphic to Z 2 ˆZ4 , so it possesses two subgroups isomorphic to Z 4 . A calculation shows that these two subgroups are of the form 1, 3, 7, 9 ( and 1, 9, 13, 17 ( . Using the table given at the beginning of the proof and (2.24) and the fact that x k is conjugate to x 20´k for all k P t1, 3, 7, 9, 11, 13, 17, 19u by (2.1), it follows in either case that there is a single Q 5 -conjugacy class of elements of order 20.

• It follows from these calculations that there is a single Q µ{2 -conjugacy class of elements of order r for all r P t1, 2, µ{2, µ, 2µu, and three Q µ{2 -conjugacy classes of elements of order 4, so r Q µ{2 " 8 for µ P t6, 10u.

• Using (2.10), we conclude that r " 1 ´rQ `rQ 2 ´rF 2 `rQ µ{2 ´rF µ{2 " 1 ´8 `p8 ´2q `p8 5q " 2 as required.

(b) Now suppose that µ " 9. Using the presentation of the form (2.1) of Dic 4µ , the following table summarises the elements of each order of Dic 4µ . order d elements of order d number of elements 1 e 1 2

x 9 1 3

x 6 , x 12 2 4

x i y, i P t0, 1, . . . , 17u 18 6

x 3 , x 15 2 9

x 2i , i P t1, 2, 4, 5, 7, 8u 6 18

x i , i P t1, 5, 7, 11, 13, 17u 6

In order to apply (2.10), we compute the number of F-conjugacy classes for each of the fields F that appear in that equation.

• By (2.1), there are two conjugacy classes of the elements of order 4, x i y i P t0, 2, . . . , 16u ( and x i y i P t1, 3, . . . , 17u ( , and the remaining conjugacy classes are of the form x i , x 18´i ( for i P t0, 1, . . . , 9u. Recall that r Q is given by the number of factors in equation (2.22), so r Q " 7 (this may also by verifying that there is a single conjugacy class of cyclic subgroups of order d for each d P t1, 2, 3, 4, 6, 9, 18u).

• If d P t1, 2, 3, 6u, there is a single conjugacy class of elements of order d. Thus there is a single Q p -conjugacy class of elements of order d by (2.24), where p P t2, 3u. Similiarly, if d P t1, 2u (resp. d P t1, 3u), there is a single F 3 -conjugacy class (resp. F 2 -conjugacy class) of elements of order d. So it suffices to determine: (i) the number of F 2 -conjugacy classes of elements of order 9.

(ii) the number of F 3 -conjugacy classes of elements of order 4. (iii) the number of Q p -conjugacy classes of elements of order d, where d P t4, 9, 18u, and p P t2, 3u.

We consider these cases in turn.

• By Theorem 24, the injective homomorphism φ : GalpF 2 pζ 9 q{F 2 q ÝÑ Z 9 is an isomorph- ism, and so there is a single F 2 -conjugacy class of elements of order 9. Now the set G 1 2 of 2-regular elements of Dic 36 is given by the union of the elements of order 1, 3 and 9, and since there is a single (usual) conjugacy class of elements of order 1 and 3, we conclude that r F 2 " 3.

• By Theorem 24, the injective homomorphism φ : GalpF 3 pζ 4 q{F 3 q ÝÑ Z 4 is an isomorphism, and so there is a single F 3 -conjugacy class of elements of order 4. Now the set G 1 3 of 3-regular elements of Dic 36 is given by the union of the elements of order 1, 2 and 4, and since there is a single (usual) conjugacy class of elements of order 1 and 2, we conclude that r F 3 " 3.

• Q 2 -conjugacy classes of elements of order 4: by Theorem 23(b), the injective homomorphism φ : GalpQ 2 pζ 4 q{Q 2 q ÝÑ Z 4 is an isomorphism, and Impφq " 1, 3 ( . Thus rys Q 2 " rys Y ry 3 s " rys Y rx 9 ys, where y is the element of Dic 36 appearing in (2.1), so rys Q 2 is the union of the two conjugacy classes of elements of order 4. Consequently, there is a single Q 2conjugacy class of elements of order 4.

• Q 2 -conjugacy classes of elements of order 9: by Theorem 23(a), the injective homomorphism φ : GalpQ 2 pζ 9 q{Q 2 q ÝÑ Z 9 is an isomorphism, and Impφq " 1, 2, 4, 5, 7, 8 ( . It follows from the above table of elements of Dic 36 and (2.24) that there is a single Q 2 -conjugacy class of elements of order 9.

• Q 2 -conjugacy classes of elements of order 18: by Theorem 23(c), GalpQ 2 pζ 18 q{Q 2 q -GalpQ 2 pζ 9 q{Q 2 q ˆGalpQ 2 pζ 2 q{Q 2 q -GalpQ 2 pζ 9 q{Q 2 q, which is cyclic of order 6. Thus the injective homomorphism φ : GalpQ 2 pζ 18 q{Q 2 q ÝÑ Z 18 is an isomorphism, and Impφq " 1, 5, 7, 11, 13, 17 ( . We conclude from the above table of elements of Dic 36 and (2.24) that there is a single Q 2 -conjugacy class of elements of order 18.

• From the above computations, for all d P t1, 2, 3, 4, 6, 9, 18u, there is a single Q 2 -conjugacy class of elements of order d, and hence r Q 2 " 7.

• Q 3 -conjugacy classes of elements of order 4: by Theorem 23(a), the injective homomorphism φ : GalpQ 3 pζ 4 q{Q 3 q ÝÑ Z 4 is an isomorphism. As in the case of the Q 2 -conjugacy classes of elements of order 4, we see that there is a single Q 3 -conjugacy class of elements of order 4.

• Q 3 -conjugacy classes of elements of order 9: by Theorem 23(b), the injective homomorphism φ : GalpQ 3 pζ 9 q{Q 3 q ÝÑ Z 9 is an isomorphism (both groups are of order 6), and Impφq " 1, 2, 4, 5, 7, 8 ( . As in the case of the Q 2 -conjugacy classes of elements of order 9, we see that there is a single Q 3 -conjugacy class of elements of order 9.

• Q 3 -conjugacy classes of elements of order 18: by Theorem 23(c), GalpQ 3 pζ 18 q{Q 3 q -GalpQ 3 pζ 9 q{Q 3 q ˆGalpQ 3 pζ 2 q{Q 3 q -GalpQ 3 pζ 9 q{Q 3 q, which as we saw above is cyclic of order 6. It follows that the injective homomorphism φ : GalpQ 3 pζ 18 q{Q 3 q ÝÑ Z 18 is an isomorphism. As in the case of the Q 2 -conjugacy classes of elements of order 18, we see that there is a single Q 3 -conjugacy class of elements of order 18.

• From the above computations, for all d P t1, 2, 3, 4, 6, 9, 18u, there is a single Q 3 -conjugacy class of elements of order d, and hence r Q 3 " 7.

• Hence the rank of K ´1pZrDic 36 sq is given by r " 1 ´rQ `rQ 2 ´rF 2 `rQ 3 ´rF 3 " 1 ´7 7 ´3 `7 ´3 " 2 as required.

We complete this section by computing K ´1pZrGsq, where G is a cyclic group of order p q or 2p q , where p is prime and q P N, or of order 12 or 20. These results will also be used in the proof of Theorem 31. Proposition 30. Let q P N, and let p be a prime number. (a) The group K ´1pZrZ p q sq is trivial.

(b) If p is odd then K ´1pZrZ 2p q sq -Z r , where r " ř q j"1

" Z pj : @ 2 D Z pj ı
, and where @ 2 D

Z pj denotes the subgroup of Z pj generated by 2.

(c) The group K ´1pZrZ 12 sq is isomorphic to Z 2 , and the group K ´1pZrZ 20 sq is isomorphic to Z 3 .

Proof. As we mentioned at the beginning of Section 2.5.1, if G is Abelian then the group K ´1pZrGsq is torsion free. So if G is one of the given groups, by (2.9), it suffices to calculate the rank r of K ´1pZrGsq.

(a) Let p be prime, and let q P N. Since Z p q is cyclic, r Q is equal to the number of divisors of p q , hence r Q " q `1. The elements of Z p q are of order p j , where j P t0, 1, . . . , qu, and the set G 1 p of p-regular elements of Z p q is equal to teu, hence r F p " 1. We now determine the number of Q p -conjugacy classes. If 1 ď j ď q, by Theorem 23(b), the injective homomorphism φ : GalpQ p pζ p j q{Q p q ÝÑ Z pj is an isomorphism, so Z p q possesses a single Q p -conjugacy of elements of order p j , and thus r Q p " q `1. Hence r " 1 ´rQ `rQ p ´rF p " 0 by (2.10), and K ´1pZrZ p q sq is trivial. (b) Let p be an odd prime, let q P N, and let x be a generator of Z 2p q . In order to apply (2.10), we compute r Q , r F 2 , r F p , r Q 2 and r Q p .

• Since Z 2p q is cyclic, we have r Q " 2pq `1q.

• The set G 1 p of p-regular elements of Z 2p q is equal to e, x p q ( , thus r F p " 2. • We now determine r Q p . Since Z 2p q possesses a single element of order m, where m P t1, 2u, it suffices to compute the number of Q p -conjugacy class of elements of order 2 ε p j , where ε P t0, 1u, and 1 ď j ď q. By Theorem 23(b), for all j P t1, . . . , qu, the injective homomorphism φ : GalpQ p pζ p j q{Q p q ÝÑ Z pj is an isomorphism, and so there is a single Q p -conjugacy class of elements of order p j . By Theorem 23(c), GalpQ p pζ 2p j q{Q p q -GalpQ p pζ p j q{Q p q ĜalpQ p pζ 2 q{Q p q -GalpQ p pζ p j q{Q p q, and by Theorem 23(b), the injective homomorphism φ : GalpQ p pζ 2p j q{Q p q ÝÑ Z 2p j is an isomorphism. Thus for all 1 ď j ď q, there is a single Q p -conjugacy class of elements of order 2p j . It follows that for every divisor m of 2p q , there is a single Q p -conjugacy class of elements of order m, whence r Q p " 2pq `1q.

• To compute r F , where F " Q 2 or F 2 , we first make the following general remark. Since Z 2p q is Abelian, for all f P Z 2p q , the (usual) conjugacy class r f s of f is equal to t f u. With the notation of Section 2.5.2, it follows from (2.24) that the cardinality of the F-conjugacy class r f s F is equal to |Impφq|, where φ is as defined in (2.23). Since the F-conjugacy classes are pairwise disjoint, if f P Z 2p q we conclude that there are rZ p m : Impφqs F-conjugacy classes of elements whose order is that of f . With this in mind, we compute r Q 2 and r F 2 .

• To calculate r Q 2 , first observe that since Z 2p q possesses a single element of order m, where m P t1, 2u, it suffices to compute the number of Q 2 -conjugacy class of elements of order 2 ε p j , where ε P t0, 1u, and 1 ď j ď q. Theorem 23(a) imples that the image of the injective homomorphism φ : GalpQ 2 pζ p j q{Q 2 q ÝÑ Z pj is equal to @ 2 D Z pj

. By the above remark, it follows that the number of Q 2 -conjugacy classes of elements of order p j is equal to " Z pj :

@ 2 D Z pj ı
. By Theorem 23(c), GalpQ 2 pζ 2p j q{Q 2 q -GalpQ 2 pζ p j q{Q 2 q ˆGalpQ 2 pζ 2 q{Q 2 q -GalpQ 2 pζ p j q{Q 2 q. So the image of the injective homomorphism φ : GalpQ 2 pζ 2p j q{Q 2 q ÝÑ Z pj is of order @ 2 D Z pj . Now Z 2p q " Z pq , and it follows from the above remark that the number of Q 2 -conjugacy classes of elements of order 2p j is also equal to

" Z pj : @ 2 D Z pj ı .
Since the elements of Z 2p q are of order 1, 2, p j or 2p j , where 1 ď j ď q, we deduce that

r Q 2 " 2 `2 ř q j"1 " Z pj : @ 2 D Z pj ı .
• To calculate r F 2 , observe that the set of 2-regular elements of Z 2p q is equal to the union of e with the elements of order p j , where j P t1, . . . , qu. By Theorem 24, for all j P t1, . . . , qu, the image of the injective homomorphism φ : GalpF 2 pζ p j q{F 2 q ÝÑ Z pj is equal to @ 2 D Z pj . Using the above remark once more, we see that the number of 2-regular F 2 -conjugacy classes of elements of order p j is given by

" Z pj : @ 2 D Z pj ı
, and thus r F 2 " 1 `řq

j"1 " Z pj : @ 2 D Z pj ı .
• To conclude, by (2.10), we have r " 1 ´rQ `rQ 2 ´rF 2 `rQ p ´rF p " ř q j"1 " Z pj :

@ 2 D Z pj ı as required.
(c) Let m " 4p, where p P t3, 5u. In order to apply (2.10), we now proceed to determine r Q , r F 2 , r F p , r Q 2 and r Q p .

• Since Z m is cyclic, r Q is equal to the number of divisors of 4p, so r Q " 6.

• The set G 1 2 of the 2-regular elements of Z 4p consists of the trivial element and the elements of order p. By Theorem 24, the injective homomorphism φ : GalpF 2 pζ p q{F 2 q ÝÑ Z p is an isomorphism, so there is a single F 2 -conjugacy class of elements of order p, and hence r F 2 " 2.

• The set G 1 p of the p-regular elements of Z 4p consists of the trivial element, the unique element of order 2, and the two elements of order 4. By Theorem 24, the injective homomorphism φ : GalpF p pζ 4 q{F p q ÝÑ Z 4 is an isomorphism if p " 3, and the image of φ is equal to 1 ( if p " 5. As in the remark regarding the F-conjugacy classes used in part (b), we conclude that there is a single F p -conjugacy class of elements of order 4 if p " 3, and two F p -conjugacy classes of elements of order 4 if p " 5. It follows that r F p " 3 if p " 3 and r F p " 4 if p " 5.

• We now compute r Q 2 . Since there is a single element of order 1 and 2, it suffices to determine the number of Q 2 -conjugacy classes of elements of order 4 and of order 2 i p, where i P t0, 1, 2u. By Theorem 23(c) (resp. Theorem 23(a)), the injective homomorphism φ : GalpQ 2 pζ 4 q{Q 2 q ÝÑ Z 4 (resp. φ : GalpQ 2 pζ p q{Q 2 q ÝÑ Z p) is an isomorphism, so there is a single Q 2 -conjugacy class of elements of order 4 (resp. of order p). If i ě 1 then GalpQ 2 pζ 2 i p q{Q 2 q -GalpQ 2 pζ p q{Q 2 q ˆGalpQ 2 pζ 2 i´1 q{Q 2 q, and so the injective homomorphism φ : GalpQ 2 pζ 2 i p q{Q 2 q ÝÑ Z 2i p is an isomorphism by Theorem 23(a) and (c). Hence there is a single Q 2 -conjugacy class of elements of order 2 i p. So for any divisor d of 4p, there is a single Q 2 -conjugacy class of elements of order d, hence r Q 2 " 6.

• We now compute r Q p . Since there is a single element of order 1 and 2, it suffices to determine the number of Q p -conjugacy classes of elements of order 4 and of order 2 i p, where i P t0, 1, 2u. By Theorem 23(a), the injective homomorphism φ : GalpQ p pζ 4 q{Q p q ÝÑ Z 4 is an isomorphism if p " 3, and the image of φ is equal to 1 ( if p " 5. As in the remark regarding the F-conjugacy classes used in part (b), we conclude that there is a single Q p -conjugacy class of elements of order 4 if p " 3, and two Q p -conjugacy classes of elements of order 4 if p " 5. Since the injective homomorphism φ : GalpQ p pζ p q{Q p q ÝÑ Z p is an isomorph- ism by Theorem 23(b), there is a single Q p -conjugacy class of elements of order p. Further, since GalpQ p pζ 2p q{Q p q -GalpQ p pζ p q{Q p q by Theorem 23(c), it follows from Theorem 23(b) that there is a single Q p -conjugacy class of elements of order 2p. Finally, by Theorem 23(c), since GalpQ p pζ 4p q{Q p q -GalpQ p pζ p q{Q p q ˆGalpQ p pζ 4 q{Q p q, it follows from Theorem 23(a) and (b) that the injective homomorphism φ : GalpQ p pζ 4p q{Q p q ÝÑ Z 4p is an isomorphism if p " 3, and the image of φ is isomorphic to Z p if p " 5. As in the remark regarding the F-conjugacy classes used in part (b), we conclude that there is a single Q p -conjugacy class of elements of order 4p if p " 3, and two Q p -conjugacy classes of elements of order 4p if p " 5. Hence r Q p " 6 if p " 3, and r Q p " 8 if p " 5.

• Using (2.10) and the above computations, the rank r of K ´1pZrZ 4p sq is given by r " 1 ŕQ `rQ 2 ´rF 2 `rQ p ´rF p , so r " 2 if p " 3 and r " 3 if p " 5 as required.

The lower algebraic K-theory of the finite subgroups of

B n pS 2 q for 4 ď n ď 11

In this section, we bring together the results of the previous sections to compute the lower algebraic K-theory of the finite subgroups of B n pS 2 q for 4 ď n ď 11. The results are summarised in the following theorem.

Theorem 31. For 4 ď n ď 11, the lower algebraic K-theory of the finite subgroups of B n pS 2 q is as given in Table 2.1.

Remark 32. It was proved in [START_REF] Carter | Localization in lower algebraic K-theory[END_REF]Theorem 3] that K ´ipZrGsq " 0 for any finite group G and for all i ě 2.

Remark 33. Although the results of Theorem 31 deal with the lower algebraic K-theory of the finite subgroups of B n pS 2 q for 4 ď n ď 11, these groups also occur as subgroups of B n pS 2 q for larger values of n. These values are given in the last column of Table 2.1, and are obtained from Theorem 2.

Finite group

G WhpGq K ´1pZrGsq r K 0 pZrGsq values of n ě 4 for which G is realised in B n pS 2 q
Z m , m P t1, 2, 3, 4u 0 0 0 all

Z 5 Z 0 0 n " 0, 1, 2 mod 5 Z 6 0 Z 0 all Z 7 Z 2 0 0 n " 0, 1, 2 mod 7 Z 8 Z 0 0 n ı 3 mod 4 Z 9 Z 2 0 0 n " 0, 1, 2 mod 9 Z 10 Z 2 Z 0 n " 0, 1, 2 mod 5 Z 11 Z 4 0 0 n " 0, 1, 2 mod 11 Z 12 Z Z 2 Z 2 n " 0, 1, 2 mod 6 Z 14 Z 4 Z 2 0 n " 0, 1, 2 mod 7 Z 16 Z 4 0 Z 2 n " 0, 1, 2 mod 8 Z 18 Z 4 Z 2 Z 3 n " 0, 1, 2 mod 9 Z 20 Z 5 Z 3 Z 5 2
n " 0, 1, 2 mod 10

Z 22 Z 8 Z Z 3 n " 0, 1, 2 mod 11 Q 8 0 0 Z 2 n even Dic 12 0 Z Z 2 n " 0, 2 mod 3 Q 16 Z Z 2 Z 2 n even Dic 20 Z 2 Z 2 ' Z Z 2 n " 0, 2 mod 5 Dic 24 Z Z 2 ' Z 2 Z 3 2 n " 0, 2 mod 6 T ˚0 Z Z 2 n even Dic 28 Z 4 Z Z 2 n " 0, 2 mod 7 Q 32 Z 4 Z 2 Z 2 n " 0, 2 mod 8 Dic 36 Z 4 Z 2 Z 2 2 n " 0, 2 mod 9 Dic 40 Z 5 Z 2 ' Z 2 Z 3 2 n " 0, 2 mod 10 Dic 44 Z 8 Z Z 2 n " 0, 2 mod 11 O ˚Z Z 2 ' Z Z 2 2
n " 0, 2 mod 6

Table 2.1: The lower algebraic K-theory of the finite subgroups of B n pS 2 q, n ď 11.

Proof of Theorem 31. By Theorem 2, the isomorphism classes of the maximal finite subgroups of B n pS 2 q, where n runs over the elements of t4, . . . , 11u, are Dic 4m , where m P t3, 4, . . . , 11u, Z 2q , where q P t4, 5, . . . , 10u, T ˚and O ˚. If we remove the condition of maximality of these subgroups, then we must also add Q 8 and Z r to the list, where r P t1, 2, . . . , 7, 9, 11, 22u. We thus obtain the groups of the first column of Table 2.1 that are subgroups of B n pS 2 q for the values of n given in the last column. We divide the proof into several parts.

(a) If G is one of the groups appearing in Table 2.1 then WhpGq is obtained by applying Proposition 10.

(b) We determine r K 0 pZrGsq, where G is one of the groups appearing in Table 2.1. By Theorem 11, r r K 0 pZrζ ν sq, where ζ ν is a primitive 2 νth root of unity, and whose kernel, denoted by W 3 in [START_REF] Kervaire | On the projective class group of cyclic groups of prime power order[END_REF], is isomorphic to Z 2 . From Theorem 11, r KpZrζ ν sq is trivial for all n " 1, . . . , 4, and so r K 0 pZrZ 16 sq -Z 2 . By Theorem 14, r K 0 pZrZ n sq is isomorphic to Z 3 if n P t18, 22u, and to Z 5 2 if n " 20. If G " Dic 4m , where 2 ď m ď 11, or if G " T ˚or O ˚then r K 0 pZrGsq is given by Theorem 12. (c) We determine K ´1pZrGsq, where G is one of the groups appearing in Table 2.1. We consider several cases.

K 0 pZrGsq is trivial if G is isomorphic to Z n ,
(i) If G " Z m ,
where m P t1, 2, 3, 4, 5, 7, 8, 9, 11, 16u, then K ´1pZrGsq is trivial by Proposition 30(a). (ii) Let G " Z m , where m P t6, 10, 14, 18, 22u. Then m is of the form m " 2p q , where q P N and p is an odd prime. By Proposition 30(b), K ´1pZrGsq -Z r , where r " ř q j"1 " Z pj :

@ 2 D Z pj ı .
A straightforward computation shows that K ´1pZrZ 6 sq -K ´1pZrZ 10 sq -K ´1pZrZ 22 sq -Z, and K ´1pZrZ 14 sq -K ´1pZrZ 18 sq -Z 2 .

(iii) If G " Z m , where m P t12, 20u, the results for K ´1pZrGsq are obtained from Proposition 30(c).

(iv) K ´1pZrGsq for G " Dic 4m , where 2 ď m ď 11: we distinguish the following cases.

• If m P t2, 4, 8u, the results are a consequence of Proposition 27.

• If m P t3, 5, 7, 11u, the results follow from Theorem 25 and Proposition 26 (observe that if m P t3, 5, 11u, 2 generates Z m, so λ " 1, while if m " 7, ´1 R @ 2 D , but @ 2 D " 3, so λ " 1 also).

• If m P t6, 9, 10u, by Propositions 20 and 29, it follows that K ´1pZrDic 36 sq -Z 2 , and

K ´1pZrDic 4m sq -Z 2 ' Z 2 if m P t6, 10u. (v) If G " T ˚or O ˚then the results follow from Proposition 28.
This completes the proof of the results given in Table 2.1.

Chapter 3

The braid group B 4 pS 2 q, and the conjugacy classes of its maximal virtually cyclic subgroups

In this chapter, we focus our attention on the braid group B 4 pS 2 q of the sphere on four strings. The aim is to understand the structure of its maximal virtually cyclic subgroups. These results will be used in Chapter 4 to compute the lower algebraic K-theory of B 4 pS 2 q, and to prove Theorem 1.

In Section 3.1, we start by recalling some properties of B 4 pS 2 q. We then study the algebraic description of the finite subgroups of B 4 pS 2 q given by Theorem 2, which enables us to prove in Proposition 37 that B 4 pS 2 q may be expressed as an amalgamated product of T ˚and Q 16 along their common normal subgroup that is isomorphic to Q 8 . This will alow us to show that B 4 pS 2 q is hyperbolic in the sense of Gromov (as we shall see in Proposition 49, B 4 pS 2 q is virtually free). In order to do this, in Section 3.2, we study the structure of the maximal virtually cyclic subgroups of B 4 pS 2 q, our main result being Theorem 41. Using [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF], in Section 3.3, we show that the maximal infinite virtually cyclic subgroups of B 4 pS 2 q possess an infinite number of conjugacy classes.

Generalities about B 4 pS 2 q

In this section, we state several results concerning B 4 pS 2 q. Some basic facts and results about Artin (pure) braid groups and (pure) braid groups of surfaces may be found in Appendix B. As we shall see, B 4 pS 2 q is rather special, and possesses some very interesting properties that will allow us to calculate its lower K-theoretical groups. Unfortunately, if n ě 5, B n pS 2 q does not share these properties. We start by recalling a presentation of B 4 pS 2 q. Theorem 34 ([30]). The group B 4 pS 2 q admits the following presentation: generators: σ 1 , σ 2 , σ 3 . relations:

σ 1 σ 3 " σ 3 σ 1 σ 1 σ 2 σ 1 " σ 2 σ 1 σ 2 σ 2 σ 3 σ 2 " σ 3 σ 2 σ 3 σ 1 σ 2 σ 2 3 σ 2 σ 1 " 1. ( 3.1)
The first three 'Artin relations' (also known as braid relations) will be used freely and without further comment in what follows. Since the given generators together with these three relations constitute a presentation of the Artin braid group B 4 (see (B.1) and (B.2)), B 4 pS 2 q is thus a quotient of B 4 .

Remark 35. It follows easily from Theorem 34 that the Abelianisation of B 4 pS 2 q is isomorphic to Z 6 , and that the Abelianisation homomorphism π : B 4 pS 2 q ÝÑ Z 6 identifies the three generators to the single generator 1 of Z 6 .

We may determine generators of representatives of the conjugacy classes of the finite subgroups of B 4 pS 2 q in terms of the generators of Theorem 34 as follows. First, according to Murasugi [68, Theorem A], any finite order element of B 4 pS 2 q is conjugate to a power of one of the following elements: a) and (b) for illustrations of the half and full twist braid on six strings). The braid ∆ 2 4 generates the centre of B 4 pS 2 q and is the unique element of B 4 pS 2 q of order 2 (this is true in general, see [START_REF] Gillette | The word problem and consequences for the braid groups and mapping class groups of the 2-sphere[END_REF]). Using (3.2), this latter fact implies that:

$ ' & ' % α 0 " σ 1 σ 2 σ 3 (of order 8) α 1 " σ 1 σ 2 σ 2 3 (of order 6) α 2 " σ 1 σ 2 2 (of order 4). (3.2)
Let ∆ 4 " σ 1 σ 2 σ 3 σ 1 σ 2 σ 1 (3.
∆ 2 4 " α 4 0 " α 3 1 " α 2 2 . (3.4) Let Q " @ α 2 0 , ∆ 4 D . By [44, Theorem 1.3(3)], Q is isomorphic to Q 8 ,
and is a normal subgroup of B 4 pS 2 q. Further, it is well known (see [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Lemma 29] for example) that:

α 0 σ i α ´1 0 " σ i`1 for i " 1, 2, and α 2 0 σ 3 α ´2 0 " σ 1 , (3.5)
and that:

∆ 4 σ i ∆ ´1 4 " σ 4´i for all i " 1, 2, 3 (3.6)
Note that relations (3.5) and (3.6) hold in B 4 , and are special cases of more general relations in B n pS 2 q.

Remarks 36.

(a) By Theorem 2, the isomorphism classes of the maximal finite subgroups of B 4 pS 2 q are T ånd Q 16 .

(b) Within B 4 pS 2 q, there is a single conjugacy class of each isomorphism class of T ˚and Q 16 [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Proposition 1.5(1)]. These subgroups may be realised algebraically as follows:

(i) Q 16 may be realised in B 4 pS 2 q as the subgroup xα 0 , ∆ 4 y (α 0 is of order 8 and ∆ 4 is of order 4) [START_REF] Gonçalves | The quaternion group as a subgroup of the sphere braid groups[END_REF]. In particular,

∆ 4 α 0 ∆ ´1 4 " α ´1 0 and α 4 0 " ∆ 2 4 . (3.7)
(ii) By [47, Remark 3.2], T ˚may be realised in B 4 pS 2 q as the subgroup

@ σ 1 σ ´1 3 , ∆ 4 D ¸@α 2 1 D - Q 8 ¸Z3
. Note that the first factor of the semi-direct product is Q, since by equations (3.2) and (3.3), we have:

α ´2 0 ∆ 4 " σ ´1 3 σ ´1 2 σ ´1 1 σ ´1 3 σ ´1 2 σ ´1 1 . σ 1 σ 2 σ 3 σ 1 σ 2 σ 1 " σ 1 σ ´1 3 . (3.8)
The action in the semi-direct product is given by α

2 1 ∆ 4 α ´2 1 " σ 1 σ ´1 3 and α 2 1 σ 1 σ ´1 3 α ´2 1 " ∆ 4 σ 1 σ ´1 3 " α ´2 0 . The only other isomorphism class of finite non-Abelian subgroups of B 4 pS 2 q is that of Q 8 : the subgroups Q " @ α 2 0 , ∆ 4 D and Q 1 " @ α 2 0 , α 0 ∆ 4 D of xα 0 , ∆ 4
y are isomorphic to Q 8 and realise the two conjugacy classes of Q 8 in B 4 pS 2 q [47, Proposition 1.5(2) and Theorem 1.6]. (c) By Theorem 2, the remaining finite subgroups are cyclic, and as we mentioned previously, are realised up to conjugacy by powers of the α i , i P t0, 1, 2u. For each finite cyclic subgroup, there is a single conjugacy class, with the exception of Z 4 , which is realised by both of the non-conjugate subgroups As we mentioned above, Q is a normal subgroup of B 4 pS 2 q. From this, we obtain the following decomposition of B 4 pS 2 q as an amalgamated product of two finite groups.

Proposition 37. B 4 pS 2 q -Q 16 ˚Q8 T ˚. Proof. Let Γ " B 4 pS 2 q{Q. Since σ 1 σ ´1 3 P Q and @ σ 1 σ ´1 3 D
is not normal in B 4 pS 2 q by Remarks 36(b)(ii), it follows that the normal closure of σ 1 σ ´1 3 in B 4 pS 2 q is Q, and that a presentation of Γ is obtained by adjoining the relation σ 1 " σ 3 to the presentation of B 4 pS 2 q given in Theorem 34. Thus Γ is generated by elements σ 1 and σ 2 , subject to the two relations

σ 1 σ 2 σ 1 " σ 2 σ 1 σ 2 and pσ 1 σ 2 σ 1 q 2 " 1. Let Λ " @ a, b a 2 " b 3 " 1 D denote the free product Z 2 ˚Z3
, and consider the map ϕ : Λ ÝÑ Γ defined on the generators of Λ by ϕpaq " σ 1 σ 2 σ 1 and ϕpbq " σ 1 σ 2 . Since pϕpbqq 3 " pσ 1 σ 2 q 3 " pσ 1 σ 2 σ 1 q 2 " pϕpaqq 2 " 1, ϕ extends to a homomorphism that is surjective since ϕpb ´1aq " σ 1 and ϕpa ´1b 2 q " σ 2 . Conversely, the map ψ : Γ ÝÑ Λ defined on the generators of Γ by ψpσ 1 q " b ´1a and ψpσ 2 q " a ´1b 2 extends to a homomorphism since: ψpσ 1 q ψpσ 2 q ψpσ 1 q " b ´1aa ´1b 2 b ´1a " a " a ´1b 2 b ´1aa ´1b 2 " ψpσ 2 q ψpσ 1 q ψpσ 2 q and pψpσ 1 q ψpσ 2 q ψpσ 1 qq 2 " a 2 " 1, and is surjective because ψpσ 1 σ 2 σ 1 q " a and ψpσ 1 σ 2 q " b. Hence ϕ is an isomorphism, and Γ -Z 2 ˚Z3 .

Let G " Q 16 ˚Q8 T ˚, and consider the following presentation of G with generators u, v, p, q, r that are subject to the relations: # p 2 " q 2 , qpq ´1 " p ´1, rpr ´1 " q, rqr ´1 " pq, r 3 " 1

u 4 " v 2 , vuv ´1 " u ´1, u 2 " p, v " q,
so that xp, q, ry -T ˚, xu, vy -Q 16 , and xp, q, ry X xu, vy " H, where H " xp, qy -Q 8 . It follows from this presentation that H is normal in G and G{H -Z 2 ˚Z3 . Let f : G ÝÑ B 4 pS 2 q be the map defined on the generators of G by f puq " α ´1 0 , f ppq " α ´2 0 , f pvq " f pqq " ∆ 4 and f prq " α 2 1 . Using Remarks 36(b), we see that f respects the relations of G, and so extends to a homomorphism that sends H isomorphically onto Q. Further, α 3 1 " ∆ 2 4 by equation (3.4). Thus α 1 " ∆ 2 4 α ´2 1 , and since B 4 pS 2 q " xα 0 , α 1 y by [41, Theorem 3], we conclude that f is surjective. We thus obtain the following commutative diagram of short exact sequences:

1 / / H / / f | H - G / / f G{H / / p f 1 1 / / Q / / B 4 pS 2 q / / Γ / / 1,
where p f is the homomorphism induced by f on the quotients. Since f is surjective, p f is too, and the isomorphisms G{H -Z 2 ˚Z3 -Γ and the fact that Z 2 ˚Z3 is Hopfian (see [START_REF] Dey | The Hopf property of free products[END_REF] for example) imply that p f is an isomorphism. The result is then a consequence of the 5-Lemma.

Remark 38. By Proposition 37, B 4 pS 2 q is isomorphic to an amalgam of finite groups. Using Bass-Serre theory of groups acting on trees, it follows that it is a virtually free group, and so is hyperbolic in the sense of Gromov (see [START_REF] Gromov | Hyperbolic groups[END_REF] and [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF]Section 1.1]). This important fact will be crucial in the computation of the lower algebraic K-theory of B 4 pS 2 q.

Maximal virtually cyclic subgroups of B 4 pS 2 q

As we mentioned at the beginning of Section 2.1, an infinite virtually cyclic group Γ is isomorphic to one of the following: (I) a semi-direct product of the form Γ -F ¸α Z, where F is a finite group and the action α belongs to HompZ, Aut pFqq. Such a group Γ surjects onto Z with finite kernel F. (II) an amalgamated product of the form Γ -G 1 ˚F G 2 , where G 1 and G 2 are finite groups containing a common subgroup F of index 2 in both G 1 and G 2 . Such a group Γ surjects onto the infinite dihedral group Dih 8 with finite kernel F. We shall say that these infinite virtually cyclic groups are of Type I or of Type II respectively.

Recall from Remarks 36(a) and (b) that up to isomorphism, the maximal finite subgroups of B 4 pS 2 q are Q 16 and T ˚, and that there exists a single conjugacy class of each. Since OutpQ 8 q -S 3 , there are three isomorphism classes of Type I groups of the form Q 8 ¸Z, and that we denote by Q 8 ¸j Z, where j P t1, 2, 3u, and for which the action is of order j [47, Definition 4(1)(e)]. More precisely, if we take the presentation of Q 8 given by (2.1) and adjoin a new generator z:

(i) Q 8 ¸1 Z is the group obtained by adding the relations rz, xs " rz, ys " 1, where ru, vs " uvu ´1v ´1 denotes the commutator of the elements u and v, so Q 8 ¸1 Z -Q 8 ˆZ. (ii) Q 8 ¸2 Z is the group obtained by adding the relations zxz ´1 " y and zyz ´1 " x (so zxyz ´1 " pxyq ´1). (iii) Q 8 ¸3 Z is the group obtained by adding the relations zxz ´1 " y and zyz ´1 " xy (so zxyz ´1 " x). Up to a finite number of exceptions, the isomorphism classes of the infinite virtually cyclic subgroups of B n pS 2 q were classified in [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF] for all n ě 4. In the case n " 4, the classification is as follows.

Theorem 39 [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Theorem 5]). Every infinite virtually cyclic subgroups of B 4 pS 2 q is isomorphic to one of the following groups: (a) subgroups of Type I: Z k ˆZ, k P t1, 2, 4u; Z 4 ¸Z for the non-trivial action; and Q 8 ¸j Z for j P t1, 2, 3u. (b) subgroups of Type II:

Z 4 ˚Z2 Z 4 , Z 8 ˚Z4 Z 8 , Z 8 ˚Z4 Q 8 , Q 8 ˚Z4 Q 8 , Q 16 ˚Q8 Q 16 .
For each of the Type II subgroups given in Theorem 39(b), abstractly there is a single isomorphism class, with the exception of Q 16 ˚Q8 Q 16 for which there are two isomorphism classes [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Proposition 11]. In this exceptional case, we recall the following result concerning the structure of the two classes, and their realisation in B 4 pS 2 q.

Proposition 40 [START_REF] Gonçalves | The virtually cyclic subgroups of the braid groups of the sphere[END_REF]Propositions 11 and 78]). Abstractly, there are exactly two isomorphism classes of the amalgamated product Q 16 ˚Q8 Q 16 , possessing the following presentations:

Γ 1 "
@ a, b, x, y a 4 " b 2 , x 4 " y 2 , bab ´1 " a ´1, yxy ´1 " x ´1, x 2 " a 2 , y " b D (3.9) and Γ 2 " @ a, b, x, y a 4 " b 2 , x 4 " y 2 , bab ´1 " a ´1, yxy ´1 " x ´1, x 2 " b, y " a 2 b D .

(3.10)

Further, for i P t1, 2u, B 4 pS 2 q possesses a subgroup G i isomorphic to Γ i .

With Chapter 4 in view, most of the rest of this chapter will be devoted to the problem of deciding which subgroups of B 4 pS 2 q are maximal within the family of virtually cyclic subgroups. In what follows, we refer to such a subgroup as a maximal virtually cyclic group (thus the word 'maximal' will be used to qualify the notion of virtually cyclic group).

Theorem 41.

(a) Let G be a maximal infinite virtually cyclic subgroup of B 4 pS 2 q. Then G is isomorphic to one of the following groups: Q 8 ¸Z for one of the three possible actions, or Q 16 ˚Q8 Q 16 . (b) If G is a subgroup of B 4 pS 2 q isomorphic to T ˚then it is maximal as a virtually cyclic subgroup. (c) For each j P t1, 2, 3u, there are subgroups of B 4 pS 2 q isomorphic to Q 8 ¸j Z that are maximal as virtually cyclic subgroups, and others that are non maximal. (d) There exist subgroups of B 4 pS 2 q isomorphic to Q 16 ˚Q8 Q 16 that are maximal as virtually cyclic subgroups, and others that are non maximal.

The proof of Theorem 41 is long, and will be split into three sections, Section 3.2.1, where we shall prove parts (a) and (b), Section 3.2.2, where we shall prove parts (c) and (d), with the exception of the case j " 1 in part (c), and Section 3.2.3, where we prove part (c) in this exceptional case. As we mentioned in Remark 38, B 4 pS 2 q is hyperbolic in the sense of Gromov. The following proposition implies that there are no infinite ascending chains of infinite virtually cyclic subgroup of B 4 pS 2 q.

Proposition 42 ([55, Propositions 5,6 and Remark 7]). Every infinite virtually cyclic subgroup of a Gromov hyperbolic group is contained in a unique maximal virtually cyclic subgroup.

Proof of parts (a) and (b) of Theorem 41

The statement of the following proposition is that of parts (a) and (b) of Theorem 41.

Proposition 43.

(a) Let G be a maximal virtually cyclic subgroup of B 4 pS 2 q. Then G is isomorphic to one of the following groups: T ˚, Q 8 ¸Z for one of the three possible actions, or Q 16 ˚Q8 Q 16 . (b) If G is a subgroup of B 4 pS 2 q isomorphic to T ˚then it is maximal as a virtually cyclic subgroup.

Before proving Proposition 43, note that if G is infinite in part (a), we will prove that G cannot be isomorphic to one of the other infinite virtually cyclic groups of B 4 pS 2 q given in Theorem 39. The question of whether there actually exist maximal virtually cyclic subgroups of B 4 pS 2 q isomorphic to Q 8 ¸Z or to Q 16 ˚Q8 Q 16 will be dealt with in Section 3.2.2.

Proof of Proposition 43. Suppose that G is a maximal virtually cyclic subgroup of B 4 pS 2 q. (a) (i) First assume that G is finite. Then G is a maximal finite subgroup of B 4 pS 2 q, so is isomorphic to either Q 16 or T ˚by Theorem 2. Suppose that G -Q 16 . Since B 4 pS 2 q possesses a single conjugacy class of subgroups isomorphic to Q 16 by Remarks 36(b), by Proposition 40, there exists a subgroup of B 4 pS 2 q isomorphic to the amalgamated product Q 16 ˚Q8 Q 16 , of which one of the factors is G, so G is not maximal as a virtually cyclic subgroup. So G must be isomorphic to T ˚.

(ii) Now assume that G is infinite. We separate the cases where G is of Type I and Type II respectively. (A) We first suppose that G is of Type I, so G " F ¸Z, for some action of Z on F, where F is finite and is the torsion subgroup of G. Suppose that F is either trivial or is isomorphic to Z 2 or Z 4 , and let u be a generator of the Z-factor of G. Up to conjugation, we claim that

F Ă @ α 2 0 D . If F is trivial or isomorphic to Z 2 then F Ă @ ∆ 2 4 D Ă @ α 2 0 D since α 4 0 " ∆ 2 4
. So suppose that F -Z 4 . By Remarks 36(c), B 4 pS 2 q admits two conjugacy classes of subgroups isomorphic to Z 4 , generated respectively by α 2 0 and α 2 . But since u normalises F and the normaliser of xα 2 y in B 4 pS 2 q is finite [47, Proposition 8(b)], it follows that F is conjugate to @ α 2 0 D . This proves the claim, and so conjugating G if necessary, we may suppose that [START_REF] Gonçalves | The lower central and derived series of the braid groups of the sphere[END_REF]Theorem 1.3(3)], the subgroup @ α 2 0 , ∆ 4 , u D is isomorphic to one of the three Type I groups Q 8 ¸j Z of Theorem 39(a), where j P t1, 2, 3u, and admits @ α 2 0 , u D as a proper subgroup. Now G is a subgroup of @ α 2 0 , u D , so G is non maximal as a virtually cyclic subgroup of B 4 pS 2 q. The result in this case is then a consequence of Theorem 39(a).

F Ă @ α 2 0 D . Since Q " @ α 2 0 , ∆ 4 D is normal in B 4 pS 2 q and Q -Q 8 by
(B) Now suppose that G is a Type II subgroup of B 4 pS 2 q that is non isomorphic to Q 16 ˚Q8 Q 16 . By Theorem 39(b), we may write G " G 1 ˚H G 2 , where either:

(1) G 1 and G 2 are subgroups of B 4 pS 2 q isomorphic to Q 8 or Z 8 , and H " G 1 X G 2 is isomorphic to Z 4 , or (2) G 1 and G 2 are subgroups of B 4 pS 2 q isomorphic to Z 4 , and

H " G 1 X G 2 " @ ∆ 2 4 D .
Note that G 1 and G 2 are not necessarily isomorphic. By Remarks 36, in B 4 pS 2 q, there are two conjugacy classes of subgroups isomorphic to Q 8 represented by Q and Q 1 , one conjugacy class of subgroups isomorphic to Z 8 , represented by xα 0 y, and two conjugacy classes of sub- groups isomorphic to Z 4 , represented by @ α 2 0 D and xα 0 ∆ 4 y (this is because the elements α 0 ∆ 4 and α 2 generate conjugate subgroups of order 4). Conjugating G if necessary, we may suppose that G 1 is equal to Q, Q 1 or xα 0 y in case [START_REF] Adem | Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften[END_REF], and is equal to @ α 2 0 D or xα 0 ∆ 4 y in case (2). Furthermore, there exists λ P B 4 pS 2 q such that G 2 " λG 1 2 λ ´1, where G 1 2 is equal to Q, Q 1 or xα 0 y in case [START_REF] Adem | Cohomology of finite groups, Grundlehren der Mathematischen Wissenschaften[END_REF], and is equal to

@ α 2 0 D or xα 0 ∆ 4 y in case (2). Set L " xα 0 , ∆ 4 y. Then G 1 and G 1 2 are subgroups of L, and Q is a subgroup of L that is normal in B 4 pS 2 q, so Q is a subgroup of L X λLλ ´1. Since L -λLλ ´1 -Q 16 and G " xG 1 Y G 2 y Ř @ L Y λLλ ´1D , it follows that @ L Y λLλ ´1D is infinite and L X λLλ ´1 " Q because Q is of index 2 in both L and λLλ ´1. We conclude from [45, Lemma 15] that @ L Y λLλ ´1D -L ˚Q L -Q 16 ˚Q8 Q 16 .
Thus G is a non-maximal virtually cyclic subgroup of B 4 pS 2 q, and it follows from Theorem 39(b) that any maximal virtually cyclic subgroup of B 4 pS 2 q of Type II must be isomorphic to Q 16 ˚Q8 Q 16 . (b) By Theorem 39, none of the infinite virtually cyclic subgroups of B 4 pS 2 q admit subgroups isomorphic to T ˚, so any subgroup of B 4 pS 2 q isomorphic to T ˚is maximal as a virtually cyclic subgroup. Combined with part (a)(i) of the proof, this shows in fact that G is a finite maximal virtually cyclic subgroup if and only if G -T ˚.

This completes the proof of parts (a) and (b) of Theorem 41.

Proof of parts (c) and (d) of Theorem 41

We now turn to parts (c) and (d) of Theorem 41, which may be regarded as a converse of part (a) in the case that G is infinite. We first prove part (c) of Theorem 41 with the exception of the existence of Q 8 ˆZ as a maximal virtually cyclic subgroup of B 4 pS 2 q, which will be dealt with in Section 3.2.3. Before doing so, we state and prove the following lemma. Lemma 44. Let π : B 4 pS 2 q ÝÑ Z 6 denote the Abelianisation homomorphism described in Remark 35.

(a) If H is a subgroup of B 4 pS 2 q that is isomorphic to either Z 8 , Q 8 or Q 16 then πpHq Ă @ 3 D . (b) If G is a subgroup of B 4 pS 2 q that is isomorphic to an amalgamated product of one of the groups

Q 16 ˚Q8 Q 16 , Q 8 ˚Z4 Q 8 , Q 8 ˚Z4 Z 8 or Z 8 ˚Z4 Z 8 then πpGq Ă @ 3 D .
Proof.

(a) Consider the subgroup K " xα 0 , ∆ 4 y. As we mentioned in the proof of Proposition 43, K contains representatives of the conjugacy classes of all subgroups of B 4 pS 2 q that are isomorphic to Z 8 , Q 8 or Q 16 . So there exists λ P B 4 pS 2 q such that λHλ ´1 Ă K. Now πpα 0 q " 3 and πp∆ 4 q " 0, thus πpKq Ă @ 3 D , which yields the result.

(b) If G is a subgroup of B 4 pS 2 q that is isomorphic to one of the given amalgamated products then by Remarks 36(c), the factors appearing in the amalgamation are subgroups of conjugates of K, and thus πpGq Ă πpKq Ă @ 3 D by part (a).

To prove Theorem 41(c), for each j P t1, 2, 3u, we shall exhibit two subgroups of B 4 pS 2 q that are isomorphic Q 8 ¸j Z, one of which is maximal as a virtually cyclic subgroup of B 4 pS 2 q, and the other of which is non maximal. For the case j " 1, the proof of the existence of a maximal virtually cyclic subgroup of B 4 pS 2 q that is isomorphic to Q 8 ˆZ is long, and will be treated separately in Section 3.2.3. With the exception of this case, the statement of the following proposition is that of parts (c) and (d) of Theorem 41.

Proposition 45.

(a) For each j P t1, 2, 3u, there are subgroups of B 4 pS 2 q isomorphic to Q 8 ¸j Z that are non maximal as virtually cyclic subgroups. (b) For each j P t2, 3u, there are subgroups of B 4 pS 2 q isomorphic to Q 8 ¸j Z that are maximal as virtually cyclic subgroups. (c) There exist subgroups of B 4 pS 2 q isomorphic to Q 16 ˚Q8 Q 16 that are maximal as virtually cyclic subgroups, and others that are non maximal.

Proof of Proposition 45. (a) By Proposition 40, for i " 1, 2, B 4 pS 2 q possesses a subgroup G i that is isomorphic to the amalgamated product Γ i given by equations (3.9) and (3.10), and so admits a presentation given by the corresponding equation. The amalgamating subgroup Γ "

@ a 2 , b D " @ x 2 , y D is isomorphic to Q 8 ,
-Q 8 ¸3 Z. In each case, H i Ř G i because rG i : H i s " 2. Now G i is isomorphic to Q 16 ˚Q8 Q 16 ,
and so H i is non maximal as a virtually cyclic subgroup of B 4 pS 2 q, which proves the statement for j P t1, 3u. It thus remains to treat the case j " 2. Using equation (3.11), note that in G 1 , the action by conjugation of xa ´1x on Γ is as follows:

$ ' & ' % xa ´1x. a 2 . x ´1ax ´1 " xa 2 x ´1 " a 2 xa ´1x. b. x ´1ax ´1 " xyx ´1 " xyx ´1y ´1y " x 2 y " a 2 b xa ´1x. a 2 b. x ´1ax ´1 " a 4 b " b ´1.
(3.12)

Now xa ´1x is of infinite order, so we conclude from equation (3.12) that the subgroup @ Γ Y xa ´1x (D is isomorphic to Q 8 ¸2 Z. Furthermore, this subgroup is contained (strictly) in G 1 , so is non maximal.

(c) The existence of subgroups of B 4 pS 2 q isomorphic to Q 16 ˚Q8 Q 16 that are non maximal as virtually cyclic subgroups is actually a consequence of the structure of the amalgamated product. Indeed, consider the following short exact sequence:

1 ÝÑ Q 8 ÝÑ Q 16 ˚Q8 Q 16 p ÝÑ Z 2 ˚Z2 ÝÑ 1.
Now Z 2 ˚Z2 is isomorphic to the infinite dihedral group Dih 8 " Z ¸Z2 . So for all n P N, n ě 2, the subgroup nZ ¸Z2 is abstractly isomorphic to Z ¸Z2 while being a proper subgroup (in other words, it is non co-Hopfian). Thus p ´1pnZ ¸Zq is isomorphic to Q 16 ˚Q8 Q 16 while being a proper subgroup (of index n). In particular, since B 4 pS 2 q contains a subgroup Γ that is isomorphic to Q 16 ˚Q8 Q 16 , Γ admits proper subgroups that are also isomorphic to Q 16 ˚Q8 Q 16 , and any one of these subgroups is a non-maximal virtually cyclic subgroup that is isomorphic to Q 16 ˚Q8 Q 16 . Conversely, let G be a subgroup of B 4 pS 2 q that is isomorphic to Q 16 ˚Q8 Q 16 . By Proposition 42, G is a contained in a subgroup M of B 4 pS 2 q that is maximal as a virtually cyclic subgroup. But Theorem 39 implies that the only isomorphism class of infinite virtually cyclic subgroups of B 4 pS 2 q that contains Q 16 is Q 16 ˚Q8 Q 16 , and so we conclude that M -Q 16 ˚Q8 Q 16 , which completes the proof. This proves parts (c) and (d) of Theorem 41, with the exception of the statement of part (c) that pertains to the existence of maximal virtually cyclic subgroups in the case j " 1.

Proof of the existence of maximal subgroups Q 8 ˆZ in part (c) of Theorem 41

We now complete the proof of Theorem 41(c) by proving the existence of maximal virtually cyclic subgroups of B 4 pS 2 q that are isomorphic to Q 8 ˆZ.

Proposition 46. The group B 4 pS 2 q contains maximal virtually cyclic subgroups that are isomorphic to Q 8 ˆZ.

In order to prove Proposition 46, we will first require two lemmas. As before, let Q denote the normal subgroup @ α 2 0 , ∆ 4 D of B 4 pS 2 q, and let H 1 " x∆ 4 y, H 2 " @ α 2 0 D and H 3 " @ α 2 0 ∆ 4 D be the three subgroups of Q isomorphic to Z 4 . Then B 4 pS 2 q acts transitively on the set H " tH 1 , H 2 , H 3 u by conjugation, and this action gives rise to the permutation representation ψ : B 4 pS 2 q ÝÑ S 3 that satisfies the following relation: for all 1 ď i, j ď 3, and for all β P B 4 pS 2 q, `βH i β ´1 " H j ˘ðñ `ψpβqpiq " j ˘.

Note that the homomorphism ψ is surjective, that ψpσ 1 q " p1, 2q by equation (3.13), and that ψpσ 2 q " p2, 3q by equation (3.14). Since σ 1 σ ´1 3 P Q by equation (3.8), and the action of the elements of Q on H is trivial, it follows that ψpσ 3 q " ψpσ 1 q. If β is of infinite order then xQ Y tβuy -Q 8 ¸Z, and the order of the action of Z on Q 8 is that of the element ψpβq. The first step is to describe Ker pψq whose elements of infinite order will give rise to subgroups of B 4 pS 2 q isomorphic to Q 8 ˆZ. Lemma 47. Ker pψq is isomorphic to the direct product of Q with a free group F 2 px, yq of rank 2, for which a basis px, yq is given by:

x " α 2 0 ∆ 4 σ 2 1 and y " ∆ 4 σ 2 2 .

(3.23)

Proof. By Remarks 36(b) and Proposition 37, B 4 pS 2 q is isomorphic to the group T ˚˚Q 8 Q 16 , where the T ˚-factor G 1 of B 4 pS 2 q is generated by Q and α 2 1 , and the Q 16 -factor G 2 of B 4 pS 2 q is generated by Q and α 0 , so G 1 X G 2 " Q. Consider the canonical projection:

ρ : B 4 pS 2 q ÝÑ B 4 pS 2 q{Q.
As in the proof of Proposition 37, we identify the quotient B 4 pS 2 q{Q with the free product Z 3 ˚Z2 , the Z 3 -(resp. Z 2 -) factor being generated by a " ρpα 1 q (resp. b " ρpα 0 q). Consider the surjective homomorphism p ψ : Z 3 ˚Z2 ÝÑ S 3 defined by p ψpaq " p1, 3, 2q and p ψpbq " p1, 3q. Since ψpα 0 q " ψpσ 1 σ 2 σ 3 q " p1, 2qp2, 3qp1, 2q " p1, 3q, ψpα 1 q " ψpσ 1 σ 2 σ 2 3 q " p1, 2qp2, 3q " p1, 3, 2q and B 4 pS 2 q " xα 0 , α 1 y by [START_REF] Gonçalves | The braid group B m,n pS 2 q and a generalisation of the Fadell-Neuwirth short exact sequence[END_REF]Theorem 3], it follows that p ψ ˝ρ " ψ, so ρ induces a homomorphism p ρ : Ker pψq ÝÑ Ker `p ψ ˘of the respective kernels. We thus obtain the following commutative diagram of short exact sequences:

1 1 Ker pp ρq / / Q 1 / / Ker pψq / / p ρ B 4 pS 2 q ρ ψ / / S 3 / / 1 1 / / Ker `p ψ ˘/ / Z 3 ˚Z2 p ψ / / S 3 / / 1, 1 1 (3.24)
as well as the equality Ker pp ρq " Q. Taking 1, a, a 2 , b, ab, a 2 b ( to be the Schreier transversal for p ψ and applying the Reidemeister-Schreier rewriting process [START_REF] Johnson | Presentation of groups[END_REF], we see that Ker `p ψ ˘is a free group of rank 2 with basis `pabq 2 , pbaq 2 ˘, which implies that Ker pψq -Q 8 ¸F2 by the commutative diagram (3.24). To determine the action of Ker `p ψ ˘on Q, note by (3.13) and (3.14) that σ 2 1 and σ 2 2 belong to Ker pψq, and that: ρpσ 2 1 q " ρpσ 2 3 q " `ρpα ´1 0 α 1 q ˘2 " pbaq 2 ρpσ 2 2 q " ρpα 0 σ 2 1 α ´1 0 q " pabq 2 , so p ρpσ 2 1 q " pbaq 2 and p ρpσ 2 2 q " pabq 2 . The same equations imply that the actions by conjugation of σ 2 1 and σ 2 2 on Q yield elements of Inn pQq, namely conjugation by α 2 0 ∆ 4 and by ∆ 4 respectively. Let s : Ker `p ψ ˘ÝÑ Ker pψq be the section for p ρ defined on the basis of Ker `p ψ by s `pbaq 2 ˘" x and s `pabq 2 ˘" y. The action of these two elements on Q is thus trivial, which shows that Ker pψq -Q 8 ˆF2 as required.

Using the definition of ψ, a transversal of Ker pψq in B 4 pS 2 q is seen to be:

T " te, σ 1 , σ 2 , σ 1 σ 2 σ 1 , σ 1 σ 2 , σ 2 σ 1 u . ( 3.25)
We now determine the action by conjugation of these coset representatives on x and y.

for all λ P Zz t0u. We conclude that if ∆ is a subgroup of Γ " xQ Y tzuy -Q 8 ¸j Z that is isomorphic to Q 8 ˆZ then ∆ " @ Q Y q. z λj (D , where z λj P pa, bq P Z 2 abpa ´bq " 0 ( relative to the basis px, yq of the Abelianisation Z 2 of F 2 px, yq.

To complete the proof of the proposition, we shall exhibit an element w P B 4 pS 2 q for which: (a) w is a non-trivial element of F 2 px, yq such that w " pc, dq, where cdpc ´dq ‰ 0. (b) πpwq R @ 3 D . Since F 2 px, yq Ă Ker pψq, the first condition implies that such an element w is a suitable generator of the Z-factor of a subgroup of B 4 pS 2 q that is isomorphic to Q 8 ˆZ, but which from the above discussion, is not contained in any subgroup that is isomorphic to Q 8 ¸j Z for j P t2, 3u. By Lemma 44(b), the second condition implies that xQ Y twuy is not contained in any subgroup of B 4 pS 2 q that is isomorphic to

Q 16 ˚Q8 Q 16 , Q 8 ˚Z4 Q 8 , Z 8 ˚Z4 Z 8 or Q 8 ˚Z4 Z 8 .
Take w " xy 3 , and let ∆ " xQ Y twuy. Then ∆ -Q 8 ˆZ since w P Ker pψq is an element of infinite order, and by Proposition 42, there exists a maximal infinite virtually cyclic subgroup M of B 4 pS 2 q that contains ∆.

Conjugacy classes of maximal infinite virtually cyclic

subgroups in B 4 pS 2 q

In order to determine the number of conjugacy classes of maximal infinite virtually cyclic subgroups, we follow the procedure given in [54, Section 2.5] based on the action of B 4 pS 2 q on a suitable tree. As in the proof of Proposition 37, we identify B 4 pS 2 q with Q 16 ˚Q8 T ˚, and the quotient B 4 pS 2 q{Q with the modular group PSLp2, Zq -Z 2 ˚Z3 " @ a, b a 2 " b 3 " 1 D . Thus we have the following short exact sequence:

1 ÝÑ Q ÝÑ B 4 pS 2 q ρ ÝÑ Z 2 ˚Z3 ÝÑ 1, (3.26)
ρ being the quotient map as in the proof of Proposition 37. There is a well-known action of PSLp2, Zq on the tree T of Figure 3.1, where the edge stabilisers are trivial and the vertex stabilisers are Z 2 and Z 3 . The quotient of T by this action is the graph:

Z 2 ' ' Z 3 .
It follows from the short exact sequence (3.26) that B 4 pS 2 q acts on T via ρ, and since Ker pρq -Q 8 , the quotient graph of this action is: We now apply the Reidemeister-Schreier rewriting process to the Abelianisation homomorphism r π : Z 2 ˚Z3 ÝÑ Z 6 . A computation similar to that given in the proof of Lemma 47 shows that the commutator subgroup Γ 2 pZ 2 ˚Z3 q of Z 2 ˚Z3 is a free group, which we denote by F 2 , of rank two with basis `ra, bs, ra, b 2 s ˘.

Q 16 ' Q 8 ' T ˚.
Proposition 49. Let r F " ρ ´1pF 2 q. Then there exists a (free) subgroup r F k of r F of rank k ě 2 that is normal and of finite index in B 4 pS 2 q.

Remark 50. The above construction gives rise to the following commutative diagram of short exact sequences: 1

1 1 / / Q / / r F ρ| r F / / F 2 / / 1 1 / / Q / / B 4 pS 2 q π ρ / / Z 2 ˚Z3 r π / / 1. Z 6 Z 6 1 1
We see that Γ 2 pB 4 pS 2 qq " r F -Q 8 ¸F2 , which yields an alternative proof of the decomposition given in [44, Theorem 1.3(3)]. groups of ZrΓs are described by the Bass-Heller-Swan formula with α " 1, which asserts that for a finite group π, there is a natural decomposition [6]: K i pZrπ ˆZsq " K i pZrπsq ' K i´1 pZrπsq ' 2 NK i pZrπsq for all i P Z,

where the i th Bass Nil group of π, denoted by NK i pZrπsq, is defined to be the kernel of the homomorphism in K-groups induced by the evaluation e : Zrπsrts ÝÑ Zrπs at t " 0. In the reduced version, equation (4.1) takes the form:

Whpπ ˆZq " Whpπq ' r K 0 pZrπsq ' 2 NK 1 pZrπsq, and r K 0 pZrπ ˆZsq " r K 0 pZrπsq ' K ´1pZrπsq ' 2 NK 0 pZrπsq.

If α ‰ 1, the group ring ZrΓs is equal to ZrF ¸α Zs -ZrFs α rt, t ´1s, the latter being the twisted Laurent polynomial ring of ZrFs, and the twisting is given by the action of α. In this case, the Bass Nil groups are replaced by the Farrell-Hsiang Nil groups NK i pZrFs, αq ' NK i pZrFs, α ´1q [START_REF] Farrell | The Whitehead group of poly-(finite or cyclic) groups[END_REF].

For virtually cyclic groups Γ of Type II, the fundamental work of Waldhausen gives rise to the following exact sequence [START_REF] Waldhausen | Algebraic K-theory of generalized free products, Part 1[END_REF]:

¨¨¨ÝÑ K n pZrFsq ÝÑ K n pZrG 1 sq ' K n pZrG 2 sq ÝÑ K n pZrΓsq{ Nil W n ÝÑ K n´1 pZrFsq ÝÑ K n´1 pZrG 1 sq ' K n´1 pZrG 2 sq ÝÑ K n´1 pZrGsq{ Nil W n´1 ÝÑ ¨¨¨,
where Nil W n denotes the Waldhausen Nil groups, denoted in [START_REF] Waldhausen | Algebraic K-theory of generalized free products, Part 1[END_REF] by: Nil W n " Nil W n pZrFs; ZrG 1 zFs, ZrG 2 zFsq.

If Γ is an infinite virtually cyclic group of Type II, there is a surjection f : Γ Ý Dih 8 whose kernel F is finite. Let T be the unique infinite cyclic subgroup of Dih 8 of index 2. Then the subgroup r Γ " f ´1pTq Ă Γ is an infinite virtually cyclic group of Type I, and r Γ is of the form F ¸α T. In this situation, it was recently established that the Waldhausen Nil groups may be identified with the Farrell-Hsiang Nil groups as follows [START_REF] Davis | Algebraic K-theory over the infinite dihedral group: an algebraic approach[END_REF][START_REF] Lafont | Relating the Farrell Nil-groups to the Waldhausen Nil-groups[END_REF]:

Nil W
n " Nil W n pZrFs; ZrG 1 zFs, ZrG 2 zFsq -NK n pZrFs, αq -NK n pZrFs, α ´1q.

In negative degrees, the Nil groups are described as follows.

Theorem 51 ([34, Theorem 2.1]). Let Γ be an infinite virtually cyclic group. Then: (a) K ´1pZrΓsq is a finitely-generated Abelian group. (b) K ´1pZrΓsq is generated by the images of K ´1pZrFsq under the maps induced by the inclusions F Ă Γ, where F runs over the representatives of the conjugacy classes of finite subgroups of Γ. (c) K ´ipZrΓsq " 0 for all i ě 2.

In summary, in order to compute the K-groups of an infinite virtually cyclic group, we need to understand the K-groups of the corresponding finite kernel F and of the associated Bass or Farrell-Hsiang Nil groups. Now R{M, S{M and S are all regular rings, so their corresponding Nil groups vanish, and it follows from the associated Mayer-Vietoris sequence that NK 0 pRq " 0. On the other hand, using the fact that 0 " NK 1 pR{Mq " NK 1 pS{Mq " NK 1 pSq and the description of the double relative group K 1 pR, S, Mq given in [37, Theorem 1.1], we obtain the isomorphism: NK 1 pR, S, Mq -NK 1 pRq.

It follows from this that NK 1 pRq is trivial by (4.4). This completes the proof of the theorem.

To prove Proposition 52, we will require some general properties of Nil groups.

Remark 54. We recall the following facts about the Nil groups NK i and Nil W i for all i P Z: In what follows, if m P N, C m will denote the cyclic group of order m.

Proof of Proposition 52. We first consider the case i " 0. Let Q 8 be equipped with the following presentation: Q 8 " @ x, y x 2 " y 2 , yxy ´1 " x

´1D

.

By [START_REF] Curtis | with applications to finite groups and orders[END_REF]Theorem 50.31,p. 266], the group ring ZrQ 8 s fits into the following Cartesian square:

ZrQ 8 s f / / q ZrC 2 ˆC2 s p R / / F 2 rC 2 ˆC2 s, (4.5)
where q is defined on the generators of Q 8 by qpxq " i and qpyq " j, and f is induced by the homomorphism Q 8 ÝÑ C 2 ˆC2 given by taking the quotient of Q 8 by its centre. The Cartesian square (4.5) gives rise to the following Mayer-Vietoris sequence: Since F 2 rC 2 ˆC2 s -F 2 rε, νs{pε 2 , ν 2 q, the ideal I " xε, ν, ενy is nilpotent in F 2 rε, νs, and it follows that NK 0 pF 2 rC 2 ˆC2 sq -NK 0 pF 2 q " 0, which yields equation (4.7). In a similar fashion, we have NK 0 pF 2 rC 2 sq " 0. As Abelian groups, Ω F 2 and V are both countable infinite direct sums of copies of Z 2 . As we saw in Theorem 53, the ring R has trivial Nil groups in degrees 0 and 1. On the other hand, observe that F 2 rC 2 ˆC2 s -F 2 rε, νs{pε 2 , ν 2 q, hence by [ ÝÑ NK 0 pZrC 2 ˆC2 sq ÝÑ 0.

NK
Since both NK 1 pF 2 rC 2 ˆC2 sq and NK 0 pZrC 2 ˆC2 sq are non-trivial infinite sums of copies of Z 2 , it follows by exactness that NK 0 pZrQ 8 sq is an infinite direct sum of copies of Z 2 , Z 4 or Z 2 ' Z 4 .

We now turn to the case i " 1. Consider the following homomorphism:

f ˚1 : NK 1 pZrQ 8 sq ÝÑ NK 1 pZrC 2 ˆC2 sq.

Exactness of the sequence (4.8) and the fact that p ˚is the trivial homomorphism imply that f ˚1 is a surjection. On the other hand, since once more both NK 2 pF 2 rC 2 ˆC2 sq and NK 1 pZrC 2 ˆC2 sq are infinite direct sum of copies of Z 2 , the latter by [2, Theorem 1.2], the result follows as before.

In order to complete the proof of Theorem 1, it remains to determine the twisted Nil groups of Q 8 . Recall from Section 3.2 that up to isomorphism, there are two non-trivial semi-direct products of the form Q 8 ¸Z, namely Q 8 ¸j Z, where j P t2, 3u. In what follows, we shall use the notation NK α i pZrQ 8 sq " NK i pZrQ 8 s, αq. Proposition 55. Let i P t0, 1u. Proof.

(a) Since the action of Z on Q 8 is of order three, there is a surjective homomorphism ϕ : Q 8 ¸3 Z Ý Q 8 ¸Z3 -T ˚defined by taking the Z-factor modulo 3. We use the technique of induction on hyper-elementary subgroups [32, proof of Theorem 3.2] that asserts that:

NK i pZrQ 8 ¸3 Zsq -lim HPHyp NK i pZrϕ ´1pHqsq,
where Hyp denotes the set of hyper-elementary subgroups of Q 8 ¸Z3 , and the limit is with respect to the morphisms induced by conjugation and inclusion in the category Hyp. Following the proof of Proposition 18, we see that the hyper-elementary subgroups of Q 8 ¸Z3 are isomorphic to one of Z 6 , Z 3 , Z 2 , Z 4 or Q 8 , and their inverse images by ϕ are isomorphic to Z 2 ˆZ, Z, Z 2 ˆZ, Z 4 ˆZ and Q 8 ˆZ respectively. With the exception of the last two, the corresponding group rings of these groups have trivial Nil groups. Further, the subgroups of Q 8 ¸3 Z that are isomorphic to Z 4 ˆZ are pairwise conjugate, and there is only one maximal element of the form Q 8 ˆZ in the limit. We thus obtain equation (4.9) using Proposition 52. (b) Consider the action α of Z on Q 8 of order 2 given by exchanging the generators x and y of Q 8 . Comparing with the Cartesian square (4.5), we observe that this action may be transposed in all the rings of (4.5), thus giving rise to the following Cartesian square of twisted polynomial rings:

ZrQ 8 s α rts f ÝÝÝÑ ZrC 2 ˆC2 s α rts q § § đ § § đ p R α rts ÝÝÝÑ F 2 rC 2 ˆC2 s α rts,
where the induced action of α exchanges the generators in all group rings, and exchanges i and j in R. By [START_REF] Farrell | The obstruction to fibering a manifold over a circle[END_REF]Theorem 1.6], the Farrell-Hsiang group NK s of R also vanishes for s " 0, 1. Moreover, let I " xε, νy be the nilpotent ideal generated by ε and ν in F 2 rC 2 ˆC2 s α rts - F 2 rε, νs α rts since F 2 rε, νs α rts{I -F 2 . By an argument similar to that given in the proof of Proposition 52, it follows that NK α 0 pF 2 rC 2 ˆC2 sq " 0. Hence we obtain the following long exact sequence: finite number (possibly zero) of punctures, the n-string braid group B n pMq may be defined geometrically simply by replacing D 2 by M. The subgroup P n pMq of n-string pure braids is defined in a manner similar to that for P n . A number of presentations of B n pMq and P n pMq may be found in the literature, see [START_REF] Bellingeri | On presentations of surface braid groups[END_REF][START_REF] Gonçalves | On the structure of surface pure braid groups[END_REF][START_REF] González-Meneses | New presentations of surface braid groups[END_REF] for example. Braid groups may also defined topologically in terms of configuration spaces as follows. Let F n pMq denote the n th configuration space of M defined by: F n pMq " pp 1 , . . . , p n q P M n | p i ‰ p j for all i, j P t1, . . . , nu, i ‰ j ( .

NK
We equip F n pMq with the topology induced by the product topology on M n . A transversality argument shows that F n pMq is a connected 2n-dimensional open manifold. There is a natural free action of the symmetric group S n on F n pMq given by permutation of coordinates.

The resulting orbit space F n pMq{S n shall be denoted by D n pMq, the n th permuted configuration space of M, and may be thought of as the configuration space of n unordered points. The associated canonical projection p : F n pMq ÝÑ D n pMq is thus a regular n!-fold covering map [52, p. 14]. Fox and Neuwirth showed that P n pMqπ 1 pF n pMqq and B n pMqπ 1 pD n pMqq [START_REF] Fox | The braid groups[END_REF]. If n " 1 then F 1 pMq " M, and thus B 1 pMq " P 1 pMq " π 1 pMq, so braid groups generalise the notion of fundamental group. The map p gives rise to the following short exact sequence:

1 ÝÑ P n pMq ÝÑ B n pMq

p Ý Ñ S n ÝÑ 1. (B.5)
In the case where M is the disc, p ˚is the surjective homomorphism σ described on page 63. This topological definition is very useful in practice, and may be used as follows to obtain fibrations involving the configurations spaces, and (short) exact sequences bringing into play the homotopy groups of these spaces. Suppose that M is a surface with empty boundary, and let m ą n ě 1. Then the map p m,n : F m pMq ÝÑ F n pMq given by p m,n px 1 , . . . , x m q " px 1 , . . . , x n q that forgets the last m ´n coordinates is a locally-trivial fibration, known as the Fadell-Neuwirth fibration, with fibre F m´n pMz tz 1 , . . . , z n uq, where pz 1 , . . . , z n q is a basepoint of F n pMq [START_REF] Fadell | Configuration spaces[END_REF]. The fibre is known to be an Eilenberg-Mac Lane space of type Kpπ, 1q. Taking the long exact sequence in homotopy of the fibration, and using Fox and Neuwirth's isomorphisms mentioned above, we obtain the Fadell-Neuwirth short exact sequence of surface pure braid groups:

1 ÝÑ π 1 pF m´n pMz tz 1 , . . . , z n uqq ÝÑ P m pMq pp m,n q Ý ÝÝÝÑ P n pMq ÝÑ 1.

(B.6)

The homomorphism pp m,n q ˚induced by the map p m,n may be visualised geometrically as the map that 'forgets' the last m ´n strings of a braid in P m pMq. Due to the fact that the higher homotopy groups of the braid groups of S 2 and RP 2 are non trivial, in order to obtain the short exact sequence (B.6) for these two surfaces, we need to suppose additionally that n ě 3 (resp. n ď 2). In particular, if m " n `1, then (B.6) becomes:

1 ÝÑ π 1 pMz tz 1 , . . . , z n uq ÝÑ P n`1 pMq pp n`1,n q Ý ÝÝÝÝÝ Ñ P n pMq ÝÑ 1.

(B.7)

The braid groups of S 2 and RP 2 are of particular interest, partly because they are the only surface braid groups to possess torsion, and as we explained in the introduction, the methods of [START_REF] Aravinda | Algebraic K-theory of pure braid groups[END_REF][START_REF] Farrell | The Whitehead groups of braid groups vanish[END_REF] cannot be applied to study their lower algebraic K-theory. The isomorphism classes of the maximal finite subgroups of B n pS 2 q are given in Theorem 2. An analogous

  6u, and g P T ˚is of order d then g and g ´1 are representatives of the two conjugacy classes of elements of order g. (b) If G " O ˚, νpdq, r 0 pdq, r 1 pdq and r 2 pdq are given by: If d " 4, and g 1 and g 2 are elements of O ˚of order 4 such that g 1 P T ˚and g 2 R T ˚then g 1 and g 2 are representatives of the two conjugacy classes of elements of order 4. If d " 8, and g P T ˚is of order 8, g and g 3 are representatives of the two conjugacy classes of elements of order 8. (c) If G " I ˚, νpdq, r 0 pdq, r 1 pdq and r 2 pdq are given by:

From [ 85 ,

 85 Example 3, Section 6], there are representations of Dic 4µ of the form U p2q α,0 , where 0 ď α ď 2µ ´1. Such representations are defined in [85, equation (8), p. 214] and induced by linear characters. Using [85, Proposition 5], the representation U p2q 1,0 gives rise to an irreducible representation of QrDic 4µ s, and the last part of [85, Example 3, Section 6] implies that its Schur index is equal to two. Since the Schur index of each of the first four components of equation (2.13) is equal to one, it follows that the simple component H 2µ of QrDic 4µ s corresponds to U p2q 1,0 . We now apply [85, Proposition 9] to U p2q 1,0 . Within our framework, the enveloping algebra env Q ´Up2q 1,0 ¯with respect to Q is isomorphic to the simple component H 2µ , and the centre E 2µ of H 2µ is isomorphic to Q ´χp2q 1,0 ¯, χ p2q 1,0 being the character of U p2q 1,0 [84, Introduction]. With the notation of [85, Proposition 9], we have d 1 "

  Remarks 19. (a) The induction/restriction arguments in the proof of Proposition 18 were inspired by those given in [73, Paragraph 14] for the r K 0 -groups. (b) Let G " O ˚or I ˚. We sketch an alternative proof of the fact that K ´1pZrGsq has non-trivial torsion that uses [73, Proposition 4.11]. The embedding of G in the Hamilton quaternions H [20, Chapter 7] induces an algebra homomorphism ψ G : QrGs ÝÑ H. By [73, Proposition 4.11 and its proof]

  Remarks 21. (a) It follows from the definition that r f s F " ď tPImpφq " f t ‰ , (2.24) in other words, an F-conjugacy class is a union of normal conjugacy classes. In particular, r f s F Ą r f s. Further, the number of F-conjugacy classes of elements of order n is bounded above by the number of usual conjugacy classes of elements of order n. (b) If F " Q then φ is an isomorphism [70, Theorem 1.5], and f , g P G are F-conjugate if and only if x f y and xgy are conjugate subgroups of G.

Theorem 22 (

 22 [START_REF] Curtis | I, with applications to finite groups and orders[END_REF] Theorems 21.5 and 21.25]). Let G be a finite group, and let F be a field of characteristic p ě 0, where p is prime if p ą 0. (a) If p " 0, then r F is equal to the number of F-conjugacy classes in G. (b) If p ą 0, then r F is equal to the number of p-regular F-conjugacy classes in G.

Proposition 26 .

 26 Let m and λ be defined as in the statement of Theorem 25. Then λ "

  10) and Theorem 22. From Remark 21(b) and the notation of Section 2.2, r Q " ÿ d |G| r 2 pdq, and it follows from Proposition 7 that r Q " 5 if G " T ˚, and r Q " 7 if G " O ˚or I ˚. These values of r Q may also be obtained from the corresponding Wedderburn decompositions given in (2.18)-(2.20).

3 )

 3 denote the 'half twist' braid on four strings. It is a square root of the full twist braid ∆ 2 4 described in (B.3) (see also Figures B.2

  (

@ α 2 0 D and xα 2 y [ 47 ,

 047 Proposition 1.5(2) and Theorem 1.6].

  Clearly condition (a) above holds, and equation (3.23) implies that condition (b) is also satisfied. It follows from the previous paragraph and Theorem 39(b) that M -Q 8 ˆZ, which completes the proof of Proposition 46. In conjunction with Proposition 45, this also proves parts (c) and (d) of Theorem 41. This proves part (c) of Theorem 41 in the the exceptional case, and bringing together Propositions 43, 45 and 46, completes the proof of Theorem 41.

Figure 3 . 1 :

 31 Figure 3.1: The tree T, showing the edge and vertex stabilisers under the action of PSLp2, Zq.

. 4 ) 53 .

 453 Notice that the group generated by u and w is isomorphic to Z 2 ' Z 2 . On the other hand, b is the only non-trivial element of S{R, b 2 " b and the relations in (4.3) become: b b z " 0. These identities imply that K 1 pR, S, Mq " 0, and so by [81, Lemma 2.1], we obtain:0 " NK 1 pR, S, Mq -K 1 pR, S, Mq b xZrxs. (4Theorem The groups NK 0 pRq and NK 1 pRq are trivial.Proof. As mentioned above, R{M and S{M are the fields of two and four elements respectively. Consider the following commutative square, where the right-hand vertical morphism is surjective: R ÝÝÝÑ S § § đ § § đ R{M ÝÝÝÑ S{M.

  (a) if NK i (resp. Nil W i ) is non trivial, it is an infinitely-generated group [64, Theorem A]. (b) if A Ă NK i (resp. A Ă Nil W i ) is a finite subgroup then NK i (resp. Nil W i ) contains an infinite direct sum of copies of A [64, Theorem B].

  (a) For the action α of Z on Q 8 of order 3,NK α i pZrQ 8 sq -NK α ´1 i pZrQ 8 sq -NK i pZrQ 8 sq.(4.9)(b) For the action α of Z on Q 8 of order 2, the twisted Nil groups are isomorphic to infinitely many copies of Z 2 , Z 4 or Z 2 ' Z 4 .

  ' 2M 3 pQq ' H 8 ' M 2 p p

	Hq, and	(2.19)

  [START_REF] Berrick | Braids: Introductory Lectures on Braids, Configurations and Their Applications[END_REF] s, and since K ´1pZrDic 12 sq is torsion free by Proposition 16, H 6 does not contribute to the torsion of K ´1pZrDic 36 sq. It thus suffices to determine the Schur and local Schur indices of the remaining factor H 18 . Following the proof of Proposition 16, we obtain m " d 1 " 18, µ " h " 9, r " 17 and u " s " v 1 " 2, and the representation U QrDic 36 s whose Schur index is equal to two. Since the Schur index of each of the first four components of equation(2.13) is equal to one, it follows that the simple component H 18 of QrDic 36 s corresponds to U p2q 1,0 . If p is a finite prime of the centre E 9 of H 18 that divides 36, then p divides 6, and hence p divides p, where p P t2, 3u. If p 3, then p " q " 3, b " 0, z " a " e p " c p " 2, t 1 " r f " f " 1, and Λ p " 2. Thus the Schur index of the simple component H 18 of the decomposition (2.22) of QrDic 36 s is equal to 2, but its local Schur indices at every finite prime of the centre of H 18 are not always odd. It follows from Theorem 15 that K ´1pZrDic 36 sq is torsion free.

	an irreducible representation of	p2q 1,0 gives rise to

  where n P t1, . . . , 11, 14u, and is non trivial if G is isomorphic to Z n , where n P t12, 16, 18, 20, 22u. By [73, page 126, line 16], if G is isomorphic to Z 12 then r K 0 pZrZ 12 sq -Z 2 . Suppose that G " Z 16 . From [58, Page 416], there is a surjective homomorphism from r K 0 pZrZ 16 sq to ś 4

	ν"1

  and the element a ´1x is a product of elements chosen alternately from the two sets xa, by z If i " 2, a similar computation shows that conjugation by a ´1x permutes cyclically a 2 , b ´1 and a ´2b ´1, and thus H 2

						@	a 2 , b	D	and xx, yy z	@	x 2 , y	D	, so is of infinite order by standard properties of
	amalgamated products. Consider the subgroup H i "	@	Γ i Y a ´1x	(D	of G i . One may check
	that	@	a ´1x	D	acts by conjugation on	@	a 2 , b	D	. If i " 1 then:
			$ '	a ´1x. a 2 . x ´1a " a 2
			&
			'
			%

a ´1x. b. x ´1a " a ´1xyx ´1a " a ´1xyx ´1y ´1ya " a ´1x 2 ya " abab ´1b " b a ´1x. a 2 b. x ´1a " a 2 b,

(3.11)

and hence H 1 -Q 8 ˆZ.

  2 pZrQ 8 sq ÝÑ NK 2 pRq ' NK 2 pZrC 2 ˆC2 sq ÝÑ NK 2 pF 2 rC 2 ˆC2 sq ÝÑ NK 1 pZrQ 8 sq ÝÑ ¨¨¨. (4.6) By [61, Lemmas 5.3 and 5.4], [81, Theorem 1.3] and [82, Lemma 2.2], we have:NK 1 pZrC 2 ˆC2 sq -Ω F 2 rxs -F 2 rxs dx, NK 0 pZrC 2 ˆC2 sq -V " xF 2 rxsNK 1 pF 2 rC 2 sq -p1 `xεF 2 rxsq ˆ-V, NK 0 pF 2 rC 2 sq " 0 NK 0 pF 2 rC 2 ˆC2 sq " 0.(4.7)

  6, Proposition 7.8] and[START_REF] Martin | Nilgroups of Finite Abelian Groups[END_REF] Theorem 3.3], we have:NK 1 pF 2 rC 2 ˆC2 sq -p1 `xεF 2 rxsq ˆˆp1 `xνF 2 rxsq ˆˆp1 `xενF 2 rxsq ˆ-V 3 .The Mayer-Vietoris sequence (4.6) thus reduces to:NK 2 pZrQ 8 sq ÝÑ NK 2 pRq ' NK 2 pZrC 2 ˆC2 sq ÝÑ NK 2 pF 2 rC 2 ˆC2 sq NK 1 pZrC 2 ˆC2 sq ÝÑ NK 1 pF 2 rC 2 ˆC2 sq is trivial since ZrC 2 ˆC2 s is reduced and NK 1 pF 2 rC 2 ˆC2 sq is Artinian [66, Theorem 3.3].Thus the part of (4.8) involving NK 0 pZrQ 8 sq is:0 ÝÑ NK 1 pF 2 rC 2 ˆC2 sq

	(4.8)
	f ˚0 ÝÑ NK 0 pZrC 2 ˆC2 sq ÝÑ 0
	(the labelled homomorphisms are discussed below). The homomorphism
	p ˚:

δ ÝÑ NK 1 pZrQ 8 sq p f ˚1q ÝÑ NK 1 pZrC 2 ˆC2 sq p Ý Ñ NK 1 pF 2 rC 2 ˆC2 sq τ ÝÑ NK 0 pZrQ 8 sq τ ÝÑ NK 0 pZrQ 8 sq f ˚0

  α 2 pZrQ 8 sq ÝÑ NK α 2 pRq ' NK α 2 pZrC 2 ˆC2 sq ÝÑ NK α 2 pF 2 rC 2 ˆC2 sq

δ ÝÑ NK α 1 pZrQ 8 sq ÝÑ NK α 1 pZrC 2 ˆC2 sq p Ý Ñ NK α 1 pF 2 rC 2 ˆC2 sq ÝÑ NK α 0 pZrQ 8 sq ÝÑ NK α 0 pZrC 2 ˆC2 sq ÝÑ 0. (4.10)

Acknowledgements

manuscript, and one referee in particular, who made a number of useful remarks and suggestions to make it easier to read.

The authors are grateful to the French-Mexican Laboratoire International Associé LAISLA for partial financial support. D. Juan-Pineda was partially supported by the CNRS, PAPIIT-UNAM and CONACYT (México), and J. Guaschi was partially supported by the ANR project TheoGar n o ANR-08-BLAN-0269-02. J. Guaschi would like to thank the CNRS for having granted him a 'délégation' during the writing of part of this paper. J. Guaschi and S. Millán also wish to thank CONACYT (México) for partial financial support through its programme 'Estancias postdoctorales y sabáticas vinculadas al fortalecimiento de la calidad del posgrado nacional'.

(b) First let j " 2. Consider the subgroup H " xQ Y tσ 1 uy of B 4 pS 2 q. By Proposition 42, H is contained in a maximal virtually cyclic subgroup M of B 4 pS 2 q. Since Q is normal in B 4 pS 2 q and σ 1 is of infinite order, H must be isomorphic to a semi-direct product of the form Q 8 ¸k Z for some k P t1, 2, 3u. To determine k, we study the action by conjugation of σ 1 on Q. Using equation (3.8), we have:

by equation (3.7)

by equation (3.6) " α ´2 0 by equation (3.8)

(3.13)

Since σ 1 is of infinite order, H is thus isomorphic to Q 8 ¸2 Z because the action fixes the subgroup @ α 2 0 ∆ 4 D of order 4 of Q, and exchanges @ α 2 0 D and ∆ 4 . But πpσ 1 q " 1 R @ 3 D , so H is not contained in any subgroup of the form Q 8 ˚Z4 Q 8 , Q 8 ˚Z4 Z 8 or Q 16 ˚Q8 Q 16 by Lemma 44(b). It cannot be contained either in a subgroup isomorphic to Q 8 ˆZ or Q 8 ¸3 Z because the actions on Q are not compatible. This implies that M, which is maximal in B 4 pS 2 q as a virtually cyclic subgroup, must also be isomorphic to Q 8 ¸2 Z. Now let j " 3. As in the case j " 2, if there exists a subgroup L of B 4 pS 2 q that is isomorphic to Q 8 ¸3 Z, it cannot be contained in a subgroup of B 4 pS 2 q isomorphic to Q 8 ˆZ or to Q 8 ¸2 Z. Moreover, by Lemma 44(b), if πpLq Ć @ 3 D then L is not contained in any subgroup of B 4 pS 2 q isomorphic to Q 8 ˚Z4 Q 8 , Z 8 ˚Z4 Z 8 , Q 8 ˚Z4 Z 8 or Q 16 ˚Q8 Q 16 . As in the previous paragraph, we conclude using Proposition 42 that L is contained in a maximal virtually cyclic subgroup of B 4 pS 2 q that must also be isomorphic to Q 8 ¸3 Z. To prove the result, we exhibit such a subgroup L. Consider the action by conjugation of σ 2 on Q:

3 α 0 by equation (3.5) " α 0 α ´2 0 ∆ 4 α 0 by equation (3.8)

In particular, σ 4 2 xσ ´4 2 " x for all x P Q. This implies that the action by conjugation of z " σ 7 2 σ 1 on the elements of Q is the same as that of σ 3 2 σ 1 . By equations (3.13) and (3.14), this action is as follows:

Hence the action by conjugation of z on Q is of order 3. Further, πpzq " 2 R @ 3 D , which shows that L " xQ Y tzuy is not contained in any subgroup isomorphic to an amalgamated product of the form Q Lemma 44(b). Ob- serve that by (B.5), the permutation associated to z is p1, 2, 3q, and so z 3 P P 4 pS 2 q. To prove that L -Q 8 ¸3 Z, it remains to show that z is of infinite order. To achieve this, we shall write z 3 in terms of the direct product decomposition (B.10) of P 4 pS 2 q, which comes down to expressing z 3 in terms of the basis pA 1,4 , A 2,4 q of the free group π 1 pS 2 z tz 1 , z 2 , z 3 u , z 4 q that is the kernel of the homomorphism pp 4,3 q ˚of (B.9). Geometrically, this homomorphism is given by forgetting the last string (see (B.6) and (B.7)). As mentioned in Appendix B, the group P 4 pS 2 q is generated by the set A i,j ( 1ďiăjď4 , where A i,j is defined by (B.4), A i,i`1 " σ 2 i for i P t1, 2, 3u, and the A i,j satisfy the 'surface relations' (B.8) (the relations are not complete). For the convenience of the reader, we write out these relations in full:

(3.18)

Using (B.3) and (B.4), one may also see that:

The reader may also convince himself or herself of the validity of this relation by drawing a picture similar to that of

from relations (3.17), (3.18) and (3.20) that

and from relations (3.16) and (3.20) that

If i P t1, 2u, it follows from the braid relations in

for all k P Z. We thus obtain:

by equations (3.20) - (3.22). But pA 1,4 , A 2,4 q is a basis of the free group π 1 pS 2 z tz 1 , z 2 , z 3 u , z 4 q, so z 3 ‰ 1, and since z 3 P xA 1,4 , A 2,4 y, it is of infinite order. We conclude that L -Q 8 ¸3 Z, which completes the proof in this case.

Lemma 48. Let τ P T z teu. Then

Proof. The action by conjugation of σ 1 and σ 2 on σ 2 1 and σ 2 2 is given by:

Using also equations (3.13) and (3.14) as well as the fact that x and y commute with the elements of Q, we see that:

from which we deduce that:

We thus obtain the relations given in the statement.

Proof of Proposition 46. To prove the proposition, we must show that there exists a maximal virtually cyclic subgroup of B 4 pS 2 q that is isomorphic to Q 8 ˆZ. Let z P B 4 pS 2 q be an element of infinite order, and suppose that Γ " xQ Y tzuy -Q 8 ¸j Z, where j P t2, 3u. Our aim is to obtain necessary conditions on the generators of the infinite cyclic factor of those subgroups of Γ that are isomorphic to Q 8 ˆZ. Thus will enable us to construct subgroups of B 4 pS 2 q that are isomorphic to Q 8 ˆZ but are not contained in any subgroup isomorphic to Q 8 ¸j Z, where j P t2, 3u. With this in mind, let ∆ be a subgroup of Γ that is isomorphic to Q 8 ˆZ.

Since the finite-order elements of Γ are precisely the elements of Q, the subgroup of ∆ that is isomorphic to Q 8 is Q. The remaining elements of Γ, of the form q. z k , where q P Q and k P Zz t0u, are of infinite order. In order that such an element belong to the centraliser of Q (and thus form a subgroup isomorphic to Q 8 ˆZ), the fact that the action of z on Q is of order j implies that k must be a multiple of j, and thus

Since the action by conjugation of z on Q is of order j, it follows from the definition of ψ that z belongs to one of the cosets τ. Ker pψq of B 4 pS 2 q where τ P T z teu, T being the transversal of equation (3.25). More precisely, z P τ. Ker pψq, where τ P tσ 1 , σ 2 , σ 1 σ 2 σ 1 u if j " 2, and τ P tσ 1 σ 2 , σ 2 σ 1 u if j " 3. Further, by Lemma 47 there exist v P Ker pψq, u P F 2 px, yq and q 1 P Q such that z " τv and v " q 1 u. Let us write u " upx, yq as a freely reduced word in F 2 px, yq:

u " x ε 1 y δ 1 ¨¨¨x ε r y δ r , where ε i , δ i P Z for all i " 1, . . . , r, and δ 1 , ε 2 , . . . , δ r´1 , ε r are non-zero. If v P Ker pψq, let v denote the image of v under projection onto the F 2 px, yq-factor, followed by Abelianisation of F 2 px, yq. We now compute z j . We have that:

" q 1 pτuq j , where q 1 P Q.

pτuτ ´1qpτ 2 uτ ´2q. τ 3 . u if j " 3. Applying Lemma 48, and using equation (3.23) as well as the fact that x and y commute with the elements of Q, it follows that there exists q 2 P Q such that pτuq j " q 1´1 q 2 w, where w P F 2 px, yq is given by: $

We have also used the fact that:

by equations (3.1), (3.7) and (3.8). Since z j " q 2 w, and q 2 commutes with w, relative to the basis px, yq of the Abelianisation Z 2 of F 2 px, yq, we obtain:

Proof of Proposition 49. By Remark 50, r F is isomorphic to a semi-direct product of the form Q 8 ¸F2 . Let s : F 2 ÝÑ r F be a section for ρ r F . Since spF 2 q is of finite index in B 4 pS 2 q, it suffices to take r F k to be the intersection of the conjugates of spF 2 q in B 4 pS 2 q.

The group F 2 acts freely on T, the resulting quotient space being a graph Γ 1 that is homotopy equivalent to a wedge of two circles. The group r F k also acts freely on T in the same way as its image ρp r F k q in Z 2 ˚Z3 , the quotient graph Γ " T{ r F k being a finite-sheeted covering space of Γ 1 .

By [54, Section 2.3], there is a bijective correspondence between: (a) the maximal infinite virtually subgroups of B 4 pS 2 q, and (b) the stabilisers of geodesics in T with infinite stabiliser. In order to determine the number of conjugacy classes of the maximal infinite virtually cyclic subgroups of B 4 pS 2 q, we observe that since the action of Q 8 on the quotient T{ r F k is trivial, it follows that π 1 ppT{ r F k q Q 8 q is free of rank k ě 2. Therefore, there are infinitely many conjugacy classes of maximal infinite virtually cyclic subgroups of the form Q 8 ¸j Z for j P t1, 2, 3u and of the form Q 16 ˚Q8 Q 16 , see [54, Section 2.5] for more details.

Chapter 4

Lower algebraic K-theory groups of the group ring ZrB 4 pS 2 qs As we mentioned in Section 2.1, B n pS 2 q is finite for all n ď 3. For these values of n, the corresponding K-groups were given in Table 2.1. This chapter is devoted to the computation of the lower K-groups of ZrB 4 pS 2 qs. The aim is to prove Theorem 1, whose statement we recall here. Theorem 1. The group B 4 pS 2 q has the following lower algebraic K-groups:

r K 0 pZrB 4 pS 2 qsq -Z 2 ' Nil 0 , and K ´1pZrB 4 pS 2 qsq -Z 2 ' Z, K ´ipZrB 4 pS 2 qsq " 0 for all i ě 2, where for i " 0, 1, the groups Nil i are isomorphic to a countably-infinite direct sum of Z 2 , Z 4 or Z 2 ' Z 4 .

The main fact that allows this computation is that B 4 pS 2 q is hyperbolic in the sense of Gromov (see Remark 38) because it is an amalgam of finite groups by Proposition 37. Hence the Farrell-Jones fibred isomorphism conjecture holds for this group, and so we may perform the K-theoretical calculations using Section 4.1 and [START_REF] Juan-Pineda | Algebraic K-theory of virtually free groups[END_REF][START_REF] Juan-Pineda | On classifying spaces for the family of virtually cyclic subgroups, in Recent developments in algebraic topology[END_REF]. All of these calculations are based on the knowledge of the lower K-theory groups of the virtually cyclic subgroups of B 4 pS 2 q. In Section 4.1, we recall some general facts about the lower K-groups of infinite virtually cyclic groups. In Section 4.2, we discuss the lower K-groups of the finite subgroups of B 4 pS 2 q and how they fit together with the infinite virtually cyclic subgroups of B 4 pS 2 q to give the lower K-groups of ZrB 4 pS 2 qs, up to computing the Nil i groups. Finally, in Section 4.3, we determine these groups, and we put together all of these ingredients to complete our calculations to prove Theorem 1.

The lower K-theory of infinite virtually cyclic groups

In this section, we provide the ingredients needed to compute the lower algebraic K-groups of infinite virtually cyclic groups. For a virtually cyclic group Γ of Type I, the algebraic K-

Preliminary K-theoretical calculations for ZrB 4 pS 2 qs

Using the hyperbolicity of B 4 pS 2 q and the results of [START_REF] Juan-Pineda | On classifying spaces for the family of virtually cyclic subgroups, in Recent developments in algebraic topology[END_REF]

'

Nil 0 , and the K ´1-group is given by:

where for i " 0, 1, Nil i splits as a direct sum of Bass or Farrell-Hsiang Nil groups over representatives of V (see Section 4.1). From Theorem 31, we have the following isomorphisms:

Moreover, by [START_REF] Swan | Projective modules over binary polyhedral groups[END_REF]Lemma 14.6], the induction r K 0 pZrQ 8 sq ÝÑ r K 0 pZrQ 16 sq is zero and the homomorphism r K 0 pZrQ 8 sq ÝÑ r K 0 pZrT ˚sq is an isomorphism by hyper-elementary induction (cf. [73, Theorem 14.1(1)]). Furthermore, by Remark 32 and Theorem 51(c), we obtain the following isomorphisms:

which proves Theorem 1 up to the computation of the Nil i terms in the first two isomorphisms. To complete the proof, we must compute the Nil groups that appear in the contribution of the conjugacy classes of maximal infinite virtually cyclic subgroups of B 4 pS 2 q.

Nil group computations

In this section, we compute the Bass Nil groups NK i pZrQ 8 sq for i " 0, 1, as well as the twisted versions. In the non-twisted case, we obtain the following result.

Proposition 52. For i " 0, 1, the groups NK i pZrQ 8 sq are isomorphic to a countable, infinite direct sum of copies of Z 2 , Z 4 or Z 2 ' Z 4 .

In order to prove Proposition 52, we first consider the ring R of Lipschitz quaternions of the form a `bi `cj `dk, where a, b, c, d P Z and i, j, k are the quaternionic roots of ´1, and compute its NK 0 and NK 1 groups. Recall that the ring S of Hurwitz quaternions consists of the quaternions of the form pa `bi `cj `dkq{2 where a, b, c and d are integers that are either all even or all odd. Hence: R " Zri, j, ks and S " Zri, j, ks

By [START_REF] Chatters | Rings which are nearly principal ideal domains[END_REF]Example 5.1], S is a non-commutative principal ideal domain, and so is a regular ring. Let M " p1 `iqS. Observe that M Ă R Ă S, and that R{M and S{M are the fields of two and four elements respectively. From this, it follows that S{R is the group with 2 elements. These computations involve the double relative term K 1 pR, S, Mq for the injection R ÝÑ S and ideal M " p1 `iqS that is described as follows [37, Theorem 0.2]:

.3)

A straightforward computation yields:

M " t a `bi `cj `dk | a `b and c `d evenu `Zri `ks M 2 " t´2b `2ai `2dj `2ck | a `b and c `d evenu `2Zri `js.

Define the elements b P S{R and u, w, z P M{M 2 to be the following cosets:

b "

We first study the groups K α s pSrC 2 ˆC2 sq. Let G be the amalgamated product defined as follows. Consider the non-trivial semi-direct products G 1 " C 4 ¸C2 and G 2 " C 4 ¸C2 , where the cyclic groups of order four are generated by a P G 1 and b P G 2 , and the cyclic groups of order 2 are generated by x P G 1 and y P G 2 . Let D 2 be the group C 2 ˆC2 generated by u and v, let D 2 ã ÝÑ G 1 be the inclusion given by u Þ ÝÑ a 2 , v Þ ÝÑ x, and let D 2 ã ÝÑ G 2 be the inclusion given by u Þ ÝÑ y and v Þ ÝÑ b 2 . Then the amalgamated product G 1 ˚D2 G 2 is a virtually cyclic group. By [START_REF] Davis | Algebraic K-theory over the infinite dihedral group: an algebraic approach[END_REF], the Farrell-Hsiang Nil groups K α pSrC 2 ˆC2 sq are isomorphic to the corresponding Waldhausen Nil groups:

for all s P Z and all rings S. Now, for the rings S " Z or F 2 , these Waldhausen Nil groups are isomorphic to infinite direct sums of copies of Z 2 . For S " Z and " 0, 1, see [61, Theorem 5.2] and [5, Section 7.2]. Hence K α pSrC 2 ˆC2 sq is isomorphic to an infinite direct sum of copies of Z 2 for " 0, 1, 2 and for S " Z or F 2 . From the exact sequence (4.10), for " 0, 1, NK α 0 pZrQ 8 sq fits into an exact sequence of the form:

where both A `1 and B are isomorphic to infinite direct sums of copies of Z 2 . The result follows using Remark 54.

Summing up, Propositions 52 and 55 give rise to the Nil i summands of equation (4.2), and the decompositions of the statement of Theorem 1 then follow.

Appendix A

The fibred isomorphism conjecture

The setup

Let S : TOP ÝÑ Ω-SPECTRA be a covariant homotopy functor. Let F be the category of continuous surjective maps: objects in F are continuous surjective maps p : E ÝÑ B, where E, B are objects in TOP, and morphisms between pairs of maps p 1 :

(A1) Within this framework, Quinn constructed a functor between F and Ω-SPECTRA [START_REF] Quinn | Ends of maps II[END_REF]. The value of this Ω-spectrum at the object pp : E ÝÑ Bq is denoted by HpB; Sppqq, and the value at the object pE ÝÑ ˚q is SpEq. The map of spectra A : HpB 1 ; Spp 1 qq ÝÑ HpB 2 ; Spp 2 qq associated to the commutative diagram (A1) is known as the Quinn assembly map. Other ingredients for the fibred isomorphism conjecture may be found in [START_REF] Farrell | Isomorphism conjectures in algebraic K-theory[END_REF].

The conjecture

Given a discrete group Γ, let E V C Γ be a universal Γ-space for the family of virtually cyclic subgroups of Γ, let B V C Γ denote the orbit space E V C Γ{Γ, and let X be a space on which Γ acts freely and properly discontinuously. If p f , gq is the following morphism in F:

the Fibred Isomorphism Conjecture for the functor S and the group Γ is the assertion that A : HpB V C Γ; Spp 1 qq ÝÑ SpX{Γq is a homotopy equivalence, and hence the induced map A ˚: π n pHpB V C Γ; Spp 1 qq ÝÑ π n pSpX{Γqq is an isomorphism for all n P Z. This conjecture was stated in [START_REF] Farrell | Isomorphism conjectures in algebraic K-theory[END_REF] for the functors S " P ˚p¨q, Kp¨q and L ´8, the pseudoisotopy, algebraic K-theory and L ´8-theory functors respectively. In this paper, we use the functor S " K ˚p¨q. The validity of this conjecture for K-theory and braid groups of S 2 is proved in [START_REF] Juan-Pineda | The K and L theoretic Farrell-Jones isomorphism conjecture for braid groups[END_REF]. Other cases in which the conjecture holds may be found in [START_REF] Wegner | The K-theoretic Farrell-Jones conjecture for CATp0q-groups[END_REF].

Appendix B

Braid groups

In this appendix, we recall briefly some basic facts and results about braid groups for the convenience of the reader. More information about braid groups may be found in [START_REF] Berrick | Braids: Introductory Lectures on Braids, Configurations and Their Applications[END_REF][START_REF] Birman | Braids, links and mapping class groups[END_REF][START_REF] González-Meneses | Basic results on braid groups[END_REF][START_REF] Hansen | Braids and coverings: Selected topics[END_REF]. We refer the reader to [START_REF] Guaschi | A survey of surface braid groups and the lower algebraic K-theory of their group rings[END_REF] for a recent survey on surface braid groups.

If n ě 1, the n-string Artin braid group, denoted B n , may be defined by the following presentation [START_REF] Artin | Theorie der Zöpfe[END_REF]: generators: σ 1 , . . . , σ n´1 (known as the Artin generators). relations: (known as the Artin relations) σ i σ j " σ j σ i if |i ´j| ě 2 and 1 ď i, j ď n ´1 (B.1)

The generator σ i may be regarded geometrically as the braid with a single positive crossing of the i th string with the pi `1q st string, while all other strings remain vertical (see Figure B.1).

It is convenient to view a geometric braid as being a collection of pairwise-disjoint arcs (or strings) in the Cartesian product D 2 ˆr0, 1s, where D 2 is the 2-disc, and each string joins two points of the form px, 0q to py, 1q, where x and y belong to a set X of n distinguished basepoints lying in the interior of D 2 . The group operation in B n corresponds to concatenation of these geometric braids. The group B 1 is trivial, B 2 is infinite cyclic generated by σ 1 , and for all n ě 2, B n is infinite. For all n P N, B n is torsion free [START_REF] Dyer | The algebraic braid groups are torsion-free: an algebraic proof[END_REF]. The map σ :

The braid σ i and its inverse.

defined on the generators by σpσ i q " pi, i `1q for all 1 ď i ď n ´1 may be seen to be a surjective homomorphism. Its kernel, denoted by P n , is known as the n-string pure Artin braid group. Thus a braid β P B n is pure if for all x P X, there is a string of β that joins px, 0q

to px, 1q. The 'half twist' braid ∆ n is defined by:

Using the braid relations, one may check that the square ∆ 2 n of ∆ n , known as the 'full twist' braid is given by: ∆ 2 n " pσ 1 ¨¨¨σ n´1 q n P B n . (B.

3)

The braids ∆ n and ∆

Geometrically, A i,j may be represented by a braid all of whose strings are vertical, with the exception of the j th string that wraps around the i th string as in all i " 1, . . . , n ´1, A i,i`1 " σ 2 i . The Artin braid groups admit many different generalisations, one being that of surface braid groups. If M is a surface, orientable or not, with or without boundary, and with a result for the braid groups of RP 2 may be found in [START_REF] Gonçalves | Surface braid groups and coverings[END_REF]. The braid groups of the sphere were initially studied by Fadell, Van Buskirk and Gillette [START_REF] Fadell | The braid groups of E 2 and S 2[END_REF][START_REF] Gillette | The word problem and consequences for the braid groups and mapping class groups of the 2-sphere[END_REF][START_REF] Van Buskirk | Braid groups of compact 2-manifolds with elements of finite order[END_REF]. A presentation of B n pS 2 q due to the first two of these authors is given in Theorem 34. From a geometric point of view, the space S 2 ˆr0, 1s in which geometric braids of the sphere are defined may be visualised as that between two concentric spheres (see [52, pp. 41, 42 and 45] or [69, n also plays an important rôle in B n pS 2 q. If n ě 3, it is the unique element of B n pS 2 q of order 2, it is the unique non-trivial torsion element of P n pS 2 q, and it generates the centre of B n pS 2 q [START_REF] Gillette | The word problem and consequences for the braid groups and mapping class groups of the 2-sphere[END_REF][START_REF] Gonçalves | The roots of the full twist for surface braid groups[END_REF]. The pure braid group P 4 pS 2 q is generated by the set A i,j ( 1ďiăjď4 , where in terms of the generators σ 1 , σ 2 and σ 3 of B 4 pS 2 q, A i,j is given by (B.4), and its geometric representation within S 2 ˆr0, 1s is as in Figure B.3. If m ě 1, a presentation of P m pS 2 q may be obtained using techniques similar to those of [START_REF] Gonçalves | The braid group B m,n pS 2 q and a generalisation of the Fadell-Neuwirth short exact sequence[END_REF]Proposition 7]. Note that if one takes n " 0 in that proposition, one does indeed obtain a presentation of P m pS 2 q whose generating set is A i,j ( 1ďiăjďm , and whose relations are given by those of [START_REF] Hansen | Braids and coverings: Selected topics[END_REF]Lemma I.4.2] for P m , and by the 'surface relations' that are of the form:

for all 1 ď j ď m. Taking M " S 2 and n " 3 in (B.7) yields:

1 ÝÑ π 1 pS 2 z tz 1 , z 2 , z 3 uq ÝÑ P 4 pS 2 q pp 4,3 q Ý ÝÝÝ Ñ P 3 pS 2 q ÝÑ 1. (B.9)

The kernel is a free group of rank 2 that may be identified with the subgroup of P 4 pS 2 q generated by pA 1,4 , A 2,4 q, and the quotient P 3 pS 2 q is equal to @ ∆ 2 3 D , and is isomorphic to Z 2 . The map s : P 3 pS 2 q ÝÑ P 4 pS 2 q defined by sp∆ 2 3 q " ∆ 2 4 is a homomorphism, and is a section for pp 4,3 q ˚since removal of the last string of ∆ 2 4 in P 4 pS 2 q yields the braid ∆ 2 3 in P 3 pS 2 q, i.e. pp 4,3 q ˚p∆ 2 4 q " ∆ 2 3 . So the short exact sequence (B.9) splits, and since ∆ 2 4 P ZpP 4 pS 2 qq, it follows that: P 4 pS 2 q -F 2 ˆZ2 . (B.10)

From this, it follows also that ZpP 4 pS 2 qq " @ ∆ 2 4 D .