A survey of surface braid groups and the lower algebraic K-theory of their group rings
AA survey of surface braid groups and the loweralgebraic K -theory of their group rings John GuaschiNormandie Université, UNICAEN,Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139,14032 Caen Cedex, Francee-mail: [email protected]
Daniel Juan-PinedaCentro de Ciencias Matemáticas,Universidad Nacional Autónoma de México, Campus Morelia,Morelia, Michoacán, México 58089e-mail: [email protected]
Abstract
In this article, we give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surfacebraid groups, such as the existence of torsion, orderability, linearity, and their relation both withmapping class groups and with the homotopy groups of the -sphere. The braid groups of the -sphere and the real projective plane are of particular interest because they possess elements offinite order, and we discuss in detail their torsion and the classification of their finite and virtuallycyclic subgroups. Finally, we outline the methods used to study the lower algebraic K-theory ofthe group rings of surface braid groups, highlighting recent results concerning the braid groupsof the -sphere and the real projective plane. K -theory, conjugacy classes, virtually cyclic subgroups, Farrell-Jones conjecture, Nil groups a r X i v : . [ m a t h . G T ] F e b ontents S and R P and periodicity . . 173.7 Orderability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 S and R P . . . . . . . . . . . . . . . 234.3 Virtually cyclic subgroups of the braid groups of S and R P . . . . . . . . . . 27 K -theory of surface braid groups 30 K -theoretic Farrell-Jones Conjecture for braid groups of aspherical surfaces 325.3 The Farrell-Jones Conjecture for the braid groups of S and R P . . . . . . . . 335.4 General remarks for computations . . . . . . . . . . . . . . . . . . . . . . . . . 345.5 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.6 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 References 39
The braid groups B n were introduced by E. Artin in 1925 [5], in a geometric and intuitivemanner, and further studied in 1947 from a more rigourous and algebraic standpoint [6, 7].These groups may be considered as a geometric representation of the standard everydaynotion of braiding strings or strands of hair. As well as being fascinating in their own right,braid groups play an important rôle in many branches of mathematics, for example in to-pology, geometry, algebra, dynamical systems and theoretical physics, and notably in thestudy of knots and links [35], in the definition of topological invariants (Jones polynomial,Vassiliev invariants) [105, 106], of the mapping class groups [26, 27, 57], and of configurationspaces [39, 52]. They also have potential applications to biology, robotics and cryptography,for example [20]. 2he Artin braid groups have been generalised in many different directions, such as Artin-Tits groups [31, 32, 48], surface braid groups, singular braid monoids and groups, and virtualand welded braid groups. One recent exciting topological development is the discovery ofa connection between braid groups and the homotopy groups of the 2-sphere via the notionof Brunnian braids [20, 21]. Although there are many surveys on braid groups [28, 91, 124,131, 135, 141, 152] as well as some books and monographs [26, 99, 113, 131], for the mostpart, the theory of surface braid groups is discussed in little detail in these works. The aimof this article is two-fold, the first being to survey various aspects of this theory and somerecent results, highlighting the cases of the 2-sphere and the real projective plane, and thesecond being to discuss current developments in the study of the lower algebraic K -theoryof the group rings of surface braid groups. In Section 2, we give various definitions of sur-face braid groups, and recall their relationship with mapping class groups. In Section 3, wedescribe a number of properties of these groups, including the existence of Fadell-Neuwirthshort exact sequences of their pure and mixed braid groups, which play a fundamental rôlein the theory. In Section 3.2, we recall some presentations of surface braid groups, and inSections 3.3 and 3.4, we survey known results about their centre and their embeddings inother braid groups. Within the theory of surface braid groups, those of the sphere S andthe real projective plane R P are interesting and important, one reason being that their con-figuration spaces are not Eilenberg-Mac Lane spaces. In Section 3.6, we study the homotopytype of these configuration spaces and the cohomological periodicity of the braid groupsof S and R P , and we describe some of the results mentioned above concerning Brunnianbraids and the homotopy groups of S . In Sections 3.7 and 3.8, we discuss orderability andlinearity of surface braid groups.Section 4 is devoted to the study of the structure of the braid groups of S and R P ,notably their torsion, their finite subgroups and their virtually cyclic subgroups. Finally,in Section 5, we discuss recent work concerning the K -theory of the group rings of surfacebraid groups. The existence of torsion in the braid groups of S and R P leads to new andinteresting behaviour in the lower algebraic K -theory of their group rings. Recent techniquesprovided by the Fibred Isomorphism Conjecture (FIC) of Farrell and Jones have broughtto light examples of of intricate group rings whose lower algebraic K -groups are trivial,see Theorem 70 for example, as well as highly-complicated algebraic K -theory groups. Afairly complete example of the latter is that of the 4-string braid group B p S q of the sphere,for which we show that K i p Z r B p S qsq is infinitely generated for i “
0, 1 (see Theorem 74).We conjecture that a similar result is probably true for all i ą
1. On the other hand, it isknown that rank p K i p Z r B p S qsqq ă 8 for all i P Z [111]. It is interesting to observe that thegeometrical aspects of a group largely determine the structure of the algebraic K -groups ofits group ring. We include up-to-date results on the algebraic K -groups of surface braidgroups, and mention possible extensions of these computations. The main obstructions toextending our results from B p S q to the general case are the lack of appropriate models fortheir classifying spaces, as well the complicated structure of the Nil groups. Acknowledgements
Both authors are grateful to the French-Mexican International Laboratory “LAISLA” for itsfinancial support. The first author was partially supported by the international CooperationCapes-Cofecub project numbers Ma 733-12 (France) and Cofecub 1716/2012 (Brazil). Thesecond author would like to acknowledge funding from CONACyT and PAPIIT-UNAM.The first author wishes to thank Daciberg Lima Gonçalves for interesting and helpful con-versations during the preparation of this paper.3
Basic definitions of surface braid groups
One of the interesting aspects about surface braid groups is that they may be defined fromvarious viewpoints, each giving a different insight into their nature [141]. The notion ofsurface braid group was first introduced by Zariski, and generalises naturally Artin’s geo-metric definition [159, 160]. Surface braid groups were rediscovered during the 1960’s byFox who proposed a powerful (and equivalent) topological definition in terms of the funda-mental group of configuration spaces. We recall these and other definitions below. Unlessstated otherwise, in the whole of this manuscript, we shall use the word surface to denote aconnected surface, orientable or non orientable, with or without boundary, and of the form M “ N z Y , where N is a compact, connected surface, and Y is a finite (possibly empty) subsetlying in the interior Int p N q of N . Let M be a surface, and let n P N . We fix once and for all a finite n -point subset X “t x , . . . , x n u of Int p M q whose elements shall be the base points of our braids. Definition. A geometric n-braid in M is a collection β “ t β , . . . , β n u consisting of n arcs β i : r
0, 1 s ÝÑ M ˆ r
0, 1 s , i “
1, . . . , n , called strings (or strands ) such that: (a) for i “
1, . . . , n , β i p q “ p x i , 0 q and β i p q P X ˆ t u (the strings join the elements of X belonging to the copies of M corresponding to t P t
0, 1 u ). (b) for all t P r
0, 1 s and for all i , j P t
1, . . . , n u , i ‰ j , β i p t q ‰ β j p t q (the strings are pairwisedisjoint). (c) for all t P r
0, 1 s , each string meets the subset M ˆ t t u in exactly one point (the strings arestrictly monotone with respect to the t -coordinate).See Figure 1 for an example of a geometric 3-braid in the 2-torus, and Figure 2 for an exampleof a geometric 3-braid that illustrates condition (c). M ˆ t u M ˆ t u x x x β β β Figure 1: A geometric 3-braid with M equal to the 2-torus.In the case where M is the plane, a braid is often depicted by a projection (taken to bein general position) onto the plane xz such as that depicted in Figure 2, so that there areonly a finite number of points where the strings cross, and such that the crossings occur atdistinct values of t . We distinguish between under- and over-crossings. Our convention isthat such a braid is to be read from top to bottom, the top of the braid corresponding to4 ˆ t u R ˆ t t u R ˆ t u β Figure 2: A 3-braid in R illustrating condition (c) of the definition of geometric braid. t “
0, and the bottom to t “
1. Similar pictures may be drawn for other surfaces of smallgenus (see [130, 131] for example).Two geometric n -braids of M are said to be equivalent if there exists an isotopy (keepingthe endpoints of the strings fixed) from one to the other through n-braids . In particular, underthe isotopy, the strings remain pairwise disjoint. This defines an equivalence relation, andthe equivalence classes are termed n-braids . The set of n -braids of M is denoted by B n p M q .By a slight abuse of terminology, we shall not distinguish between a braid and its geometricrepresentatives.The product of two n -braids α and β , denoted αβ , is their concatenation, defined by glue-ing the endpoints of α to the respective initial points of β (formally, α should be ‘squashed’into the slab 0 ď t ď , and β into the slab ď t ď α and β , and that it is associat-ive . The identity element Id of B n p M q is the braid all of whose strings are vertical. The inverse of an n -braid β “ tp β p t q , . . . , β n p t qqu t Pr s is given by β ´ “ tp β p ´ t q , . . . , β n p ´ t qqu t Pr s (its mirror image with respect to M ˆ ! ) ). Equipped with this operation, B n p M q is thus agroup, which we call the n-string braid group of M .To each n -braid β “ p β , . . . , β n q , one may associate a permutation τ n p β q P S n defined by β i p q “ p x τ n p β qp i q , 1 q , and the following correspondence: τ n : B n p M q ÝÑ S n β ÞÝÑ τ n p β q (1)is seen to be a surjective group homomorphism. The kernel P n p M q of τ n is known as the n-string pure braid group of M , and so β P P n p M q if and only if β i p q “ i for all i “
1, . . . , n .Clearly P n p M q is a normal subgroup of B n p M q of index n !, and we have the following shortexact sequence: 1 ÝÑ P n p M q ÝÑ B n p M q τ n ÝÑ S n ÝÑ
1. (2)It is well known that if M is equal to R or to the 2-disc D then B n p M q and P n p M q areisomorphic to the usual Artin braid groups B n and P n [99, Theorem 1.5]. Remark 1.
The exact sequence (2) is frequently used to reduce the study of certain problemsin B n p M q to that in P n p M q (see for example Theorems 2, 15, 61, 62 and 63, as well as Propos-ition 17). The group B n p M q is also sometimes known as the permuted or full braid group of M , and P n p M q as the unpermuted or coloured braid group.5 .2 Surface braids as trajectories of non-colliding particles Definition.
Consider n particles which move on the surface M , whose initial points are γ i p q “ x i for i “
1, . . . , n , and whose trajectories are γ i p t q for t P r
0, 1 s . A braid is thusthe collection γ “ p γ p t q , . . . , γ n p t qq t Pr s of trajectories satisfying the following two condi-tions: (a) the particles do not collide, i.e. for all t P r
0, 1 s and for all i , j P t
1, . . . , n u , i ‰ j , γ i p t q ‰ γ j p t q . (b) they return to their initial points, but possibly undergoing a permutation: γ i p q P X forall i P t
1, . . . , n u .There is a clear bijective correspondence between this definition of braid and the defin-ition of geometric n -braid in Section 2.1. Indeed, if γ “ p γ p t q , . . . , γ n p t qq t Pr s is such abraid then β “ p β , . . . , β n q is a geometric n -braid, where for all i “
1, . . . , n and t P r
0, 1 s , β i p t q “ p γ i p t q , t q . Conversely, we may obtain the ‘particle’ notion of braid by reparametrisingeach string β “ p β , . . . , β n q of a geometric n -braid so that β i p t q is of the form p γ i p t q , t q for i “
1, . . . , n and t P r
0, 1 s , where γ “ p γ p t q , . . . , γ n p t qq t Pr s satisfies conditions (a) and (b).The transition from a geometric n -braid to the ‘particle notion’ may thus be realised geomet-rically by projecting the strings lying in M ˆ r
0, 1 s onto the surface M .It is easy to check that two geometric braids are homotopic (in the sense of Section 2.1) ifand only if the braids defined in terms of trajectories are homotopic. It thus follows that theset of homotopy classes of the latter class of braids may be equipped with a group structure,and that the group thus obtained is isomorphic to B n p M q . In this setting, the identity braidis represented by the configuration where all particles are stationary, and the inverse of abraid is given by running through the trajectories in reverse. This point of view proves to beuseful when working with braid groups of surface of higher genus, notably in determiningpresentations [16, 24, 70, 89, 145]. Configuration spaces are important and interesting in their own right [38, 52], and havemany applications, for example to the study of polynomials in C r X s [99]. The followingdefinition is due to Fox [66] (according to Magnus [124], the idea first appeared in the workof Hurwitz), and has very important consequences. The motivation for the definition em-anates from condition (c) of the definition of geometric n -braid given in Section 2.1, and isillustrated by Figure 2. Definition.
Let F n p M q denote the n th configuration space of M defined by: F n p M q “ (cid:32) p p , . . . , p n q P M n ˇˇ p i ‰ p j for all i , j P t
1, . . . , n u , i ‰ j ( .We equip F n p M q with the topology induced by the product topology on M n . A transversalityargument shows that F n p M q is a connected 2 n -dimensional open manifold. There is a naturalfree action of the symmetric group S n on F n p M q by permutation of coordinates. The resultingorbit space F n p M q{ S n shall be denoted by D n p M q , the n th permuted configuration space of M ,and may be thought of as the configuration space of n unordered points. The associatedcanonical projection p ρ n : F n p M q ÝÑ D n p M q is thus a regular n !-fold covering map.We may thus describe F n p M q as M n z ∆ , where ∆ denotes the ‘fat diagonal’ of M n : ∆ “ (cid:32) p p , . . . , p n q P M n ˇˇ p i “ p j for some 1 ď i ă j ď n ( .6f M “ R then ∆ “ ď ď i ă j ď n H i , j , where H i , j is the hyperplane defined by: H i , j “ ! p p , . . . , p n q P p R q n ˇˇˇ p i “ p j ) .The following theorem is fundamental, and brings in to play a topological definition ofthe braid groups that will be very important in what follows. The proof is a good illustrationof the use of the short exact sequence (2). Theorem 2 (Fox and Neuwirth [66]) . Let n P N . Then P n p M q – π p F n p M qq and B n p M q – π p D n p M qq . Remarks 3. (a)
Since F p M q “ M , we have that B p M q – P p M q – π p M q . The braid groups of M maythus be seen as generalisations of its fundamental group. (b) The fact that F n p M q (resp. D n p M q ) is connected implies that the isomorphism class of π p F n p M qq (resp. π p D n p M qq ) does not depend on the choice of basepoint. We thus havetwo finite-dimensional topological spaces F n p M q (resp. D n p M q ) whose fundamental groupsare P n p M q (resp. B n p M q ). As we shall see in Section 3.1, the relations between configurationspaces and braid groups play a fundamental rôle in the study of the latter, notably via thefact that we may form certain natural fibre spaces of the former. (c) The definitions of surface braid groups given in Sections 2.1–2.3 generalise to any topo-logical space. It was shown in [53, Theorem 9] that for connected manifolds of dimension r ě
3, there is no braid theory, as it is formulated here.The natural inclusion ι : F n p M q ã ÝÑ M n induces a homomorphism of the correspondingfundamental groups: ι : P n p M q ÝÑ p π p M qq n ,and the inclusion j : D ã ÝÑ Int p M q of a topological disc D in the interior of M induces ahomomorphism j : P n ÝÑ P n p M q that is an embedding for most surfaces: Proposition 4 ([24]) . Let M be a compact, orientable surface different from S . Then the inclusionj : D ã ÝÑ M induces an embedding P n ã ÝÑ P n p M q . Proposition 4 extends first to the non-orientable case [69], with the exception of M “ R P ,and secondly, to the full braid groups by applying equation (2). If M is different from S and R P then Goldberg showed that the following sequence is short exact [69]:1 ÝÑ xx Im p j qyy P n p M q ã ÝÑ P n p M q ι ÝÑ p π p M qq n ÝÑ
1, (3)where xx H yy G denotes the normal closure of a subgroup H in a group G . This sequence wasanalysed in [92] in order to study Vassiliev invariants of braid groups of orientable surfaces.In the case of R P , Ker p ι q was computed and the homotopy fibre of ι was determinedin [86]. Let M be a compact, connected, orientable (resp. non-orientable) surface, possibly withboundary B M , and for n ě
0, let Q n be a finite subset of Int p M q consisting of n distinct7oints (so Q “ ∅ ). Let H p M , Q n q denote the group Homeo ` p M , Q n q (resp. Homeo p M , Q n q )of orientation-preserving homeomorphisms (resp. of homeomorphisms) of M under com-position that leave Q n invariant (so we allow the points of Q n to be permuted), and that fix B M pointwise. We equip H p M , Q n q with the compact-open topology. Let H p M , Q n q de-note the path component of Id M in H p M , Q n q . The n th mapping class group of M , denoted by MC G p M , n q , is defined to be the set of isotopy classes of the elements of Homeo ` p M , Q n q (resp. Homeo p M , Q n q ), in other words, MC G p M , n q “ H p M , Q n q{ H p M , Q n q “ π p H p M , Q n qq .It is straightforward to check that MC G p M , n q is indeed a group whose isomorphism classdoes not depend on the choice of Q n . If n “ H p M q and MC G p M q forthe corresponding groups. The mapping class groups have been widely studied and play animportant rôle in low-dimensional topology. Some good general references are [26, 57, 103].The mapping class groups are closely related to braid groups. If M “ D then it is wellknown that they coincide: Theorem 5 ([26, 99, 113]) . B n – MC G p D , n q . The proof of Theorem 5 makes use of Artin’s representation of B n as a subgroup ofthe automorphism group of the free group F n of rank n , the free group in question beingidentified with π p D z Q n q . In the general case, the relationship between MC G p M , n q and B n p M q arises in a topological setting as follows [25, 26, 145]. Let n ě
1, and fix a basepoint Q n P D n p M q . Then the map Ψ : H p M q ÝÑ D n p M q defined by Ψ p f q “ f p Q n q is a locally-trivial fibre bundle [25, 127], whose fibre over Q n is equal to H p M , Q n q . Taking the longexact sequence in homotopy of this fibration yields: ¨ ¨ ¨ ÝÑ π p H p M , Q n qq ÝÑ π p H p M qq ÝÑ π p D n p M qq ÝÑ π p H p M , Q n qq ÝÑ π p H p M qq ÝÑ
1. (4)If M is different from S , R P , the torus or the Klein bottle then π p H p M qq “ ÝÑ B n p M q ÝÑ MC G p M , n q ÝÑ MC G p M q ÝÑ
1. (5)The braid group B n p M q is thus isomorphic to the kernel of the homomorphism that corres-ponds geometrically to forgetting the marked points. We recover Theorem 5 by noting that MC G p D q “ t u using the Alexander trick. If M “ S (resp. R P ) and n ě n ě π p H p M , Q n qq “ π p H p M qq – Z [97, 98], which is a manifestation ofthe fact that the fundamental group of SO p q is isomorphic to Z [51, 99, 132]. In this case,we obtain the following short exact sequence:1 ÝÑ Z ÝÑ B n p M q ÝÑ MC G p M , n q ÝÑ
1. (6)As we shall see in Section 4.1, viewed as an element of B n p M q , the generator of the kernel isthe full twist braid ∆ n [54, 151]. In particular, B n p M q{ @ ∆ n D – MC G p M , n q . In the case of S ,the short exact sequence (6) may be obtained by combining the presentation of MC G p S , n q due to Magnus [123, 125] with Fadell and Van Buskirk’s presentation of B n p S q (see The-orem 32). It plays an important part, notably in the study of the centralisers and conjugacyclasses of the finite order elements, and of the finite subgroups of B n p M q (see Section 4.2).Finally, if M is the torus T or the Klein bottle then (4) yields a six-term exact sequence start-ing and ending with 1. In the case of T , this exact sequence involves MC G p T q , which isisomorphic to SL p Z q . 8 Some properties of surface braid groups
In this section, we describe various properties of surface braid groups. We start with one ofthe most important, that makes use of the definition of Section 2.3 in terms of configurationspaces.
Let M be a connected surface. For n P N , we equip F n p M q with the topology induced by theproduct topology on the n -fold Cartesian product M n . For m ě
0, let Q m be as in Section 2.4,and set F m , n p M q “ F n p M z Q m q and D m , n p M q to be the quotient space of F m , n p M q by the freeaction of S n , so that the projection F m , n p M q ÝÑ D m , n p M q is a covering map. Note that thetopological type of F m , n p M q does not depend on the choice of Q m , and that as special cases,we obtain F n p M q “ F n p M q and F m ,1 p M q “ M z Q m . We have the following important resultconcerning the topological structure of the spaces F m , n p M q . Theorem 6 (Fadell and Neuwirth [53, 99, 113]) . Let ď r ă n and m ě . Suppose that M is asurface with empty boundary. Then the map p n , r : F m , n p M q ÝÑ F m , r p M qp x , . . . , x n q ÞÝÑ p x , . . . , x r q (7) is a locally-trivial fibration with fibre F m ` r , n ´ r p M q . One may then take the long exact sequence in homotopy of the fibration (7): ¨ ¨ ¨ ÝÑ π k p F m ` r , n ´ r p M qq ÝÑ π k p F m , n p M qq ÝÑ π k p F m , r p M qq ÝÑ π k ´ p F m ` r , n ´ r p M qq ÝÑ ¨ ¨ ¨ ÝÑ π p F m ` r , n ´ r p M qq ÝÑ π p F m , n p M qq ÝÑ π p F m , r p M qq ÝÑ π p F m ` r , n ´ r p M qq ÝÑ π p F m , n p M qq ÝÑ π p F m , r p M qq ÝÑ
1. (8)Since F m ` n ` i ´ p M q has the homotopy type of a bouquet of circles for all 0 ď i ď n ´
2, itfollows that: π k p F m , n p M qq – π k p F m , n ´ p M qq – ¨ ¨ ¨ – π k p F m ,1 p M qq “ π k p M z Q m q for all k ě π p F m , n ´ i p M qq ÝÑ π p F m , n ´ i ´ p M qq is injective for all such i .Thus π p F m , n p M qq is isomorphic to a subgroup of π p F m ,1 p M qq , which is in turn isomorphicto π p M z Q m q . Since π p F m , n p M qq – P n p M z Q m q by Theorem 2, we recover the followingresult: Theorem 7 ([51, 53, 54, 151]) . (a) Let n P N and m ě . We suppose additionally that M is different from S and R P if m “ .Then the spaces F m , n p M q and D m , n p M q are Eilenberg-Mac Lane spaces of type K p P n p M z Q m q , 1 q andK p B n p M z Q m q , 1 q respectively.(b) If n ě (resp. n ě ) then π p F n p S qq “ and π p F n p R P qq “ .(c) Let ď r ă n and m ě . If m “ then we suppose that r ě if M “ S , and that r ě ifM “ R P . Then the Fadell-Neuwirth fibration (8) induces a short exact sequence: ÝÑ P n ´ r p M z Q m ` r q ÝÑ P n p M z Q m q p p n , r q ÝÝÝÝÑ P r p M z Q m q ÝÑ
1. (9)9 emarks 8. (a)
The short exact sequence (9) is known as the
Fadell-Neuwirth short exact sequence of sur-face braid groups.
It plays a central rôle in the study of surface (pure) braid groups. It wasused to study mapping class groups in [136], and in work on Vassiliev invariants for braidgroups [92]. (b)
Theorem 7(b) was proved in [51, 54, 151] by showing that π p F p S qq “ π p F p R P qq “ (c) The projection P n p M z Q m q ÝÑ P r p M z Q m q may be interpreted geometrically as the epi-morphism that ‘forgets’ the last n ´ r strings. (d) In order to prove that (7) is a locally-trivial fibration, one needs to suppose that M iswithout boundary. However, the long exact sequence (8) exists even if M has boundary, andthus Theorem 7 holds for any connected surface. To see this, let M be a surface with bound-ary, and let M “ M zB M . Then M is a surface with empty boundary, and so Theorems 6and 7 hold for M . The inclusion of M in M is not only a homotopy equivalence between M and M , but it also induces a homotopy equivalence between their n th configuration spaces.In particular, (8) and Theorem 7 are valid also for M , and the n th (pure) braid groups of M and M are isomorphic. (e) Let n ě M “ S , n ě M “ R P , and n ě (i) m “
0, in which case the short exact sequence (9) becomes:1 ÝÑ P n ´ r p M z Q r q ÝÑ P n p M q p p n , r q ÝÝÝÝÑ P r p M q ÝÑ
1. (10) (ii) m “ r “ n ´
1, in which case the short exact sequence (9) becomes:1 ÝÑ π p M z Q n ´ q ÝÑ P n p M q p p n , n ´ q ÝÝÝÝÝÝÑ P n ´ p M q ÝÑ
1. (11)In particular, each element of Ker pp p n , n ´ q q may be interpreted as an n -string braid whosefirst n ´ K p π , 1 q are torsion free. Thisimplies immediately the sufficiency of the following assertions: Corollary 9 ([53, 54, 151]) . Let M be a surface. Then the braid groups P n p M z Q m q and B n p M z Q m q are torsion free if and only if either:(a) m ě , or(b) m “ and M is a surface different from S and R P . As for the necessity of the conditions, we already mentioned in Section 2.4 that the fulltwist ∆ n is an element of P n p M q of order 2 if M “ S or R P . The existence of torsionin the braid groups of S and R P is a fascinating phenomenon to which we shall returnin Sections 4.1 and 4.2, and makes for interesting and intricate K -theoretical structure (seeSection 5.5). More will be said about the homotopy groups of the configuration spaces of theexceptional surfaces, S and R P , in Section 3.6.We remark that a purely algebraic proof of the fact that the Artin braid groups are torsionfree was given later by Dyer [49]. We shall see another proof in Section 3.7.10he short exact sequences (9)–(11) do not extend directly to the full braid groups, but maybe generalised as follows to certain subgroups that lie between P n p M q and B n p M q . Oncemore, let 1 ď r ă n , and suppose that r ě M “ S and r ě M “ R P . Weconsider the space obtained by taking the quotient of F n p M q by the subgroup S r ˆ S n ´ r of S n . If M is without boundary then as in Theorem 6 we obtain a locally-trivial fibration q n , r : F n p M q{p S r ˆ S n ´ r q ÝÑ D r p M q , defined by forgetting the last n ´ r coordinates. We set B r , n ´ r p M q “ π ` F n p M q{p S r ˆ S n ´ r q ˘ , which is often termed a ‘mixed’ braid group, and isdefined whether or not M has boundary. As in the pure braid group case, we obtain thefollowing generalisation of (10) [71]:1 ÝÑ B n ´ r p M z Q r q ÝÑ B r , n ´ r p M q p q n , r q ÝÝÝÝÑ B r p M q ÝÑ
1, (12)known as a generalised Fadell-Neuwirth short exact sequence of mixed braid groups. Such braidgroups are very useful, and have been studied in [19, 71, 73, 83, 121, 126, 136] for example.Further generalisations are possible by taking quotients by direct products of the form S i ˆ¨ ¨ ¨ ˆ S i r , where r ÿ j “ i j “ n . We recall the classical presentation of the Artin braid groups:
Theorem 10 (Artin, 1925 [5]) . For all n ě , the braid group B n admits the following presenta-tion generators : σ , . . . , σ n ´ (known as the Artin generators ). relations : (known as the Artin relations ) σ i σ j “ σ j σ i if | i ´ j | ě and ď i , j ď n ´ σ i σ i ` σ i “ σ i ` σ i σ i ` for all ď i ď n ´ . (14)The generator σ i may be regarded geometrically as the braid with a single positive cross-ing of the i th string with the p i ` q st string, while all other strings remain vertical (see Fig-ure 3). It follows from this presentation that B “ t u and B “ x σ y – Z . Adding the ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ i ´ i i ` i ` n i ´ i i ` i ` n σ i σ ´ i Figure 3: The braid σ i and its inverse.relations σ i “ i “
1, . . . , n ´
1, to those of Theorem 10 yields the Coxeter presentation of S n . If 1 ď i ă j ď n , the pure braid defined by: A i , j “ σ j ´ ¨ ¨ ¨ σ i ` σ i σ ´ i ` ¨ ¨ ¨ σ ´ j ´ , (15)may be represented geometrically by the braid all of whose strings are vertical, with theexception of the j th string, that wraps around the i th string (see Figure 4). Such elementsgenerate P n : 11 ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ i ´ i j j ` n Figure 4: The element A i , j of B n . Proposition 11 ([99]) . For all n ě , P n is generated by (cid:32) A i , j (cid:12)(cid:12) ď i ă j ď n ( whose elements aresubject to the following relations:A ´ r , s A i , j A r , s “ $’’’’&’’’’% A i , j if i ă r ă s ă j or r ă s ă i ă jA r , j A i , j A ´ r , j if r ă i “ s ă jA r , j A s , j A i , j A ´ s , j A ´ r , j if i “ r ă s ă jA r , j A s , j A ´ r , j A ´ s , j A i , j A s , j A r , j A ´ s , j A ´ r , j if r ă i ă s ă j. One interesting fact that may be deduced immediately from the presentation of Propos-ition 11 is that the action by conjugation of P n on itself induces the identity on the Abelian-isation of P n , and via the short exact sequence (11) in the case where M “ R , implies that P n is an almost-direct product of F n ´ and P n ´ . This plays an important rôle in various aspectsof the theory, for example in the proof of the fact that P n is residually nilpotent [55, 56].A number of presentations are known for surface (pure) braid groups [16, 24, 70, 79,89, 120, 121, 145, 159, 160], the first being due to Birman and Scott. We recall those due toBellingeri for B n p N q , where N is a connected surface of the form M z Q m , M being compactand without boundary, and orientable in the first case, and non-orientable in the second.One way to find such presentations is to apply standard techniques to obtain presentationsof group extensions [104]. One first uses induction and the short exact sequence (11) toobtain presentations of the pure braid groups, and then (2) yields presentations of the fullbraid groups. Theorem 12 ([16]) . Let M be a compact, connected, orientable surface without boundary of genusg, where g ě , and let m ě . Then B n p M z Q m q admits the following presentation:Generators: σ , . . . , σ n ´ , a , . . . , a g , b , . . . , b g , z , . . . , z m ´ .Relations:(a) the Artin relations (13) and (14).(b) a r σ i “ σ i a r , b r σ i “ σ i b r and z j σ i “ σ i z j for all ď r ď g, ď i ď n ´ and ď j ď m ´ .(c) p σ ´ a r q “ p a r σ ´ q , p σ ´ b r q “ p b r σ ´ q and p σ ´ z j q “ p z j σ ´ q for all ď r ď g and ď j ď m ´ .(d) σ ´ a s σ a r “ a r σ ´ a s σ , σ ´ b s σ b r “ b r σ ´ b s σ , σ ´ a s σ b r “ b r σ ´ a s σ and σ ´ b s σ a r “ a r σ ´ b s σ for all ď s ă r ď g.(e) if n ě , σ ´ z j σ a r “ a r σ ´ z j σ and σ ´ z j σ b r “ b r σ ´ z j σ for all ď r ď g and ď j ď m ´ .(f) σ ´ z j σ z l “ z l σ ´ z j σ for all ď j ă l ď m ´ .(g) σ ´ a r σ ´ b r “ b r σ ´ a r σ for all ď r ď g.(h) if m “ then r a , b ´ s ¨ ¨ ¨ r a g , b ´ g s “ σ ¨ ¨ ¨ σ n ´ σ n ´ σ n ´ ¨ ¨ ¨ σ , where r a , b s “ aba ´ b ´ . heorem 13 ([16]) . Let M be a compact, connected, non-orientable surface without boundary ofgenus g, where g ě , and let m ě . Then B n p M z Q m q admits the following presentation:Generators: σ , . . . , σ n ´ , a , . . . , a g , z , . . . , z m ´ .Relations:(a) the Artin relations (13) and (14).(b) a r σ i “ σ i a r for all ď r ď g and ď i ď n ´ .(c) p σ ´ a r q “ a r σ ´ a r σ and p σ ´ z j q “ p z j σ ´ q for all ď r ď g and ď j ď m ´ .(d) σ ´ a s σ a r “ a r σ ´ a s σ for all ď s ă r ď g.(e) z j σ i “ σ i z j for all ď i ď n ´ and ď j ď m ´ .(f) if n ě , σ ´ z j σ a r “ a r σ ´ z j σ for all ď r ď g and ď j ď m ´ .(g) σ ´ z j σ z l “ z l σ ´ z j σ for all ď j ă l ď m ´ .(h) if m “ then a ¨ ¨ ¨ a g “ σ ¨ ¨ ¨ σ n ´ σ n ´ σ n ´ ¨ ¨ ¨ σ . Remarks 14. (a)
Geometrically, the generators a , . . . , a g , b , . . . , b g (resp. a , . . . , a g ) of B n p M q given in The-orem 12 (resp. Theorem 13) correspond to a standard set of generators of π p M q based at thefirst basepoint of the braid, and in both cases, the generator z i , i P t
1, . . . , m ´ u , corres-ponds to the braid all of whose strings are vertical, with the exception of the first string thatwraps around the i th puncture. (b) By Remarks 8(d), it follows that we may also take some or all of the punctures to beboundary components. In other words, Theorems 12 and 13 yield presentations of the braidgroups of any surface as defined at the beginning of Section 2.Presentations for B n p S q and B n p R P q will be given in Section 4.1. Results on the minimalcardinality of different types of generating sets of B n p M q , where M “ D , S or R P , aregiven in [84]. Positive presentations of braid groups of orientable surfaces were obtainedin [18]. Braid groups of the annulus, which are Artin-Tits groups of type B n , were studiedin [42, 78, 114, 121, 126, 136]. In terms of the presentation of Theorem 10, the ‘full twist’ braid ∆ n of B n is defined by: ∆ n “ p σ ¨ ¨ ¨ σ n ´ q n P B n . (16)It has a special rôle in the theory of Artin braid groups. Since τ n p σ ¨ ¨ ¨ σ n ´ q “ p n , n ´
1, . . . , 2 q , we see that ∆ n belongs to P n , and in terms of the generators of P n of equation (15),one may check that: ∆ n “ p A qp A A q ¨ ¨ ¨ p A n A n ¨ ¨ ¨ A n ´ n q .The parenthesised terms in this expression commute pairwise – geometrically, this is obvi-ous. This braid is the square of the well-known Garside element (or ‘half-twist’) ∆ n of B n (seeFigure 5), defined by: ∆ n “ p σ σ ¨ ¨ ¨ σ n ´ qp σ σ ¨ ¨ ¨ σ n ´ q ¨ ¨ ¨ p σ σ qp σ q .The notion of Garside element is important in the study of braid groups, notably in the13igure 5: The Garside element ∆ of B .resolution of the conjugacy problem in B n [26, 67], and in a more general setting, in thetheory of Garside groups and monoids [46, 113]. By [26, Lemma 2.5.1], we have: ∆ n σ i ∆ ´ n “ σ n ´ i for all 1 ď i ď n ´
1, (17)and the n th root σ ¨ ¨ ¨ σ n ´ of ∆ n that appears in equation (16) satisfies [5, 85, 129]: p σ ¨ ¨ ¨ σ n ´ q σ i p σ ¨ ¨ ¨ σ n ´ q ´ “ σ i ` for all 1 ď i ď n ´
2, and (18) p σ ¨ ¨ ¨ σ n ´ q σ n ´ p σ ¨ ¨ ¨ σ n ´ q ´ . (19)From equations (18)–(19), it follows that ∆ n commutes with all of the generators of The-orem 10, and so belongs to the centre of B n and of P n . A straightforward argument using theshort exact sequences (2) and (11) with M “ R enables one to show that ∆ n generates thecentre of the Artin braid groups. Theorem 15 (Chow [37]) . Let n ě . Then Z p B n q “ Z p P n q “ @ ∆ n D . Remark 16.
From Section 3.2, we know that B “ P “ t u , and B and P are infinite cyclic: Z p B q “ x σ y , and Z p P q “ x A y “ @ ∆ D .A small number of surface braid groups possess non-trivial centre: (a) If M “ S (resp. M “ R P ) and n ě Z p B n p M qq is cyclic of order 2 [68, 130, 151](see Section 4.1). (b) Let T denote the 2-torus. Then Z p B n p T qq is free Abelian of rank 2 [26, 136].Apart from these cases and a few other exceptions, most surface braid groups have trivialcentre. With the aid of Corollary 9, one may once more use the short exact sequences (2)and (11) to prove the following: Proposition 17 ([71, 136]) . Let M be a compact surface different from the disc and the sphere whosefundamental group has trivial centre. Then for all n ě , Z p B n p M qq is trivial. Remark 18.
The only compact surfaces whose fundamental group does not have trivialcentre are the real projective plane, the annulus, the torus, the Möbius band and the Kleinbottle.
One possible approach in the study of surface braid groups is to determine relationshipsbetween braid groups of different surfaces. The first result in this direction is the embedding14f P n in P n p M q given by Proposition 4 and its extensions to non-orientable surfaces and tothe full braid groups. The proof of Proposition 4 uses induction and the Fadell-Neuwirthshort exact sequences (9) and (11).Let N be a subsurface of M , and let m ě
0. Paris and Rolfsen studied the homomorphism B n p N q ÝÑ B n ` m p M q of braid groups induced by inclusion of N in M , and gave necessaryand sufficient conditions for it to be injective [136]. In another direction, it is reasonableto ask whether it is possible to obtain embeddings of braid groups of surfaces that are notinduced by inclusions (see [26, page 216, Problem 1] for example). The answer is affirmativein the case of covering spaces: Theorem 19 ([83]) . Let M be a compact, connected surface, possibly with a finite set of pointsremoved from its interior. Let d , n P N , and let Ă M be a d-fold covering space of M. Then the coveringmap induces an embedding of the n th braid group B n p M q of M in the dn th braid group B dn p Ă M q of Ă M. To prove Theorem 19, note that the inverse image of the covering map induces a mapbetween the permuted configuration spaces of M and Ă M . By restricting first to F n p M q , oneshows that this map induces the embedding mentioned in the statement of Theorem 19.Note however that the embedding does not restrict to the corresponding pure braid groups:the image of P n p M q is a subgroup of the ‘mixed’ subgroup π ` F dn p Ă M q{p S ˆ ¨ ¨ ¨ ˆ S q ˘ thatis not contained in P dn p Ă M q . Although the map in question appears at first sight to be verynatural, to our knowledge, it does not seem to have been studied previously in the literature.Theorem 19 should prove to be useful in the analysis of the structure of surface braid groups.As examples of this, one may deduce the linearity of the braid and mapping class groupsof R P (see Section 3.8), and one may classify their finite subgroups (see Section 4.2). Thefollowing is an immediate consequence of Theorem 19: Corollary 20.
Let n P N . The n th braid group of a non-orientable surface embeds in the n th braidgroup of its orientable double covering. In particular, B n p R P q embeds in B n p S q . Using the covering map, one may write down explicitly the images in B n p S q of elementsof B n p R P q . In this case, we see once more that such an embedding does not restrict to anembedding of the corresponding pure braid subgroups since if n ě P n p R P q has torsion4 (see Proposition 38(b)), while P n p S q has torsion 2 (see Proposition 34). Corollary 20 (andTheorem 19 in a more general context) would seem to be a significant step towards theresolution of the problem of Birman mentioned above concerning the relationship betweenthe braid groups of a non-orientable surface and those of its orientable double covering. Let n ě M “ R , and consider the short exact sequence (11):1 ÝÑ F n ´ ÝÑ P n p n ÝÑ P n ´ ÝÑ
1, (20)where we set p n “ p p n , n ´ q and we identify F n ´ naturally with the free group Ker p p n q – π p R z Q n ´ , x n q , where t x n u “ Q n z Q n ´ . Recall that geometrically, p n ‘forgets’ the n th string of a braid in P n , and using Proposition 11, it may be seen easily that p n admits asection s n : P n ´ ÝÑ P n given geometrically by adding a vertical string (in terms of thegenerators of Proposition 11, s n maps A i , j , 1 ď i ă j ď n ´
1, considered as an element15f P n ´ to A i , j , considered as an element of P n ). It follows that P n is isomorphic to the semi-direct product F n ´ ¸ ϕ P n ´ , where the action ϕ is given by conjugation via s n . By inductionon n , P n may be written as an iterated semi-direct product of free groups, known as the Artinnormal form : P n – F n ´ ¸ F n ´ ¸ ¨ ¨ ¨ ¸ F ¸ F . (21)The procedure for obtaining the Artin normal form of a pure braid β is known as Artin comb-ing , and involves writing β in the form β “ β n ´ ¨ ¨ ¨ β , where β i P F i . Since this expansionis unique and the word problem in free groups is soluble, this yields a (finite) algorithmto solve the word problem in P n . Furthermore, P n is of finite index in B n , and it is then aneasy matter to solve the word problem in B n also. The decomposition (21) is one of the fun-damental results in classical braid theory, and is frequently used to prove assertions about P n by induction, such as the study of the lower central series and the residual nilpotenceof P n [63], the bi-orderability of P n (see Theorem 30) and the fact that P n is poly-free (seeSection 5.2). Another application is obtained by taking M “ R and r “ p p n ,2 q : P n ÝÑ P sends ∆ n to the generator ∆ of P : Proposition 21 ([71]) . Let n ě . Then P n – P n ´ p R z Q q ˆ Z . The problem of deciding whether a decomposition of the form (21) exists for surfacebraid groups is thus fundamental. This was indeed a recurrent and central question duringthe foundation of the theory and its subsequent development during the 1960’s [24, 51, 53,54, 151]. An interesting and natural question, to which we shall refer henceforth as the split-ting problem , is that of whether the short exact sequences (9)–(12) split. Clearly, the existenceof a geometric cross-section on the level of configuration spaces implies that of a section onthe algebraic level, and in most cases the converse is true. Indeed, if M is aspherical, thisfollows from [15, 157], while if M “ S or R P , one may consult [72, 73]. We sum up thesituation as follows. Proposition 22.
Let M be a compact, connected surface (so m “ in equation (9)). Let ď r ă n,and suppose that r ě if M “ S and r ě if M “ R P . Then the Fadell-Neuwirth fibrationp n , r : F n p M q ÝÑ F r p M q (resp. q n , r : F n p M q{p S r ˆ S n ´ r q ÝÑ D r p M q ) admits a cross-section if andonly if the short exact sequence (9) (resp. (12)) splits. In the case of the pure braid groups, the splitting problem for (9) has been studied forother surfaces besides the plane. Fadell and Neuwirth gave various sufficient conditions forthe existence of a geometric section for p n , r [53]. If m ě B M ‰ ∅ ) then p n , r alwaysadmits a cross-section, and hence p p n , r q does too [70, 79]. So suppose that m “
0. If M “ S and r ě p n , r admits a cross-section [54], and thus the short exact sequence (10) splits. Inthe case M “ R P , Van Buskirk showed that the fibration p admits a cross-section [151](and hence so does the corresponding homomorphism p p q ), but that for n ě
2, neitherthe fibration p n ,1 nor the homomorphism p p n ,1 q admit a section (this is one of the casesnot covered by Proposition 22), this being a consequence of the fact that R P has the fixedpoint property. If M is the 2-torus then Birman exhibited an explicit cross-section for p n , n ´ if n ě n . This impliesthat (10) splits for all 1 ď r ă n . In the case of orientable surfaces without boundary of genusat least two, the question of the splitting of (11) was posed explicitly by Birman in 1969 [24],and was finally answered in [70]: 16 heorem 23 ([70]) . If M is a compact orientable surface without boundary of genus g ě , the shortexact sequence (10) splits if and only if r “ . For the remaining cases, the problem was studied in a series of papers [72, 73, 75], and acomplete solution to the splitting problem for (10) was given in [79]:
Theorem 24 ([79]) . Let ď r ă n and m ě , and let M be a connected surface.(a) If m ą or if M has non-empty boundary then p p n , r q admits a section.(b) Suppose that m “ and that B M “ ∅ . Then p p n , r q admits a section if and only if one of thefollowing conditions holds:(i) M “ S , the -torus T or the Klein bottle K .(ii) M “ R P , n “ and r “ .(iii) M ‰ R P , S , T , K and r “ . To obtain a positive answer to the splitting problem, it suffices of course to exhibit an ex-plicit section. However, in general it is very difficult to prove directly that the (generalised)Fadell-Neuwirth short exact sequences do not split. One of the principal methods that wasused in the proof of Theorem 24 is based on the following observation: let G be a group,and let K , H be normal subgroups of G such that H is contained in K . If the extension 1 ÝÑ K ÝÑ G ÝÑ Q ÝÑ ÝÑ K { H ÝÑ G { H ÝÑ Q ÝÑ
1. Thecondition on H is satisfied for example if H is an element of either the lower central series p Γ i p K qq i P N or of the derived series of K . In many parts of the proof of Theorem 24, it sufficesto take H “ Γ p K q , in which case K { H is the Abelianisation of K , to show that this secondextension does not split, which then implies that the first extension does not split.From the point of view of the splitting problem, it is thus helpful to know the lowercentral and derived series of the braid groups occurring in these group extensions. Theseseries have been calculated in many cases [17, 19, 77, 78, 82, 94]. The splitting problemfor the generalised Fadell-Neuwirth short exact sequence (12) has been studied in the case M “ S [73]. S and R P and period-icity As we saw in Theorem 7(a), the configuration spaces of surfaces different from S and R P are Eilenberg-Mac Lane spaces of type K p π , 1 q . For the two exceptional cases of S and R P ,the situation is very different, and in view of the relation with the homotopy groups of S (and S ), motivates the study of their configuration spaces. In the case of S , the followingproposition may be found in [29, 64]. An alternative proof was given in [85]. Proposition 25 ([29, 64]) . (a) The space F p S q (resp. D p S q ) has the homotopy type of S (resp. of R P ). Hence the universalcovering space of D p S q is F p S q .(b) If n ě , the universal covering space of F n p S q or of D n p S q has the homotopy type of the –sphere S . A similar result holds for the configuration spaces of R P : Proposition 26 ([72]) . (a) The universal covering of F p R P q is S .(b) For n ě , the universal covering space of F n p R P q or of D n p R P q has the homotopy type of S . n ě M “ S and that n ě M “ R P . From Propositions 25and 26, the universal covering space X of F n p M q is a finite-dimensional complex that hasthe homotopy type of S . Thus any finite subgroup of B n p M q acts freely on X , and so hasperiod 2 or 4 by [33, Proposition 10.2, Section 10, Chapter VII]. It thus follows that such asubgroup must be one of the subgroups that appear in the Suzuki-Zassenhaus classificationof periodic groups [1]. We shall come back to the finite subgroups of B n p M q in Section 4.2.Using results of [2, Section 2] allows one to obtain a periodicity result for any subgroup of B n p M q : Proposition 27 ([85]) . Let M “ S or R P , let n ě if M “ S and n ě if M “ R P , andlet G be a group abstractly isomorphic to a subgroup of B n p M q . Then there exists r ě such thatH r p G ; Z q – H r ` p G ; Z q for all r ě r . The connections between surface braid groups and the homotopy groups of S do notend there. If M is a surface, recall that an element of P n p M q is said to be Brunnian if itbecomes trivial after removing any one of its n strings. The subgroup Brun n p M q of Brunnianbraids may thus be seen to be the intersection Ş ni “ Ker p d i : P n p M q ÝÑ P n ´ p M qq , where d i corresponds geometrically to removing the i th string. The study of the homomorphisms d i allows one to introduce a simplicial structure on the pure braid groups of M . In this way,the following result was proved in [21]: Theorem 28 ([21]) . Let n ě . Then there is an exact sequence of the form: ÝÑ Brun n ` p S q ÝÑ Brun n ` p D q ÝÑ Brun n ` p S q ÝÑ π n p S q ÝÑ R P in [10], and to other surfacesin [133]. The hope is that one might understand better the homotopy groups of S using thestructure of Brunnian braid groups. A group G is said to be left orderable (resp. right orderable ) if it admits a total ordering ă thatis invariant under left (resp. right) multiplication in G . In other words, @ x , y , z P G , x ă y ùñ zx ă zy (resp. x ă y ùñ xz ă yz ).Any left ordering may be converted into a right ordering by considering inverses of ele-ments, but the two orderings will in general be different. A group is said to be biorderable if there exists a total ordering ă for which G is both right and left orderable. The classesof left orderable and biorderable groups are closed under subgroups, direct products andfree products (so free groups are biorderable), and that the class of left orderable groups isalso closed under extensions. It is an easy exercise to show that a left orderable group istorsion free. Further, a biorderable group has no generalised torsion (a group G is said tohave generalised torsion if there exist g , h , . . . , h k P G , g ‰
1, such that h gh ´ ¨ ¨ ¨ h k gh ´ k “ B n isleft orderable: Theorem 29 (Dehornoy [45, 47, 112, 113]) . B n is left orderable. B n is torsion free. Inthe wake of Dehornoy’s paper, a group of topologists came up with a different way of inter-preting his ordering of B n in terms of MC G p D , n q [65]. Short and Wiest described anotherapproach due to Thurston using the action of the mapping class group on the hyperbolicplane which in fact defines uncountably many different orderings on B n [146]. The readeris referred to the monograph [47] for a full description of these different points of view, aswell as to [113, Chapter 7]. These results have led to renewed interest in orderable groups,notably in the case of 3-manifold groups [30].If n ě B n is not biorderable since it has generalised torsion. Indeed, by equa-tion (17), we have ∆ n p σ ´ n ´ σ q ∆ ´ n “ p σ ´ n ´ σ q ´ . However: Theorem 30 (Falk and Randell, Kim and Rolfsen [47, 55, 113, 115, 143]) . P n is biorderable. Falk and Randell’s result is a consequence of the residual nilpotence of P n , and the factthat its lower central series quotients are torsion free. Kim and Rolfsen’s proof gives anexplicit biordering, and makes use of equation (20) and an ordering emanating from theMagnus expansion of free groups.Theorems 29 and 30 motivated the study of the (bi)orderability of surface braid groups.We summarise the known results as follows. (a) Since the braid groups of the S and R P have torsion (see Remark 16 and Section 4.1),they are not left orderable. (b) As was pointed out in [142], the short exact sequence (11) implies that the braid groups ofany compact surface different from S and R P are left orderable. Pure braid groups of com-pact, orientable surfaces without boundary of genus g ě g ě (c) If n ě M is a compact surface different from S and R P then the generalisationof Proposition 4 to B n p M q and the fact that B n is not biorderable imply that B n p M q is notbiorderable. Using equation (5) and the fact that mapping class groups of surfaces withnon-empty boundary are left orderable [144], it follows that the braid groups of any surfacewith boundary are left orderable. If M is without boundary and n ě B n p M q is left orderable. A group is said to be linear if it admits a faithful representation in a multiplicative groupof matrices over some field. The linearity of the braid groups is a classical problem (see[24, page 220, Problem 30] and [9, Question 1] for example). Krammer [117, 118] and Bi-gelow [23] showed that B n is linear. The question of the linearity of surface braid groups hasbeen the subject of various papers during the last few years [8, 9, 23, 28, 116]. The linearityof MC G p S , n q was proved in [8, 9, 28, 116], and that of B n p S q was obtained in [8, 9, 28].If n “ M . If n ď B n p R P q is linear because it is finite, while B p R P q is known to beisomorphic to a subgroup of GL p Z q [9]. With the help of Corollary 20 and the short exactsequence (6), we have the following results. 19 heorem 31 ([83]) . Let n P N .(a) Let M be a compact, connected surface, possibly with boundary, of genus zero if M is orientable,and of genus one if M is non-orientable. Then B n p M q is linear.(b) The mapping class groups MC G p R P , n q are linear.(c) Let T denote the -torus, and let x P T . Then B n ` p T q is linear if and only if B n p T z t x uq islinear. Consequently, B p T q is linear. In particular, the braid groups of R P and the Möbius band are linear. To our knowledge,very little is known about the linearity of braid groups of other surfaces. Together with the braid groups of R P , the braid groups of S are of particular interest, not-ably because they have non-trivial centre (see Proposition 33), and torsion (see Theorem 35).In Section 4.1, we begin by recalling some of their basic properties, including the character-isation of their torsion elements. In Section 4.2, we give the classification of the isomorph-ism classes of their finite subgroups, and in Section 4.3, this is extended to the isomorphismclasses of the virtually cyclic subgroups of their pure braid groups and of B n p S q . As well asbeing interesting in their own right, these results play an important rôle in the determinationof the lower algebraic K -theory of the group rings of the braid groups of these two surfaces(see Section 5). From this point of view, it is also necessary to have a good understanding ofthe conjugacy classes of the finite order elements and the finite subgroups. In this section, we recall briefly some of the basic properties of the braid groups of S and R P . We first consider B n p S q . The reader may consult [51, 54, 68, 71, 151] for more details.Consider the group homomorphism j : B n ÝÑ B n p S q of Section 2.3 induced by an inclusion j : D ÝÑ S . If β P B n then we shall denote its image j p β q simply by β . A presentation of B n p S q is as follows: Theorem 32 ([54]) . The following constitutes a presentation of the group B n p S q :generators: σ , . . . , σ n ´ .relations:(i) relations (13) and (14).(ii) the ‘surface relation’ of B n p S q : σ ¨ ¨ ¨ σ n ´ σ n ´ σ n ´ ¨ ¨ ¨ σ “ . (22)The surface relation may be seen geometrically to indeed represent the trivial elementof B n p S q (see [131, page 194] for example). It follows from Theorem 32 that B n p S q is aquotient of B n , and that its Abelianisation is isomorphic to Z p n ´ q . The first three spherebraid groups are finite: B p S q is trivial, B p S q is cyclic of order 2, and B p S q is isomorphicto Z ¸ Z , the action being the non-trivial one. For n ě B n p S q is infinite. Just as forthe Artin braid groups, the full twist braid of B n p S q plays an important part, and has someinteresting additional properties. 20 roposition 33 ([68, 71]) . Let n ě . Then:(a) ∆ n is the unique element in P n p S q of finite order, and is the unique element of B n p S q of order .(b) ∆ n generates the centre Z p B n p S qq of B n p S q . Taking M “ S , m “ r “ Proposition 34 ([71]) . Let n ě . Then P n p S q – P n ´ p S z Q q ˆ Z . From this and Proposition 17, it follows that Z p P n p S qq “ @ ∆ n D for all n ě n ě
3. Fadell and Van Buskirk showed that the element α “ σ ¨ ¨ ¨ σ n ´ σ n ´ is oforder 2 n in B n p S q [54]. Gillette and Van Buskirk later proved that if k P N then B n p S q hasan element of order k if and only if k divides one of 2 n , 2 p n ´ q or 2 p n ´ q [68]. UsingSeifert fibre space theory, Murasugi characterised the finite order elements of B n p S q and B n p R P q . In the case of the sphere, B n p S q , up to conjugacy and powers, there are preciselythree torsion elements: Theorem 35 ([130]) . Let n ě . Then the torsion elements of B n p S q are precisely the conjugates ofpowers of the three elements α , α “ σ ¨ ¨ ¨ σ n ´ σ n ´ and α “ σ ¨ ¨ ¨ σ n ´ σ n ´ , which are of order n, p n ´ q and p n ´ q respectively. Theorem 35 implies Gillette and Van Buskirk’s result, and in conjunction with Proposi-tion 33(a), yields the useful relation: ∆ n “ α n ´ ii for all i P t
0, 1, 2 u , (23)which implies that α i is an p n ´ i q th root of ∆ n . Since the permutation τ n p α i q consists ofan p n ´ i q -cycle and i fixed elements, we see that the α i are pairwise non conjugate. Oneinteresting fact about the group B n p S q is that it is generated by α and α [73], and so istorsion generated in the sense of [84]. Equations (18)–(19) also hold in B n p S q , and moregenerally, for i P t
0, 1, 2 u we have [85]: α li σ j α ´ li “ σ j ` l for all j , l P N satisfying j ` l ď n ´ i ´
1, (24) σ “ α i σ n ´ i ´ α ´ i (25)in B n and so also in B n p S q , in other words, conjugation by α i permutes the n ´ i elements σ , . . . , σ n ´ i ´ , α i σ n ´ i ´ α ´ i cyclically. These relations prove to be very useful in the study ofthe finite and virtually cyclic subgroups of B n p S q .We now turn to the braid groups of the projective plane. Some basic references are [71,73, 81, 82, 151]. We first recall a presentation of B n p R P q due to Van Buskirk [151]: Theorem 36 ([151]) . The following constitutes a presentation of the group B n p R P q :generators: σ , . . . , σ n ´ , ρ , . . . , ρ n .relations:(i) relations (13) and (14).(ii) σ i ρ j “ ρ j σ i for j ‰ i , i ` .(iii) ρ i ` “ σ ´ i ρ i σ ´ i for ď i ď n ´ .(iv) ρ ´ i ` ρ ´ i ρ i ` ρ i “ σ i for ď i ď n ´ .(v) ρ “ σ σ ¨ ¨ ¨ σ n ´ σ n ´ σ n ´ . . . σ σ . ρ i corresponds geometrically to an element of the fundamentalgroup of R P based at the i th basepoint. A presentation of P n p R P q was given in [73]. Fromthese presentations, we see that the first two braid groups of R P are finite: B p R P q “ P p R P q – Z , P p R P q is isomorphic to the quaternion group Q of order 8, and B p R P q is isomorphic to the generalised quaternion group of order 16 [151]. For n ě B n p R P q isinfinite. If n ě
2, the Abelianisation of B n p R P q is Z , while that of P n p R P q is Z n . If M “ R P and m “
0, the map p of equation (7) admits a geometric section given by taking the vectorproduct of two directions, and so by equation (10), P p R P q is isomorphic to a semi-directproduct of a free group of rank 2 by Q [151]; an explicit action was given in [71, 82].We recall that the virtual cohomological dimension of a group is equal to the (common)cohomological dimension of its torsion-free subgroups of finite index [33, page 226]. As anapplication of the Fadell-Neuwirth short exact sequence (10), Proposition 34 and the fact that P p R P q – Q , one may compute the virtual cohomological dimension of the braid groupsof S and R P : Proposition 37 ([86]) . Let M be equal to S (resp. R P ), and let n ě (resp. n ě ). Then thevirtual cohomological dimension of B n p M q and of P n p M q is equal to n ´ (resp. n ´ ). For n ě
2, Murasugi showed that ∆ n generates the centre of B n p R P q [130]. The followingproposition summarises some other basic results concerning the torsion of the braid groupsof R P . Proposition 38 ([71, 81]) . Let n ě . Then:(a) B n p R P q has an element of order k if and only if k divides either n or p n ´ q .(b) the (non-trivial) torsion of P n p R P q is precisely and .(c) the full twist ∆ n is the unique element of B n p R P q of order . If M “ S or R P , it follows from Propositions 33 and 38 that the kernel of the short exactsequence (6) is generated by ∆ n . In [71, Proposition 26], it was proved that the followingelements of B n p R P q : a “ σ ´ n ´ ¨ ¨ ¨ σ ´ ρ b “ σ ´ n ´ ¨ ¨ ¨ σ ´ ρ are of order 4 n and 4 p n ´ q respectively. By [71, Remark 27], we have α “ a n “ ρ n ¨ ¨ ¨ ρ β “ b n ´ “ ρ n ´ ¨ ¨ ¨ ρ . (26)It is clear that α and β are pure braids of order 4. The finite order elements of B n p R P q had previously been characterised in [130], but the results are less transparent than in thecase of S given by Theorem 35. For example, it is not clear what the orders of the giventorsion elements are, even for elements of P n p R P q . In [80], Murasugi’s characterisation wassimplified somewhat as follows. Theorem 39 ([80]) . Let n ě , and let x P B n p R P q . Then x is of finite order if and only if thereexist i P t
1, 2 u and ď r ď n ` ´ i such that x is a power of a conjugate of the following element: p ρ r σ r ´ ¨ ¨ ¨ σ q r { l p σ r ` ¨ ¨ ¨ σ n ´ σ i ´ r ` q p { l (27) where p “ p n ` ´ i q ´ r and l “ gcd p p , 2 r q . Further, this element is of order l. a (resp. b ) is one of the above elementsby taking r “ n and i “ r “ n ´ i “ P n p R P q of order 4: Proposition 40 ([80]) . Let n ě , and let x P P n p R P q be an element of order .(a) In B n p R P q , x is conjugate to an element of (cid:32) α , β , α ´ , β ´ ( .(b) The centraliser Z P n p R P q p x q of x in P n p R P q is equal to x x y . It was shown in [84] that if n ě
2, there are p n ´ q ! p n ´ q conjugacy classes of ele-ments of order 4 in P n p R P q (there is a misprint in the statement of [84, Proposition 11], B n p R P q should read P n p R P q ). The analysis of the conjugacy classes of finite order elementsof B n p R P q is the subject of work in progress [87].The elements a and b have some interesting properties that mirror those of equations (24)–(25) that may be used to study the structure of B n p R P q . From [71, pages 777–778], conjuga-tion by a ´ permutes cyclically the elements of the following sets: ! σ , . . . , σ n ´ , a ´ σ n ´ a , σ ´ , . . . , σ ´ n ´ , a ´ σ ´ n ´ a ) and ! ρ , . . . ρ n , ρ ´ , . . . , ρ ´ n ) ,and conjugation by b ´ permutes cyclically the following elements: σ , . . . , σ n ´ , b ´ σ n ´ b , σ ´ , . . . , σ ´ n ´ , b ´ σ ´ n ´ b .Note that there is a typographical error in line 16 of [71, page 778]: it should read ‘. . . showsthat b ´ σ n ´ b “ σ ´ . . . ’, and not ‘. . . shows that b ´ σ n ´ b “ σ ´ . . . ’. By [83, pages 865–866], we also have that: ∆ n a ∆ ´ n “ a ´ and p ∆ n a ´ q b p a ∆ ´ n q “ b ´ for all i “
1, . . . , n ´
1. (28)As for S , such relations are very useful in the study of the finite and virtually cyclic sub-groups of B n p R P q . S and R P We start by considering the pure braid groups of S and R P . In the case of P n p S q , thereare only two finite subgroups for n ě t e u and thatgenerated by the full twist ∆ n . In the case of P n p R P q , there are more possibilities: Proposition 41 ([80]) . Up to isomorphism, the maximal finite subgroups of P n p R P q are:(a) Z if n “ .(b) Q if n “
2, 3 .(c) Z if n ě . As we mentioned above, in Proposition 40, we know the numebr of conjugacy classes ofthe elements of P n p R P q of order 4, both in P n p R P q and in B n p R P q .We now turn to B n p S q and B n p R P q . The results of Theorem 35 and Proposition 38 implythat we know the isomorphism classes of their finite cyclic subgroups. This leads naturallyto the question as to which isomorphism classes of finite groups are realised as subgroupsof these two groups. From [73], if n ě B n p S q contains an isomorphic copy of the23nite group B p S q of order 12 if and only if n ı B n p S q , it was observed that the commutator subgroup Γ ` B p S q ˘ of B p S q is isomorphic to a semi-direct product of Q by a free group of rank 2 [77] (see also [95]). Thequestion of the realisation of Q as a subgroup of B n p S q was posed explicitly by R. Brown [3]in connection with the Dirac string problem and the fact that the fundamental group ofSO p q is isomorphic to Z [51, 99, 132]. The existence of a subgroup of B p S q isomorphic to Q was studied by J. G. Thompson [149]. It was shown in [74] that if n ě B n p S q containsa subgroup isomorphic to Q if and only if n is even. The construction of Q given in [74]may be generalised. If m ě
2, let Dic m denote the dicyclic group of order 4 m . It admits apresentation of the form: A x , y (cid:12)(cid:12)(cid:12) x m “ y , yxy ´ “ x ´ E . (29)If in addition m is a power of 2 then we will refer to the dicyclic group of order 4 m as the generalised quaternion group of order 4 m , and denote it by Q m . For example, if m “ Q . For i P t
0, 2 u , we have: ∆ n α i ∆ ´ n “ α i , where α i “ α α i α ´ “ α i { α i α ´ i { , (30)and the group Dic p n ´ i q is realised in terms of the generators of B n p S q by the subgroup @ α i , ∆ n D , which we shall call the standard copy of Dic p n ´ i q in B n p S q [77, 79]. Let T ˚ (resp. O ˚ ,I ˚ ) denote the binary tetrahedral group of order 24 (resp. the binary octahedral group of order48, the binary icosahedral group of order 120). The groups T ˚ , O ˚ and I ˚ , to which we refercollectively as the binary polyhedral groups , admit presentations of the form [40, 41]: x p , 3, 2 y “ A A , B (cid:12)(cid:12)(cid:12) A p “ B “ p AB q E ,where p “
3, 4, 5 respectively, and the element A p is central and is the unique element oforder 2. The group T ˚ also admits the following presentation [158, page 198]: A P , Q , X (cid:12)(cid:12)(cid:12) X “ P “ Q , PQP ´ “ Q ´ , XPX ´ “ Q , XQX ´ “ PQ E , (31)and thus T ˚ is a semi-direct product of x P , Q y – Q by x X y – Z . Also, T ˚ is abstractly asubgroup of O ˚ and of I ˚ . We refer the reader to [1, 40, 41, 85, 158] for more properties ofthe binary polyhedral groups. One important property that they share with the family ofcyclic and dicyclic groups is that they possess a unique element of order 2 (except for cyclicgroups of odd order), which is a ramification of the fact that they are periodic in the senseof Section 3.6, and that in the non-cyclic case, this element generates the centre of the group.Further, the quotient by the unique subgroup of order 2 induces a correspondence betweenthe family of even-order cyclic, dicyclic and binary polyhedral groups with the finite sub-groups of SO p q , the dicyclic group Dic m being associated with the dihedral group Dih m of order 2 m , and T ˚ , O ˚ and I ˚ being associated respectively with the polyhedral groups A , S and A . Using Kerckhoff’s solution to the Nielsen realisation problem, Stukow classifiedthe isomorphism classes of the finite subgroups of MC G p S , n q , showing that they are finitesubgroups of SO p q , with appropriate restrictions on n [147]. The analysis of equation (6)then leads to the complete classification of the isomorphism classes of the finite subgroupsof B n p S q . 24 heorem 42 ([76]) . Let n ě . The isomorphism classes of the maximal finite subgroups of B n p S q are as follows:(a) Z p n ´ q if n ě .(b) Dic n .(c) Dic p n ´ q if n “ or n ě .(d) T ˚ if n ” .(e) O ˚ if n ”
0, 2 mod 6 .(f) I ˚ if n ”
0, 2, 12, 20 mod 30 . The geometric realisation of the finite subgroups of B n p S q may be obtained by letting thecorresponding finite subgroup of MC G p S , n q act by homeomorphisms on S (see [76, Sec-tion 3.2] for more details). Concretely, consider the geometric definition given in Section 2.1.We visualise the space S ˆ r
0, 1 s as that confined between two concentric spheres (see [99,page 41] for example). For the (maximal) subgroups Dic n , Z p n ´ q and Dic p n ´ q , we at-tach strings, each representing the constant path in terms of the definition of Section 2.2, to n (resp. n ´ n ´
2) equally-spaced points on the equator, and 0 (resp. 1, 2) points at thepoles. For T ˚ , O ˚ and I ˚ , the n strings are attached symmetrically with respect to the asso-ciated regular polyhedron. We now let the corresponding finite subgroup of MC G p S , n q act on the inner sphere as a group of homeomorphisms, so that the set of basepoints is leftinvariant globally. This yields a subgroup of B n p S q , and one may check that it is exactly thegiven finite subgroup of Theorem 42. In particular, a complete rotation of the inner spheregives rise to the full twist braid ∆ n , and is a manifestation of the famous ‘Dirac string trick’(see [51, Section 6], [99, page 43] or [123, page 628]).Algebraic representations of some of the binary polyhedral groups have been found:see [76, Remarks 3.2 and 3.3] for realisations of T ˚ in B p S q and B p S q . Note however thatin the second case there is a misprint, and the expression for δ should read δ “ σ ´ σ ´ σ ´ σ ´ σ ´ σ ´ σ σ σ σ σ σ .By [76, Proposition 1.5], there are at most two conjugacy classes of each isomorphismclass of the finite subgroups of B n p S q , and there is a single conjugacy class for each maximalfinite subgroup.As another application of Corollary 20, we obtain the classification of the finite subgroupsof B n p R P q . Theorem 43 ([83]) . Let n ě . The isomorphism classes of the finite subgroups of B n p R P q are thesubgroups of the following groups:(a) Dic n .(b) Dic p n ´ q if n ě .(c) O ˚ if n ”
0, 1 p mod 3 q .(d) I ˚ if n ”
0, 1, 6, 10 p mod 15 q . Although the groups involved in the statements of Theorems 42 and 43 are basically thesame, there is a difference in terms of those that are maximal. The finite groups described inTheorem 43(a)–(d) are maximal in an abstract sense, while those of Theorem 42 are maximalwith respect to inclusion. This is partly related to the fact that up to powers and conjugacy, B n p S q has just three conjugacy classes of finite order elements, while B n p R P q has manymore. It could happen that a subgroup of B n p R P q that is abstractly isomorphic to a propersubgroup of one of the groups given in Theorem 43 be maximal with respect to inclusion.This is the subject of work in progress [87]. 25he proof of Theorem 43 is obtained by combining Corollary 20 with Theorem 42. Inthis way, we establish a list of possible finite subgroups of B n p R P q . Some of these possibil-ities are not realised (notably T ˚ is not realised if n ” p mod 3 q , despite apparently beingcompatible with the embedding). The final step is to prove that the subgroups given in thestatement of Theorem 43 are indeed realised for the given values of n . This is achieved in asimilar manner to that of the finite subgroups of B n p S q . As for S , it is also possible to giveexplicit algebraic realisations of the dicyclic subgroups of B n p R P q . For example, we obtain x a , ∆ n y – Dic n and @ b , ∆ n a ´ D – Dic p n ´ q using equation (28) [83, Proposition 15]. Explicitrealisations of T ˚ and O ˚ have been found in B p R P q [87], and applying Corollary 20 tothem yields isomorphic copies in B p S q .As an application of Theorem 43 and the short exact sequence (6) for R P , one may alsoobtain an alternative proof of the classification of the finite subgroups of MC G p R P , n q dueto Bujalance, Cirre and Gamboa [34]. Theorem 44 ([34]) . Let n ě . The finite subgroups of MC G p R P , n q are abstractly isomorphic tothe subgroups of the following groups:(a) the dihedral group Dih n of order n.(b) the dihedral group Dih p n ´ q if n ě .(c) S if n ”
0, 1 p mod 3 q .(d) A if n ”
0, 1, 6, 10 p mod 15 q . One useful fact that is used to classify the virtually cyclic subgroups of B n p S q is theknowledge of the centraliser and normaliser of its maximal finite cyclic and dicyclic sub-groups. Note that if i P t
0, 1 u , the centraliser of α i , considered as an element of B n , is equalto x α i y [22, 93]. A similar equality holds in B n p S q and is obtained using equation (6) and thecorresponding result for MC G p S , n q , which is due to Hodgkin [101]. Proposition 45 ([85]) . Let i
P t
0, 1, 2 u , and let n ě .(a) The centraliser of x α i y in B n p S q is equal to x α i y , unless i “ and n “ , in which case it is equalto B p S q .(b) The normaliser of x α i y in B n p S q is equal to: $’’&’’% x α , ∆ n y – Dic n if i “ A α , α ´ ∆ n α E – Dic p n ´ q if i “ x α y – Z p n ´ q if i “ ,unless i “ and n “ , in which case it is equal to B p S q .(c) If i P t
0, 2 u , the normaliser of the standard copy of Dic p n ´ i q in B n p S q is itself, except when i “ and n “ , in which case the normaliser is equal to α ´ σ ´ x α , ∆ y σ α , and is isomorphic to Q . A related problem is that of knowing which powers of α i are conjugate in B n p S q , for each i P t
0, 1, 2 u . The answer is that such powers are either equal or inverse: Proposition 46 ([85]) . Let n ě and i P t
0, 1, 2 u , and suppose that there exist r , m P Z such that α mi and α ri are conjugate in B n p S q .(a) If i “ then α m “ α r .(b) If i P t
0, 2 u then α mi “ α ˘ ri . MC G p S , n q [101]. Using The-orem 35, Proposition 46 implies that if F is a finite cyclic subgroup of B n p S q then that theonly possible actions of Z on F are the trivial action and multiplication by ´
1. This also hasconsequences for the possible actions of Z on dicyclic subgroups of B n p S q . S and R P In view of the Farrell-Jones Fibred Isomorphism Conjecture (see Section 5.1), in order tocalculate the lower algebraic K -theory of the group rings of the braid groups of S and R P ,it is necessary to know their virtually cyclic subgroups. Recall that a group is said to be virtually cyclic if it contains a cyclic subgroup of finite index. It is clear from the definitionthat any finite subgroup is virtually cyclic, hence it suffices to concentrate on the infinite virtually cyclic subgroups of these braid groups, which are in some sense their ‘simplest’infinite subgroups. The classification of the virtually cyclic subgroups of these braid groupsis an interesting problem in its own right, and helps us to understand better the structure ofthese two groups. For the whole of this section, we refer the reader to [85] for more details.Recall that by results of Epstein and Wall [50, 154], any infinite virtually cyclic group G is isomorphic to F ¸ Z or to G ˚ F G , where F is finite and r G i : F s “ i P t
1, 2 u . Weshall say that G is of Type I or Type II respectively. This enables us to establish a list of thepossible infinite virtually cyclic subgroups of a given infinite group Γ , providing one knowsits finite subgroups (which by Theorems 42 and 43 is the case for our braid groups). Thereal difficulty lies in deciding whether the groups belonging to this list are indeed realisedas subgroups of Γ .Let n ě
4. In the case of P n p S q , as we saw in Section 4.2, @ ∆ n D is the only non-trivialfinite subgroup, and since it is equal to the centre of P n p S q by Proposition 33(b), it is theneasy to see that the infinite virtually cyclic subgroups of P n p S q are isomorphic to Z or to Z ˆ Z . The classification of the virtually cyclic subgroups of P n p R P q was obtained in [80],using Proposition 41. Although the structure of the finite subgroups of P n p R P q differs for n “ n ě
4, up to isomorphism, the infinite virtually cyclic subgroups of P n p R P q arethe same for all n ě Theorem 47 ([80]) . Let n ě . The isomorphism classes of the infinite virtually cyclic subgroups ofP n p R P q are Z , Z ˆ Z and Z ˚ Z Z . One obtains the classification of the virtually cyclic subgroups of P n p R P q as a immediatecorollary of Proposition 41 and Theorem 47 [80]. One of the key results needed in the proofof Theorem 47 is that P n p R P q has no subgroup isomorphic to Z ˆ Z , which follows in astraightforward manner from Proposition 40(b). This fact allows us to eliminate severalpotential Type I and Type II subgroups.We now turn to the case of B n p S q . As we observed previously in Section 4.1, if n ď B n p S q is a known finite group, and so we shall suppose in what follows that n ě
4. If G is a group, let Aut p G q (resp. Out p G q ) denote the group of its automorphisms (resp. outerautomorphisms). We define the following two families of virtually cyclic groups. Definition.
Let n ě (1) Let V p n q be the family comprised of the following Type I virtually cyclic groups: (a) Z q ˆ Z , where q is a strict divisor of 2 p n ´ i q , i P t
0, 1, 2 u , and q ‰ n ´ i if n ´ i is odd. (b) Z q ¸ ρ Z , where q ě p n ´ i q , i P t
0, 2 u , q ‰ n ´ i if n is odd, and ρ p q P Aut ` Z q ˘ is multiplication by ´
1. 27 c) Dic m ˆ Z , where m ě n ´ i and i P t
0, 2 u . (d) Dic m ¸ ν Z , where m ě n ´ i , i P t
0, 2 u , p n ´ i q{ m is even, and where ν p q P Aut p Dic m q is defined by: ν p qp x q “ x ν p qp y q “ xy (32)for the presentation (29) of Dic m . (e) Q ¸ θ Z , for n even and θ P Hom p Z , Aut p Q qq , for the following actions: (i) θ p q “ Id. (ii) θ “ α , where α p q P Aut p Q q is given by α p qp i q “ j and α p qp j q “ k , where Q “t˘ ˘ i , ˘ j , ˘ k u . (iii) θ “ β , where β p q P Aut p Q q is given by β p qp i q “ k and β p qp j q “ j ´ . (f) T ˚ ˆ Z for n even. (g) T ˚ ¸ ω Z for n ”
0, 2 mod 6, where ω p q P Aut p T ˚ q is the automorphism defined in termsof the presentation (31) by: $’&’% P ÞÝÑ
QPQ
ÞÝÑ Q ´ X ÞÝÑ X ´ . (33) (h) O ˚ ˆ Z for n ”
0, 2 mod 6. (i) I ˚ ˆ Z for n ”
0, 2, 12, 20 mod 30. (2)
Let V p n q be the family comprised of the following Type II virtually cyclic groups: (a) Z q ˚ Z q Z q , where q divides p n ´ i q{ i P t
0, 1, 2 u . (b) Z q ˚ Z q Dic q , where q ě p n ´ i q{ i P t
0, 2 u . (c) Dic q ˚ Z q Dic q , where q ě n ´ i strictly for some i P t
0, 2 u . (d) Dic q ˚ Dic q Dic q , where q ě n ´ i for some i P t
0, 2 u . (e) O ˚ ˚ T ˚ O ˚ , where n ”
0, 2 mod 6.Finally, let V p n q be the family comprised of the elements of V p n q and V p n q . In what follows, ρ , ν , α , β and ω will denote the actions defined in parts (1)(b), (1)(d), (1)(e)(ii), (1)(e)(iii) and(1)(g) respectively.Up to a finite number of exceptions, we may then classify the infinite virtually cyclicsubgroups of B n p S q . Theorem 48 ([85]) . Suppose that n ě .(1) If G is an infinite virtually cyclic subgroup of B n p S q then G is isomorphic to an element of V p n q .(2) Conversely, let G be an element of V p n q . Assume that the following conditions hold:(a) if G – Q ¸ α Z then n R t
6, 10, 14 u .(b) if G – T ˚ ˆ Z then n R t
4, 6, 8, 10, 14 u .(c) if G – O ˚ ˆ Z or G – T ˚ ¸ ω Z then n R t
6, 8, 12, 14, 18, 20, 26 u .(d) if G – I ˚ ˆ Z then n R t
12, 20, 30, 32, 42, 50, 62 u .(e) if G – O ˚ ˚ T ˚ O ˚ then n R t
6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38 u .Then there exists a subgroup of B n p S q isomorphic to G.(3) Let G be equal to T ˚ ˆ Z (resp. O ˚ ˆ Z ) if n “ (resp. n “ ). Then B n p S q has no subgroupisomorphic to G. emark 49. Together with Theorem 42, Theorem 48 yields a complete classification of thevirtually cyclic subgroups of B n p S q with the exception of a the thirty-eight cases for whichthe problem of their existence is open, given by the excluded values of n in the above condi-tions (2)(a)–(e) but removing the two cases of part (3) which we know not to be realised.The proof of Theorem 48 is divided into two stages. In conjunction with Theorem 42,Epstein and Wall’s results give rise to a family V C of virtually cyclic groups with the prop-erty that any infinite virtually cyclic subgroup of B n p S q belongs to V C . The first stage is toshow that any such subgroup belongs in fact to the subfamily V p n q of V C . This is achievedin several ways: the analysis of the centralisers and normalisers of the finite order elementsof B n p S q given in Propositions 45 and 46; the study of the (outer) automorphism groups ofthe finite subgroups of Theorem 42; and the periodicity of B n p S q given by Proposition 27.Putting together these reductions allows us to prove Theorem 48(1). The structure of the fi-nite subgroups of B n p S q imposes strong constraints on the possible Type II subgroups, andthe proof in this case is more straightforward than that for the Type I subgroups. The secondstage of the proof consists in proving the realisation of the elements of V p n q as subgroups of B n p S q and to proving parts (2) and (3) of Theorem 48. The construction of the elements of V p n q involving finite cyclic and dicyclic groups as subgroups of B n p S q is largely algebraic,and relies heavily on equations (24) and (25) that describe the action by conjugation of the α i on the generators of B n p S q . In contrast, the realisation of the elements of V p n q involvingthe binary polyhedral groups is geometric in nature, and occurs on the level of mappingclass groups via the relation (6) and the constructions of the finite subgroups of B n p S q ofTheorem 42.Since the open cases of Remark 49 only occur for even values of n , the complete classi-fication of the infinite virtually cyclic subgroups of B n p S q for all n ě Theorem 50 ([85]) . Let n ě be odd. Then up to isomorphism, the following groups are the infinitevirtually cyclic subgroups of B n p S q .(I) (a) Z m ¸ θ Z , where θ p q P t Id , ´ Id u , m is a strict divisor of p n ´ i q , for i P t
0, 2 u , and m ‰ n ´ i.(b) Z m ˆ Z , where m is a strict divisor of p n ´ q .(c) Dic m ˆ Z , where m ě is a strict divisor of n ´ i for i P t
0, 2 u .(II) (a) Z q ˚ Z q Z q , where q divides p n ´ q{ .(b) Dic q ˚ Z q Dic q , where q ě is a strict divisor of n ´ i, and i P t
0, 2 u Since in Theorem 48 we are considering the realisation of the various subgroups up toisomorphism, one may ask whether each of the given elements of V p n q is unique up toisomorphism. It turns out that with with the exception of Q ˚ Q Q , abstractly there isonly one way (up to isomorphism) to embed the amalgamating subgroup in each of thetwo factors, in other words for all of the other elements of V p n q , the group is unique up toisomorphism [85]. Note that this result refers to abstract isomorphism classes of the givenType II groups, and does not depend on the fact that the amalgamated products occurring aselements of V p n q are realised as subgroups of B n p S q . In the exceptional case of Q ˚ Q Q ,abstractly there are two isomorphism classes defined respectively by: K “ A x , y , a , b (cid:12)(cid:12)(cid:12) x “ y , a “ b , yxy ´ “ x ´ , bab ´ “ a ´ , x “ a , y “ b E .and K “ A x , y , a , b (cid:12)(cid:12)(cid:12) x “ y , a “ b , yxy ´ “ x ´ , bab ´ “ a ´ , x “ b , y “ a b E .29f n ě K and K are realised as subgroups of B n p S q , with the possibleexception of K if n P t
6, 14, 18, 26, 30, 38 u [85].Using equation (6), another consequence of Theorem 48 is the classification of the vir-tually cyclic subgroups of MC G p S , n q , with a finite number of exceptions (see [85, The-orem 14] for more details).A similar analysis of the isomorphism classes of the infinite virtually cyclic subgroups of B n p R P q is the subject of work in progress [87, 88]. K -theory of surface braid groups In this section, we indicate how the results of the previous sections may be used to computethe lower algebraic K -theory of the group rings of surface braid groups. In Section 5.1, westart by recalling two conjectures of Farrell and Jones, whose validity for a given groupprovides a recipe to calculate its lower K -groups. In Section 5.2, we outline the proof of thefact that surface braid groups of aspherical surfaces satisfy the Farrell-Jones conjecture, andin Section 5.3, we shall see how to extend this result to the braid groups of S and R P . Inorder to calculate the lower algebraic K -theory of a group using this approach, one needs tobe able to determine the lower K -groups of its virtually cyclic subgroups, as well as certain Nil groups that are related to these subgroups. In Section 5.4, we recall some general methodsthat one may use to determine these lower K - and Nil groups. Finally, in Section 5.5, we stateand outline the proofs of the known results, namely the lower K -groups of braid groups ofaspherical surfaces, and of P n p S q , P n p R P q and B p S q . Let G be a discrete group and let Z r G s denote its integral group ring. The approach tothe algebraic K -theoretical calculations of Z r G s , which we outline in this section, consists inusing the Farrell-Jones (Fibred) Isomorphism Conjecture that proposes to compute the K -groups of Z r G s from two sources: first, the algebraic K -theory of the class of virtually cyclicsubgroups of G , and secondly, homological data. Definition.
A collection F of subgroups of G is called a family if: (a) if H P F and A ď H then A P F , and (b) if H P F and g P G then gHg ´ P F .The collection of finite subgroups of G , denoted F in , and that of the virtually cyclicsubgroups of G , denoted V C , are examples of families of G . Given a family F of subgroupsof G , a universal space for G with isotropy in F is a G -space E F that satisfies the followingproperties: (a) the fixed set E F H is non empty and contractible for all H P F , and (b) the fixed set E F H is empty for all H R F .Universal spaces exist and are unique up to G -homotopy [150]. If F consists of the trivialsubgroup of G , the corresponding universal space is the universal space for principal G -bundles, and if F “ F in , the corresponding universal space is the universal space for properactions. If F “ V C , we denote the corresponding universal space by EG . Although universalspaces exist for any family of subgroups of G , models for E V C that are suitable for makingcomputations are still sparse, but there are some constructions for hyperbolic groups [108]and
CAT p q groups [58, 122]. 30et R be a ring with unit, and let Or F is the orbit category of the group G restricted to thefamily F . J. Davis and W. Lück constructed a functor K : Or F p G q ÝÑ Spectra [44], whosevalue at the orbit G { H is the non-connective algebraic K -theory spectrum of Pedersen-Weibel [137], and which satisfies the fundamental property that π i p K p G { H qq “ K i p Z r H sq .The K -theoretical formulation of the Farrell-Jones isomorphism conjecture is as follows (onemay consult [44, 111] for more details). Isomorphism Conjecture (IC).
Let G be a discrete group. Then the assembly mapA VC : H Gn p EG ; K q ÝÑ H Gn p pt ; K q – K n p Z r G qs , induced by the projection EG ÝÑ pt is an isomorphism, where H Gn p´ ; K q is a generalised equivarianthomology theory with local coefficients in the functor K , and EG is a model for the universal spacefor the family V C . A version of IC that is suitable for more general situations is the
Fibred Farrell-Jones Conjec-ture (FIC), which we now describe. Given a group homomorphism ϕ : K ÝÑ G and a family F of subgroups of a group G that is also closed under finite intersections, the induced familyon K by ϕ is defined by: ϕ ˚ F “ t H ď K | ϕ p H q P F u . Fibred Isomorphism Conjecture (FIC) ([11]) . Let G be a discrete group and let F be a family ofsubgroups of G. The pair p G , F q is said to satisfy the Fibred Isomorphism Conjecture if for allgroup homomorphisms ϕ : K ÝÑ G, the assembly mapA ϕ ˚ F : H Kn p E ϕ ˚ F ; K q ÝÑ H Kn p pt ; K q is an isomorphism for all n P Z . Note that the validity of FIC implies that of IC by taking K “ G and ϕ “ Id. Two of thefundamental properties of FIC are as follows.
Theorem 51 ([12]) . If G is a group that satisfies FIC and H is a subgroup of G then H also satisfiesFIC.
Theorem 52 ([12]) . Let f : G ÝÑ Q be a surjective group homomorphism. Assume that p Q , V C p Q qq satisfies FIC and that IC is satisfied for all H P f ˚ V C p Q q . Then p G , V C p G qq satisfies FIC. The Fibred Isomorphism Conjecture has been verified for word hyperbolic groups byA. Bartels, W. Lück and H. Reich [11], for
CAT p q groups by C. Wegner [155], and forSL n p Z q , n ě
3, by A. Bartels, W. Lück, H. Reich and H. Rueping [13]. We record two ofthese results for future reference.
Theorem 53 ([11]) . If G is a hyperbolic group in the sense of Gromov then G satisfies FIC.
Theorem 54 ([155]) . If G is a
CAT p q group then G satisfies FIC. The validity of the Fibred Isomorphism Conjecture has recently been shown for braidgroups by D. Juan-Pineda and L. Sánchez [111] (see Theorems 61, 62 and 63). We will sketchthe proofs in Sections 5.2 and 5.3. The original isomomorphism conjecture by T. Farrell andL. Jones was stated in [61]. They proved several cases of the conjecture for the pseudoisotopy functor. Here we shall only treat the case of the conjecture for the algebraic K -theory functor.31 .2 The K -theoretic Farrell-Jones Conjecture for braid groups of aspher-ical surfaces In this section, we outline the ingredients needed to prove that braid groups of the plane ora compact surface other than the sphere or the projective plane satisfy FIC. The main toolsthat we shall require are the concepts of poly-free and strongly poly-free groups, which wenow recall.
Definition.
A group G is said to be poly-free if there exists a filtration 1 “ G Ă G Ă ¨ ¨ ¨ Ă G n “ G of normal subgroups such that each quotient G i ` { G i is a finitely-generated freegroup.The following result is due to D. Juan-Pineda and L. Sánchez [111]. Theorem 55 ([111]) . If G is a poly-free group then G satisfies FIC.
The proof uses induction on the length of the filtration and the fact that the initial induc-tion step is applied to a hyperbolic group.Suppose first that M is either the complex plane or a compact surface with non-emptyboundary. Taking r “ F m ` n ´ p Int p M qq ÝÑ F m , n p Int p M qq ÝÑ F m ,1 p Int p M qq ,so by Theorem 7, we obtain the short exact sequence (9):1 ÝÑ P n ´ p M z Q m ` q ÝÑ P n p M z Q m q ÝÑ π p M z Q m q ÝÑ i P
1, . . . , n , P i ´ p M z Q n ´ i ` m ` q is normal in P i p M z Q n ` m ´ i q , andthe corresponding quotient is isomorphic to the free group π p M z Q m q that is of finite rank.Setting G i “ P i p M z Q n ´ i q for all i P
0, 1, . . . , n gives rise to a filtration that yields a poly-freestructure for P n p M q , and applying Theorem 55, we obtain the following: Theorem 56 ([111]) . Assume that M “ C or that M is a compact surface with non-empty boundary.Then the pure braid group P n p M q is poly-free, and thus satisfies FIC. Now suppose that M is a compact aspherical surface with empty boundary. Taking m “ r “ F n ´ p M z Q q ÝÑ F n p M q p ÝÑ F p M q “ M , and by Theorem 7 induces the following short exact sequence:1 ÝÑ P n ´ p M z Q q ÝÑ P n p M q p ÝÑ π p M q ÝÑ M is aspherical, the group π p M q is finitely-generated Abelian or hyperbolic, and sosatisfies FIC by Theorems 53 and 54. Now Ker p p q – P n ´ p M z Q q is poly-free and p ´ p C q – P n ´ p M z Q q ¸ C where C is any cyclic subgroup of π p M q , which is also poly-free, hencein both cases they satisfy FIC. Theorem 52 then implies that P n p M q satisfies FIC. Puttingtogether the two cases gives: Theorem 57 ([111]) . Assume that M “ C or that M is a compact surface other than the sphere orthe projective plane. Then the pure braid group P n p M q satisfies FIC for all n ě . The next step is to go from P n p M q to B n p M q . The idea is to embed the given group in alarger group (a wreath product in fact) that satisfies FIC and then apply Theorem 51. Westart by adding one more property to the definition of poly-free group.32 efinition ([4]) . A group G is called strongly poly-free (SPF) if it is poly-free and the follow-ing condition holds: for each g P G there exists a compact surface M and a diffeomorphism f : M ÝÑ M such that the action C g by conjugation of g on G i ` { G i may be realised geomet-rically, i.e. the following diagram commutes: π p M q f (cid:47) (cid:47) ϕ (cid:15) (cid:15) π p M q G i ` { G i C g (cid:47) (cid:47) G i ` { G i ϕ ´ (cid:79) (cid:79) where ϕ is a suitable isomorphism.The following result was proved in [4]. Theorem 58 ([4]) . Assume that M “ C or that M is a compact surface with non-empty boundary.Then P n p M q is an SPF group for all n ě . One of the main theorems in [111] is the following:
Theorem 59 ([111]) . Let G be an SPF group, and let H be a finite group. Then the wreath productG (cid:111)
H satisfies FIC.
We also recall the following result due to A. Bartels, W. Lück and H. Reich [12].
Lemma 60 ([12]) . Let ÝÑ K ÝÑ G ÝÑ Q ÝÑ be a short exact sequence of groups. Assumethat K is virtually cyclic and that Q satisfies FIC. Then G satisfies FIC. Moreover, given a finite extension of a group of the form1 ÝÑ G ÝÑ Γ ÝÑ H ÝÑ H is a finite group, it follows that there is an injective homomorphism Γ ã ÝÑ G (cid:111) H [63,Algebraic Lemma]. Since P n p M q is of finite index in B n p M q by equation (2), it follows fromTheorems 58 and 59 and the above observation that: Theorem 61 ([111]) . Assume that M “ C or that M is a compact surface other than the sphere orthe projective plane. Then the full braid group B n p M q satisfies FIC for all n ě . S and R P The results of Section 5.2 treat the case of the braid groups of all surfaces with the exceptionof S and R P . In this section, we outline the proof of the fact that the braid groups of thesetwo surfaces also satisfy FIC.Let n P N . Recall from Section 4.1 that P n p S q is trivial for n “
1, 2, and that P p S q – Z ,hence these groups satisfy trivially FIC. So suppose that n ą
3. Taking m “ r “ M “ S in equation (7), we obtain the following fibre bundle: F n ´ p C q « F n ´ p S q ÝÑ F n p S q ÝÑ F p S q ,and by Theorem 7, its long exact sequence in homotopy yields the Fadell-Neuwirth shortexact sequence: 1 ÝÑ P n ´ p C z Q q ÝÑ P n p S q ÝÑ P p S q ÝÑ G “ P n ´ p C z Q q is an SPF group as it is part of the filtration of P n ´ p C q , henceTheorems 51 and 59 imply that π p F n p S qq “ P n p S q satisfies FIC. In [128], S. Millán-Vosslerproved that B n p S q fits in an extension of the form:1 ÝÑ G ÝÑ B n p S q{x ∆ n y ÝÑ S n ÝÑ B n p S q{x ∆ n y satisfiesFIC by Theorems 51 and 59. Taking M “ S in equation (6) and applying Lemma 60, we seethat B n p S q satisfies FIC. Summing up these considerations, we obtain: Theorem 62 ([111]) . Both P n p S q and B n p S q satisfy FIC for all n ě . The situation for R P is similar. Consider first the case of P n p R P q . By Section 4.1, P p R P q – Z , P p R P q – Q and P p R P q – F ¸ Q . It follows that P p R P q and P p R P q satisfy FIC as they are finite, and that P p R P q also satisfies FIC by Theorem 53 since it is(virtually) hyperbolic. Now let n ą
3. Taking the short exact sequence (10) with M “ R P and r “ ÝÑ G ÝÑ P n p R P q ÝÑ Q ÝÑ G “ P n ´ p R P z Q q is an SPF group. It follows once more from Theorems 59 and 51that P n p R P q satisfies FIC for all n ą
3. Passing to the case of B n p R P q , note that B p R P q – Z and B p R P q – Q by Section 4.1. Now G is not normal in B n p R P q , but the intersection H of its conjugates in B n p R P q is a finite-index normal subgroup of both G and B n p R P q , andfor all n ě B n p R P q fits in a short exact sequence:1 ÝÑ H ÝÑ B n p R P q ÝÑ B n p R P q{ H ÝÑ B n p R P q{ H is finite. Since G is SPF, it follows from [128] that H is also SPF, and weconclude from Theorem 59 and [63, Algebraic Lemma] that B n p R P q satisfies FIC. We recordthese results as follows. Theorem 63 ([111]) . Both P n p R P q and B n p R P q satisfy FIC for all n ě . As we mentioned before, the validity of FIC should, in principle, furnish the necessary toolsneeded to compute the algebraic K -groups of the group rings for surface braid groups. Wewill concentrate in this section on lower K -groups, that is K i p´q for i ď
1. Recall that thedomain of the assembly map in the statement of IC is H Gn p EG ; K q . (34)This is an extraordinary equivariant homology theory whose coefficients are the functor K . The input of K consists of the orbits of the type G { V , where V varies over the virtu-ally cyclic subgroups of G , and its values at these orbits are the spectra K p G { V q whosehomotopy groups are given by π i p K p G { V qq – K i p Z r V sq . On the other hand, there is anAtiyah-Hirzebruch-type spectral sequence that computes the equivariant homology groupsof equation (34) whose E -term is given by: E p , q – H p p BG ; (cid:32) K q ( q ,34here this is now an ordinary homology theory whose local coefficients are the algebraic K -groups of the virtually cyclic subgroups of G , and which appear as isotropy at differentsubcomplexes of BG “ EG { G . In summary, in order to compute H Gn p EG ; K q , we need tounderstand the following: (a) the algebraic K -groups K i p Z r V sq for all i ď n and all virtually cyclic subgroups V of G . (b) the spaces EG and BG . (c) how these groups and spaces are assembled together. This is encoded in the spectralsequence.Let V be a virtually cyclic group. As indicated in Section 4.3, V is either finite, of Type I(so is isomorphic to a semidirect product of the form F ¸ Z , where F is finite), or of Type II(so is isomorphic to an amalgam of the form G ˚ F G where F is of index 2 in both G and G .). In the Type II case, V fits in a short exact sequence of the form:1 ÝÑ F ÝÑ V ÝÑ Dih ÝÑ F is a finite group and Dih is the infinite dihedral group. The computation of thealgebraic K -theory groups for each of these cases is currently an active area of study. Ingeneral, finite groups may be treated with induction-restriction methods, see [134]. We shallcomment on the case of the finite subgroups of B n p S q later on. In order to study the algebraic K -groups of Type I and Type II groups, we need some background.Let R be an associative ring with unit, and let R r t s denote its polynomial ring. Let ε : R r t s ÝÑ R be the augmentation map induced by t ÞÝÑ
1, and let ε ˚ : K i p R r t sq ÝÑ K i p R q be the homomorphism induced on K -groups. Definition.
Let R be an associative ring with unit. The Bass Nil groups of R are defined by: NK i p R q “ Ker p ε ˚ q .The Bass Nil groups appear in the study of K -groups of virtually cyclic groups via theBass-Heller-Swan fundamental theorem: Theorem 64 (Bass, Heller and Swan [14]) . Let R be an associative ring with unit, and let R r t , t ´ s be its Laurent polynomial ring. Then for all i P Z ,K i p R r t , t ´ sq – K i p R q ‘ K i ´ p R q ‘ NK i p R q ‘ NK i p R q . (35)Observe that if a group G is of the form F ˆ Z for some group F , its group ring may bedescribed as follows: Z r G s “ Z r F ˆ Z s – Z r F sr t , t ´ s .From Theorem 64, we thus obtain: Corollary 65.
The algebraic K-groups of a group V “ F ˆ Z are of the form:K i p Z r V sq – K i p Z r F sq ‘ K i ´ p Z r F sq ‘ NK i p Z r F sq ‘ NK i p Z r F sq . (36)If V is as above and virtually cyclic, so F is finite, equation (36) tells us that we needto compute the K -groups of the group ring Z r F s as well as the Bass Nil groups. If on theother hand, V is a non-trivial semi-direct product of the form V “ F ¸ α Z , where α denotesthe action of Z on F , the corresponding group ring is the twisted Laurent polynomial ring Z r F s α r t , t ´ s . This case has been studied by T. Farrell and W. C. Hsiang in [60]. They found35 formula similar to that of equation (35) of Bass-Heller-Swan, but the terms NK i p Z r F sq ‘ NK i p Z r F sq should be replaced by: NK i p Z r F s , α q ‘ NK i p Z r F s , α ´ q ,which are similar groups that take into account the action of Z on F . These are now knownas Farrell-Hsiang twisted Nil groups. Together with the Bass Nil groups, these Nil groupsare the subject of investigation, full computations are few and far between, and they are ingeneral very large groups due to the following fact: Theorem 66 ([59, 140]) . Let R be a ring. Then both the Bass Nil and Farrell-Hsiang Nil groups areeither trivial or are not finitely generated.
The case of virtually cyclic groups of the form V “ A ˚ F B is handled by the foundationalwork of F. Waldhausen [153]. There is a long exact sequence of the form: ¨ ¨ ¨ ÝÑ K n p Z r F sq ÝÑ K n p Z r A sq ‘ K n p Z r B sq ÝÑ K n p Z r V sq{ Nil Wn ÝÑ K n ´ p Z r F sqÝÑ K n ´ p Z r A sq ‘ K n ´ p Z r B sq ÝÑ K n ´ p Z r V sq{ Nil Wn ´ ÝÑ ¨ ¨ ¨ ,where the term
Nil Wn denotes the Waldhausen Nil groups defined by:
Nil Wn “ Nil Wn p Z r F s ; Z r A z F s , Z r B z F sq .A somewhat better description of the Waldhausen Nil groups Nil Wn is given in the workof J. Davis, K. Khan and A. Ranicki [43] who identify these groups with Farrell-Hsiang Nilgroups of a group of the form F ¸ Z for a suitable subgroup isomorphic to Z of the infinitedihedral group Dih “ V { F .Some general results for algebraic K -groups for group rings of finite groups are known.We record some of them in the following proposition. Proposition 67.
Let F be a finite group. Then:(a) The groups K i p Z r F sq are finitely-generated Abelian groups for all i ě ´ .(b) The groups K i p Z r F sq vanish for i ă ´ .(c) The groups NK i p Z r F sq vanish for i ă . The first part is proved in [119] if i ě i “ ´
1, the second part is provedin [36], and the third part in [36, 62].On the other hand, the NK i p Z r F sq are non trivial for i “
0, 1 even for simple finite vir-tually cyclic groups, such as F “ Z ˆ Z or Z [156]. It is therefore a challenge to de-cide whether the algebraic K -groups of infinite virtually cyclic groups are finitely-generatedgroups. The only known case that is always finitely generated is in degree ´ Proposition 68 ([62]) . Let V be a virtually cyclic group. Then:(a) K ´ p Z r V sq is a finitely-generated group that is generated by the images of the homomorphismsK ´ p Z r G sq ÝÑ K ´ p Z r V sq induced by the inclusions G ã ÝÑ V, where G runs over the conjugacyclasses of the finite subgroups of V.(b) the groups K i p Z r V sq are trivial for i ă ´ . We finish this section by recalling the lower K -groups of the integers Z , which is funda-mental for many of the calculations that follow.36 roposition 69. For the ring Z , the following results hold:(a) K i p Z q is a finitely-generated Abelian group for all i P Z .(b) K p Z q “ Z and K p Z q “ Z .(c) K i p Z q “ for all i ă .(d) NK i p Z q “ for all i P Z .(e) K i p Z r Z sq – K i p Z q for all i P Z . The proof of (a) may be found in [139], and that of (d) follows from the regularity of Z andthe work of D. Quillen who showed that the Nil groups of regular rings vanish [138]. Part (b)is a consequence of the fact that K p Z q is just the units of Z , and that every finitely-generatedprojective module over Z is free, and part (c) follows from the equality dim p Z q “
0. Finally,part (e) is implied by the previous results and the Bass-Heller-Swan theorem (Theorem 64).We are interested in the non-trivial lower K -groups. Given a group G , we define r K i p Z r G sq to be the Whitehead group Wh p G q if i “
1, the reduced K -group r K p Z r G sq if i “
0, and theusual K i -groups if i ă
0. The results stated are valid for these reduced groups and for i ď
1, and some of the computational results will be given for these reduced groups. In thiscontext, we may reinterpret Proposition 69 by saying that r K i p Z q “ r K i p Z r Z sq “ i ď We now gather together the information obtained in the preceding sections. We start withthe case of torsion-free braid groups, which by Corollary 9 are precisely the braid groupsof the complex plane or compact surfaces other than S or R P . In this case, the only virtu-ally cyclic subgroups of G are trivial or infinite cyclic. By Proposition 69, the reduced lower K -groups of Z and of Z r Z s vanish, and the coefficients of the spectral sequence needed tocompute the equivariant homology groups of equation (34), whose coefficients are the re-duced K -groups, are all trivial, so this spectral sequence collapses, thus yielding the trivialgroup. Hence: Theorem 70 ([4, 111]) . Let G be the braid group (pure or full) of the complex plane or of a compactsurface without boundary different from S and R P . Then r K i p Z r G sq “ for all i ď . We now turn to the case of the pure braid groups of S and R P . From the discussionjust before the statement of Theorem 47, if n ě
4, the infinite virtually cyclic subgroups V of P n p S q are isomorphic to Z or Z ˆ Z and it is well known that r K i p Z r V sq “ P p S q and P p S q aretrivial and P p S q “ Z and the reduced lower K -groups of these groups also vanish, wehave the following: Theorem 71 ([109]) . For all i ď and n ě , r K i p Z r P n p S qsq “ . The case of P n p R P q is somewhat more involved. The reason is that by Proposition 41, Q is realised as a subgroup of P n p R P q if n P t
2, 3 u , and its reduced K -group is non trivialin degree 0. More precisely, if i ď r K i p Z r Q sq “ Z if i “
00 otherwise.37ince P p R P q – Z and P p R P q – Q , we thus obtain the lower K -groups of these twogroups. So assume that n ě
3. With the exception of Q , the reduced lower K -groups of theother finite subgroups of P n p R P q , as well as those of the infinite virtually cyclic subgroupsgiven by Theorem 47, are trivial. From this, one may show that the reduced lower algebraic K -groups of P n p R P q are as follows. Theorem 72 ([110]) . Suppose that n ě and i ď . Then: r K i p Z r P n p R P qsq “ Z if n “ and i “ otherwise. The situation for the braid groups of both S and R P is currently the subject of invest-igation. By Theorem 48, the virtually cyclic subgroups of B n p S q are known for all n ą K -groups of thefinite subgroups of B n p S q have been carried out. The r K -groups of the binary polyhedralgroups and of the dicyclic groups Dic m , m ď
13, were computed in [148]. The Whiteheadgroup of all finite subgroups of B n p S q and the K ´ -groups of the binary polyhedral groupsand of many dicyclic groups were determined in [95]. We remark that these K ´ -groups ex-hibit new structural phenomena that had not appeared previously in the study of the loweralgebraic K -theory of other groups, such as the existence of torsion. These calculations aresomewhat involved and require techniques from different areas.Passing to the case of the computation of the lower algebraic K -theory of B n p S q , n ě n “ B p S q is isomorphic to an amalgamated product of the form Q ˚ Q T ˚ [95]. By Theorem 42 and [76, Proposition 1.5], the maximal finite subgroups of B p S q are isomorphic to T ˚ or Q , and there is a single conjugacy class of each. Moreover,we obtain the infinite virtually cyclic subgroups of B p S q from Theorem 48, and from this,one may deduce the maximal virtually cyclic subgroups of B p S q : Theorem 73 ([95]) . (a) Every infinite maximal virtually cyclic subgroup of B p S q is isomorphic to Q ˚ Q Q orto Q ¸ Z for one of the three possible actions (see part (e) of the definition of the family V p n q inSection 4.3).(b) If V is a finite maximal cyclic subgroup of B p S q then V – T ˚ .(c) Let G be a group that is isomorphic to Q ¸ Z for one of the three possible actions, or to Q ˚ Q Q . Then B p S q possesses both maximal and non-maximal virtually cyclic subgroups that areabstractly isomorphic to G. Calculations of the reduced lower algebraic K -groups of the groups given in Theorem 73may be found in [95]. The next step is to find a model for EB p S q . Since B p S q is anamalgam of finite groups, it follows that it is Gromov hyperbolic. If G is a hyperbolic group,D. Juan-Pineda and I. Leary found a model for EG [108]. In our case, this can be describedas: EB p S q “ T ˚ D ,which is the join of a suitable tree T and a countable discrete set D . From this description, italso follows that the equivariant homology groups of equation (34) are isomorphic to: H B p S q n p T ; t K uq ‘ ¨˝ à V P Max p VC p B p S qqq N I L n p V q ˛‚ ,38here N I L n denotes one of the Nil groups described above according to the type of infinitevirtually cyclic group involved, and Max p V C p B p S qqq is a set of representatives of the con-jugacy classes of maximal infinite virtually cyclic subgroups of B p S q . We summarise thefinal result for B p S q as follows. Theorem 74 ([95]) . The reduced lower algebraic K-groups for B p S q are given by r K i p Z r B p S qsq “ $’’’&’’’% Z ‘ Nil , if i “ Z ‘ Nil , if i “ Z ‘ Z if i “ ´ if i ă ´ ,where for i “
0, 1 , Nil i – à r p Z q ‘ W s ,2 p Z q denotes two infinite countable direct sums of copies of Z , and W is an infinitely-generatedAbelian group of exponent or . Since the groups Q ¸ Z and Q ˚ Q Q that appear in the statement of Theorem 73appear as maximal subgroups of B p S q , they contribute in a non-trivial manner via the Bass,Farrell-Hsiang and Waldhausen Nil groups to the reduced lower K -groups of Z r B p S qs . (a) We have concentrated on the lower algebraic K i -groups, that is, in degrees i ď
1. This isdue to our lack of knowledge about K i p Z r V sq if V is a virtually cyclic group if i ą
1. Little isknown about the K i -groups for i ą
1, even for finite groups. One example for i “ (b) In [95], J. Guaschi, D. Juan-Pineda and S. Millán-López developed techniques to computereduced lower algebraic K -groups of many of the finite subgroups of B n p S q , in particularfor small values of n . Some other results concerning these computations will appear in [96].How these subgroups are assembled to build up all of the reduced lower K -groups of aspecific braid group B n p S q for n ą EB n p S q . Note that the amalgamatedproduct structure of B p S q is specific to this case, and we cannot hope for it to be carriedover to braid groups with more strings. (c) The case of B n p R P q is also still open if n ě
3. However, many features are currentlybeing studied: the classification of the virtually cyclic subgroups of B n p R P q [87, 88], as wellas their K -groups and models for the corresponding universal spaces. (d) In work in progress, it has been proved by D. Juan-Pineda and L. Sánchez that if G is a hyperbolic group, then rank p K i p Z r G sqq ă 8 for all i P Z . From this we have thatrank p K i p B p Z r S sqqq ă 8 for all i P Z . References [1] A. Adem and R. J. Milgram, Cohomology of finite groups, Springer-Verlag, New York-Heidelberg-Berlin (1994).
2] A. Adem and J. H. Smith, Periodic complexes and group actions,
Ann. Math. (2001), 407–435.[3] Algebraic topology discussion list, January 2004, .[4] C. S. Aravinda, F. T. Farrell and S. K. Roushon, Algebraic K -theory of pure braid groups, AsianJ. Math. (2000), 337–343.[5] E. Artin, Theorie der Zöpfe, Abh. Math. Sem. Univ. Hamburg (1925), 47–72.[6] E. Artin, Theory of braids, Ann. Math. (1947), 101–126.[7] E. Artin, Braids and permutations, Ann. Math. (1947), 643–649.[8] V. Bardakov, Linear representations of the group of conjugating automorphisms and the braidgroups of some manifolds, Siberian Math. J. (2005), 13–23.[9] V. Bardakov, Linear representations of the braid groups of some manifolds, Acta Appl. Math. (2005), 41–48.[10] V. Bardakov, R. Mikhailov, V. V. Vershinin and J. Wu, Brunnian braids on surfaces, Algebr. Geom.Topol. (2012) 1607–1648.[11] A. Bartels, W. Lück and H. Reich, The K -theoretic Farrell-Jones conjecture for hyperbolicgroups, Invent. Math. (2008), 29–70.[12] A. Bartels, W. Lück and H. Reich, On the Farrell-Jones conjecture and its applications,
J. Topol. (2008), 57–86.[13] A. Bartels, W. Lück, H. Reich and H. Rueping, K - and L -theory of group rings over GL n p Z q ,preprint, arXiv:1204.2418 .[14] H. Bass, Algebraic K -theory, W. A. Benjamin Inc., New York-Amsterdam, 1968.[15] H. J. Baues, Obstruction theory on homotopy classification of maps, Lecture Notes in Math-ematics , Springer-Verlag, Berlin, 1977.[16] P. Bellingeri, On presentation of surface braid groups, J. Algebra (2004), 543-563.[17] P. Bellingeri, S. Gervais and J. Guaschi, Lower central series of Artin-Tits and surface braidgroups,
J. Algebra (2008), 1409–1427.[18] P. Bellingeri and E. Godelle, Positive presentations of surface braid groups,
J. Knot TheoryRamif. (2007), 1219–1233.[19] P. Bellingeri, E. Godelle and J. Guaschi, Exact sequences, lower central series and representa-tions of surface braid groups, preprint, arXivmath:1106.4982 .[20] A. J. Berrick, F. R. Cohen, E. Hanbury, Y.-L. Wong and J. Wu, Braids: Introductory Lectures onBraids, Configurations and Their Applications, Lecture Notes Series, Institute for Mathemat-ical Sciences, National University of Singapore, Vol. 19, World Scientific, 2010.[21] A. J. Berrick, F. R. Cohen, Y.-L. Wong and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. (2006), 265–326.[22] D. Bessis, F. Digne and J. Michel, Springer theory in braid groups and the Birman-Ko-Leemonoid, Pacific J. Math. (2002), 287–309.
23] S. Bigelow, Braid groups are linear,
J. Amer. Math. Soc. (2001), 471–486.[24] J. S. Birman, On braid groups, Comm. Pure Appl. Math. (1969), 41–72.[25] J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl.Math. (1969), 213–238.[26] J. S. Birman, Braids, links and mapping class groups, Ann. Math. Stud. , Princeton UniversityPress, 1974.[27] J. S. Birman, Mapping class groups of surfaces, in Braids (Santa Cruz, CA, 1986), 13–43, Con-temp. Math. , Amer. Math. Soc., Providence, RI, 1988.[28] J. S. Birman and T. E. Brendle, Braids: a survey, in Handbook of knot theory, 19–103, edited byW. Menasco and M. Thistlethwaite, Elsevier B. V., Amsterdam, 2005.[29] C.-F. Bödigheimer, F. R. Cohen and M. D. Peim, Mapping class groups and function spaces,Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. , 17–39,Amer. Math. Soc., Providence, RI, 2001.[30] S. Boyer, D. Rolfsen and B. Wiest, Orderable 3-manifold groups,
Ann. Inst. Fourier (2005),243–288.[31] E. Brieskorn, Sur les groupes de tresses (d’après V. I. Arnol’d), Séminaire Bourbaki, 24èmeannée (1971/1972), Exp. No. 401, Lecture Notes in Mathematics , Springer, Berlin, 1973,21–44.[32] E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math. (1972), 245–271.[33] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics , Springer-Verlag, NewYork-Berlin (1982).[34] E. Bujalance, F. J. Cirre and J. M. Gamboa, Automorphism groups of the real projective planewith holes and their conjugacy classes within its mapping class group, Math. Ann. (2005),253–275.[35] G. Burde and H. Zieschang, Knots, Second edition, de Gruyter Studies in Mathematics, ,Walter de Gruyter & Co., Berlin, 2003.[36] D. W. Carter, Lower K -theory of finite groups, Comm. Algebra (1980), 1927–1937.[37] W.-L. Chow, On the algebraical braid group, Ann. Math. (1948) 654–658.[38] F. R. Cohen, Introduction to configuration spaces and their applications, in [20], 183–261.[39] F. R. Cohen and S. Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. (2002), 1705–1748.[40] H. S. M. Coxeter, Regular complex polytopes, Second edition, Cambridge University Press,Cambridge, 1991.[41] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Ergebnisseder Mathematik und ihrer Grenzgebiete, Vol. 14, Fourth edition, Springer-Verlag, Berlin, 1980.[42] J. Crisp, Injective maps between Artin groups, in Geometric group theory down under, 1996,119–137, Eds. J. Cossey, C. F. Miller III, W. D. Neumann, M. Shapiro, de Gruyter, 1999.
43] J. Davis, K. Khan and A. Ranicki, Algebraic K -theory over the infinite dihedral group: analgebraic approach, Algebr. Geom. Topol. (2011), 2391–2436.[44] J. Davis and W. Lück, Spaces over a category and assembly maps in isomorphism conjecturesin K - and L -theory, K-Theory (1998), 201–252.[45] P. Dehornoy, Braid groups and left distributive operations, Trans. Amer. Math. Soc. (1994),115–150.[46] P. Dehornoy, F. Digne, E. Godelle, D. Krammer and J. Michel, Garside Theory, in preparation.[47] P. Dehornoy, I. Dynnikov, D. Rolfsen and B. Wiest, Ordering braids, Mathematical Surveys andMonographs , American Mathematical Society, Providence, RI, 2008.[48] P. Deligne, Les immeubles des groupes de tresses généralisés,
Invent. Math. (1972), 273–302.[49] J. L. Dyer, The algebraic braid groups are torsion-free: an algebraic proof, Math. Z. (1980),157–160.[50] D. B. A. Epstein, Ends, in Topology of 3-manifolds and related topics (Proc. Univ. GeorgiaInstitute, 1961), 110–117, Prentice-Hall, Englewood Cliffs, N.J., 1962.[51] E. Fadell, Homotopy groups of configuration spaces and the string problem of Dirac,
DukeMath. J. (1962), 231–242.[52] E. Fadell and S. Y. Husseini, Geometry and topology of configuration spaces, Springer Mono-graphs in Mathematics, Springer-Verlag, Berlin, 2001.[53] E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand. (1962), 111–118.[54] E. Fadell and J. Van Buskirk, The braid groups of E and S , Duke Math. J. (1962), 243–257.[55] M. Falk and R. Randell, The lower central series of a fiber-type arrangement, Invent. Math. (1985), 77–88.[56] M. Falk and R. Randell, Pure braid groups and products of free groups, in Braids (Santa Cruz,CA, 1986), 217–228, Contemp. Math. , Amer. Math. Soc., Providence, RI, 1988.[57] B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series ,Princeton University Press, Princeton, NJ, 2012.[58] D. Farley, Constructions of EVC and
EFBC for groups acting on
CAT p q spaces, Algebr. Geom.Topol. (2010), 2229–2250.[59] F. T. Farrell, The nonfiniteness of Nil, Proc. Amer. Math. Soc. (1977), 215–216.[60] F. T. Farrell and W. C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. LondonMath. Soc. (1981), 308–324.[61] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic K -theory, J. Amer. Math. Soc. (1993), 249–297.[62] F. T. Farrell and L. E. Jones, The lower algebraic K -theory of virtually infinite cyclic groups, K-Theory (1995), 13–30.[63] F. T. Farrell and S. K. Roushon, The Whitehead groups of braid groups vanish, Inter-nat. Math. Res. Notices (2000), 515–526.
64] E. M. Feichtner and G. M. Ziegler, The integral cohomology algebras of ordered configurationspaces of spheres,
Doc. Math. (2000), 115–139.[65] R. A. Fenn, M. T. Greene, D. Rolfsen, C. Rourke and B. Wiest, Ordering the braid groups, Pac.J. Math. (1999), 49–74.[66] R. H. Fox and L. Neuwirth, The braid groups,
Math. Scand. (1962), 119–126.[67] F. A. Garside, The braid group and other groups, Quart. J. Math. Oxford (1969), 235–254.[68] R. Gillette and J. Van Buskirk, The word problem and consequences for the braid groups andmapping class groups of the 2-sphere, Trans. Amer. Math. Soc. (1968), 277–296.[69] C. H. Goldberg, An exact sequence of braid groups,
Math. Scand. (1973), 69–82.[70] D. L. Gonçalves and J. Guaschi, On the structure of surface pure braid groups, J. Pure Appl.Algebra (2003), 33–64 (due to a printer’s error, this article was republished in its entiretywith the reference (2004) 187–218).[71] D. L. Gonçalves and J. Guaschi, The roots of the full twist for surface braid groups,
Math. Proc.Camb. Phil. Soc. (2004), 307–320.[72] D. L. Gonçalves and J. Guaschi, The braid groups of the projective plane,
Algebr. Geom. Topol. (2004), 757–780.[73] D. L. Gonçalves and J. Guaschi, The braid group B n , m p S q and the generalised Fadell-Neuwirthshort exact sequence, J. Knot Theory Ramif. (2005), 375–403.[74] D. L. Gonçalves and J. Guaschi, The quaternion group as a subgroup of the sphere braidgroups, Bull. London Math. Soc. (2007), 232–234.[75] D. L. Gonçalves and J. Guaschi, The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence, Geom. Dedicata (2007), 93–107.[76] D. L. Gonçalves and J. Guaschi, The classification and the conjugacy classes of the finite sub-groups of the sphere braid groups,
Algebr. Geom. Topol. (2008), 757–785.[77] D. L. Gonçalves and J. Guaschi, The lower central and derived series of the braid groups of thesphere, Trans. Amer. Math. Soc. (2009), 3375–3399.[78] D. L. Gonçalves and J. Guaschi, The lower central and derived series of the braid groups of thefinitely-punctured sphere,
J. Knot Theory Ramif. (2009), 651–704.[79] D. L. Gonçalves and J. Guaschi, Braid groups of non-orientable surfaces and the Fadell-Neuwirth short exact sequence, J. Pure Appl. Algebra (2010), 667–677.[80] D. L. Gonçalves and J. Guaschi, Classification of the virtually cyclic subgroups of the purebraid groups of the projective plane,
J. Group Theory (2010), 277–294.[81] D. L. Gonçalves and J. Guaschi, The Borsuk-Ulam theorem for maps into a surface, TopologyAppl. (2010), 1742–1759.[82] D. L. Gonçalves and J. Guaschi, The lower central and derived series of the braid groups of theprojective plane,
J. Algebra (2011), 96–129.[83] D. L. Gonçalves and J. Guaschi, Surface braid groups and coverings,
J. London Math. Soc. (2012), 855–868.
84] D. L. Gonçalves and J. Guaschi, Minimal generating and normally generating sets for the braidand mapping class groups of the disc, the sphere and the projective plane,
Math. Z. , to appear.[85] D. L. Gonçalves and J. Guaschi, The classification of the virtually cyclic subgroups of thesphere braid groups, monograph to appear in the series SpringerBriefs in Mathematics, arXivmath:1110.6628 .[86] D. L. Gonçalves and J. Guaschi, Some homotopy properties of the inclusion F n p S q ã ÝÑ S n for S either S or R P and the virtual cohomological dimension of B n p S q and P n p S q , work inprogress.[87] D. L. Gonçalves and J. Guaschi, Conjugacy classes of finite subgroups of the braid groups ofthe projective plane, work in progress.[88] D. L. Gonçalves and J. Guaschi, The classification of the virtually cyclic subgroups of the braidgroups of the projective plane, work in progress.[89] J. González-Meneses, New presentations of surface braid groups, J. Knot Theory Ramif. (2001), 431–451.[90] J. González-Meneses, Ordering pure braid groups on closed surfaces, Pac. J. Math. (2002),369–378.[91] J. González-Meneses, Basic results on braid groups,
Ann. Math. Blaise Pascal (2011), 15–59.[92] J. González-Meneses and L. Paris, Vassiliev invariants for braids on surfaces, Trans. Amer. Math.Soc. (2004), 219–243.[93] J. González-Meneses and B. Wiest, On the structure of the centralizer of a braid,
Ann. Sci. ÉcoleNorm. Sup. (2004), 729–757.[94] E. A. Gorin and V. J. Lin, Algebraic equations with continuous coefficients and some problemsof the algebraic theory of braids, Math. USSR Sbornik (1969), 569–596.[95] J. Guaschi, D. Juan-Pineda and S. Millán-López, The lower algebraic K -theory of the braidgroups of the sphere, preprint, arXiv:1209.4791 .[96] J. Guaschi, D. Juan-Pineda and S. Millán-López, The lower algebraic K -theory of the finitesubgroups of the braid groups of the sphere, work in progress.[97] M.-E. Hamstrom, Homotopy properties of the space of homeomorphisms on P and the Kleinbottle, Trans. Amer. Math. Soc. (1965), 37–45.[98] M.-E. Hamstrom, Homotopy groups of the space of homeomorphisms on a 2-manifold,
IllinoisJ. Math. (1966), 563–573.[99] V. L. Hansen, Braids and Coverings: selected topics, London Math. Society Student Text , Cam-bridge University Press, 1989.[100] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002.[101] L. Hodgkin, K -theory of mapping class groups: general p -adic K -theory for punctured spheres, Math. Z. (1995), 611–634.[102] S. T. Hu, Homotopy theory, Pure and Applied Mathematics, Vol. VIII, Academic Press, NewYork, 1959. (1976), Cambridge UniversityPress.[105] V. F. R. Jones, Braid groups, Hecke algebras and type II factors, in Geometric methods inoperator algebras (Kyoto, 1983), Pitman Res. Notes Math. Ser., , 242–273, Longman Sci.Tech., Harlow, 1986.[106] V. F. R. Jones, Hecke algebra representation of braid groups and link polynomials, Ann. Math. (1987), 335–388.[107] D. Juan-Pineda, J.-F. Lafont, S. Millán-Vossler and S. Pallekonda, Algebraic K -theory of virtu-ally free groups, Proc. Roy. Soc. Edinburgh Sect. A (2011), 1295–1316.[108] D. Juan-Pineda and I. Leary, On classifying spaces for the family of virtually cyclic subgroups,in Recent developments in algebraic topology,
Contemp. Math. (2001), 135–145.[109] D. Juan-Pineda and S. Millán-López, Invariants associated to the pure braid groups of thesphere,
Bol. Soc. Mat. Mexicana (2006), 27–32.[110] D. Juan-Pineda and S. Millán-López, The Whitehead group and the lower algebraic K -theoryof braid groups of S and R P , Algebr. Geom. Topol. (2010), 1887–1903.[111] D. Juan-Pineda and J. Sánchez, The K -theoretic Farrell-Jones Isomorphism conjecture for braidgroups, submitted for publication.[112] C. Kassel, L’ordre de Dehornoy sur les tresses, in Séminaire Bourbaki, Vol. 1999/2000, As-térisque (2002), 7–28.[113] C. Kassel and V. Turaev, Braid groups, Graduate Texts in Mathematics , Springer, NewYork, 2008.[114] R. P. Kent IV and D. Peifer, A geometric and algebraic description of annular braid groups,
Int. J. Algebra and Computation (2002) 85–97.[115] D. M. Kim and D. Rolfsen, An ordering for groups of pure braids and fibre-type hyperplanearrangements, Canad. J. Math. (2003), 822–838.[116] M. Korkmaz, On the linearity of certain mapping class groups, Turkish J. Math. (2000), 367–371.[117] D. Krammer, The braid group B is linear, Invent. Math. (2000), 451–486.[118] D. Krammer, Braid groups are linear,
Ann. Math. (2002), 131–156.[119] A. Kuku, Higher algebraic K -theory, Handbook of Algebra , 3–74, Elsevier/North-Holland,Amsterdam, 2006.[120] Y. Ladegaillerie, Groupes de tresses et problème des mots dans les groupes de tresses, Bull. Sci.Math. (2) (1976), 255–267.[121] S. Lambropoulou, Braid structures in knot complements, handlebodies and 3-manifolds, inKnots in Hellas ’98 (Delphi), 274–289, Ser. Knots Everything, , World Sci. Publishing, RiverEdge, NJ, 2000. p q -groups. Münster J. Math. (2009), 201–214.[123] W. Magnus, Über Automorphismen von Fundamentalgruppen berandeter Flächen, Math.Ann. (1934), 617–646.[124] W. Magnus, Braid groups: a survey, in Proceedings of the Second International Conferenceon the Theory of Groups (Australian Nat. Univ., Canberra, 1973), 463–487, Lecture Notes inMaths., Vol. 372, Springer, Berlin, 1974.[125] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory, Second revised edition,Dover Publications Inc., New York, 1976.[126] S. Manfredini, Some subgroups of Artin’s braid group,
Topology Appl. (1997), 123–142.[127] G. S. McCarty Jr., Homeotopy groups, Trans. Amer. Math. Soc. (1963), 293–304.[128] S. Millán-Vossler, The lower algebraic K -theory of braid groups on S and R P , VDM VerlagDr. Muller, Germany, 2008.[129] S. Moran, The mathematical theory of knots and braids, an introduction, North-Holland Math-ematics Studies , North-Holland Publishing Co., Amsterdam, 1983.[130] K. Murasugi, Seifert fibre spaces and braid groups, Proc. London Math. Soc. (1982), 71–84.[131] K. Murasugi and B. I. Kurpita, A study of braids, Mathematics and its Applications ,Kluwer Academic Publishers, Dordrecht, 1999.[132] M. H. A. Newman, On a string problem of Dirac, J. London Math. Soc. (1942), 173–177.[133] O. Ocampo Uribe, Grupos de tranças brunnianas, Ph.D thesis, Universidade de São Paulo,Brazil, work in progress.[134] R. Oliver, Whitehead groups of finite groups, London Mathematical Society Lecture NoteSeries , Cambridge University Press, Cambridge, 1988.[135] L. Paris, Braid groups and Artin groups, in Handbook of Teichmüller theory Vol. II, 389–451, IRMA Lect. Math. Theor. Phys. , Eur. Math. Soc., Zürich, 2009.[136] L. Paris and D. Rolfsen, Geometric subgroups of surface braid groups Ann. Inst. Fourier (1999), 417–472.[137] E. Pedersen and C. Weibel, A non-connective delooping of algebraic K -theory, Springer Lec-ture Notes in Math. , Springer-Verlag, Berlin-Heidelberg-New York, 166–181, 1985.[138] D. Quillen, Higher algebraic K -theory: I, Cohomology of groups and algebraic K-theory, 413–478, Adv. Lect. Math. , Int. Press, Somerville, MA, 2010.[139] D. Quillen, Finite generation of the groups K i of rings of algebraic integers, Cohomology ofgroups and algebraic K -theory, 479–488, Adv. Lect. Math. , , Int. Press, Somerville, MA, 2010.[140] R. Ramos, Non-finiteness of twisted nils, Bol. Soc. Mat. Mexicana (2007), 55–64.[141] D. Rolfsen, Tutorial on the braid groups, in [20], 1–30.[142] D. Rolfsen and B. Wiest, Free group automorphisms, invariant orderings and topological ap-plications, Algebr. Geom. Topol. (2001), 311–320. J. Knot Theory Ramif. (1998), 837–841.[144] C. Rourke and B. Wiest, Order automatic mapping class groups, Pacific J. Math. (2000),209–227.[145] G. P. Scott, Braid groups and the group of homeomorphisms of a surface,
Proc. Camb. Phil. Soc. (1970), 605–617.[146] H. Short and B. Wiest, Orderings of mapping class groups after Thurston, Enseign. Math. (2000), 279–312.[147] M. Stukow, Conjugacy classes of finite subgroups of certain mapping class groups, Seifert fibrespaces and braid groups, Turkish J. Math. (2004), 101–110.[148] R. G. Swan, Projective modules over binary polyhedral groups, J. Reine Angew. Math. (1983), 66–172.[149] J. G. Thompson, Note on H p q , Comm. Algebra (1994), 5683–5687.[150] T. tom Dieck, Transformation groups and representation theory, Springer Lecture Notes inMathematics , Springer, Berlin, 1979.[151] J. Van Buskirk, Braid groups of compact 2-manifolds with elements of finite order, Trans. Amer.Math. Soc. (1966), 81–97.[152] V. V. Vershinin, Braids, their properties and generalizations, Handbook of Algebra ,Elsevier/North-Holland, Amsterdam, 2006, 427–465.[153] F. Waldhausen, Algebraic K -theory of generalized free products, Part I, Ann. Math. (1978),135-–204.[154] C. T. C. Wall, Poincaré complexes I,
Ann. Math. (1967), 213–245.[155] C. Wegner, The K -theoretic Farrell-Jones conjecture for CAT p q -groups, Proc. Amer. Math. Soc. (2012), 779–793.[156] C. Weibel, NK and NK of the groups C and D , Comment. Math. Helv. (2009), 339–349(addendum).[157] G. W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics , Springer-Verlag, New York, 1978.[158] J. A. Wolf, Spaces of constant curvature, sixth edition, AMS Chelsea Publishing, vol. 372, 2011.[159] O. Zariski, On the Poincaré group of rational plane curves, Amer. J. Math. (1936), 607–619.[160] O. Zariski, The topological discriminant group of a Riemann surface of genus p , Amer. J. Math. (1937), 335–358.(1937), 335–358.