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The Classification and the Conjugacy Classes of the Finite Subgroups Of The classification and the conjugacy classes of the finite subgroups of the sphere braid groups DACIBERG LIMA GONÇALVES Departamento de Matemática - IME-USP, Caixa Postal 66281 - Ag. Cidade de São Paulo, CEP: 05311-970 - São Paulo - SP - Brazil. e-mail: JOHN GUASCHI Laboratoire de Mathématiques Nicolas Oresme UMR CNRS 6139, Université de Caen BP 5186, 14032 Caen Cedex, France. e-mail: ºÖ 21st November 2007 Abstract Let n ¥ 3. We classify the finite groups which are realised as subgroups of the sphere braid 2 Õ group Bn ÔS . Such groups must be of cohomological period 2 or 4. Depending on the 2 Ô Õ ¡ Õ value of n, we show that the following are the maximal finite subgroups of Bn S : Z2Ô n 1 ; ¡ Õ the dicyclic groups of order 4n and 4Ô n 2 ; the binary tetrahedral group T1; the binary octahedral group O1; and the binary icosahedral group I. We give geometric as well as 2 Õ some explicit algebraic constructions of these groups in Bn ÔS , and determine the number arXiv:0711.3968v1 [math.GT] 26 Nov 2007 of conjugacy classes of such finite subgroups. We also reprove Murasugi’s classification of 2 2 Õ Ô Õ the torsion elements of Bn ÔS , and explain how the finite subgroups of Bn S are related 2 Õ to this classification, as well as to the lower central and derived series of Bn ÔS . 1 Introduction The braid groups Bn of the plane were introduced by E. Artin in 1925 [A1, A2]. Braid groups of surfaces were studied by Zariski [Z]. They were later generalised by Fox to braid groups of arbitrary topological spaces via the following definition [FoN]. Let M be a compact, connected surface, and let n È N. We denote the set of all ordered n-tuples of distinct points of M, known as the n configuration space of M, by: ´ Õ Ô Õ È Fn Ô M p1,..., pn pi M and pi pj if i j . 2000 AMS Subject Classification: 20F36 (primary), 20F50, 20E45, 57M99 (secondary). 1 Configuration spaces play an important rôle in several branches of mathematics and have been extensively studied, see [CG, FH] for example. Õ The symmetric group Sn on n letters acts freely on Fn Ô M by permuting coordinates. Õ Ô Õ The corresponding quotient will be denoted by Dn Ô M . The n pure braid group Pn M Õ Ô Õ (respectively the n braid group Bn Ô M ) is defined to be the fundamental group of Fn M Õ (respectively of Dn Ô M ). Together with the real projective plane RP2, the braid groups of the 2-sphere S2 are of particular interest, notably because they have non-trivial centre [GVB, GG1], and torsion elements [VB, Mu]. Indeed, Van Buskirk showed that among the braid 2 2 Õ Ô Õ groups of compact, connected surfaces, Bn ÔS and Bn RP are the only ones to have 2 Õ torsion [VB]. Let us recall briefly some of the properties of Bn ÔS [FVB, GVB, VB]. 2 2 2 ÝÑ Ô Õ If D S is a topological disc, there is a group homomorphism ι : Bn Bn S Ô Õ induced by the inclusion. If β È Bn, we shall denote its image ι β simply by β. Then 2 Ô Õ Bn S is generated by σ1,..., σn ¡1 which are subject to the following relations: ¡ ¥ ¤ ¤ ¡ σiσj σjσi if |i j| 2 and1 i, j n 1 ¤ ¤ ¡ σiσi 1σi σi 1σiσi 1 for all 1 i n 2, and 2 ¤ ¤ ¤ ¤ ¤ ¤ ¡ ¡ σ1 σn 2σn ¡1σn 2 σ1 1. 2 Õ Consequently, Bn ÔS is a quotient of Bn. The first three sphere braid groups are finite: 2 2 2 Õ Ô Õ Ô Õ B1 ÔS is trivial, B2 S is cyclic of order 2, and B3 S is a ZS-metacyclic group (a group whose Sylow subgroups, commutator subgroup and commutator quotient group are all cyclic) of order 12, isomorphic to the semi-direct product Z3 « Z4 of cyclic groups, the action being the non-trivial one, which in turn is isomorphic to the dicyclic group Dic12 2 Ô Õ ¡ Õ of order 12. The Abelianisation of Bn S is isomorphic to the cyclic group Z2Ô n 1 . The 2 Ô Õ ÝÑ Ô Õ ¡ Õ kernel of the associated projection ξ : Bn S Z2Ô n 1 (which is defined by ξ σi 1 2 ¨ 2 ¤ ¡ Ô Õ È Ô Õ Ô Õ for all 1 ¤ i n 1) is the commutator subgroup Γ2 Bn S . If w Bn S then ξ w ¡ Õ is the exponent sum (relative to the σi) of w modulo 2Ô n 1 . 2 È Ô Õ Gillette and Van Buskirk showed that if n ¥ 3 and k N then Bn S has an element ¡ Õ Ô ¡ Õ of order k if and only if k divides one of 2n, 2Ô n 1 or 2 n 2 [GVB]. The torsion 2 2 2 Õ Ô Õ Ô Õ elements of Bn ÔS and Bn RP were later characterised by Murasugi [Mu]. For Bn S , these elements are as follows: 2 Ô Õ THEOREM 1 ([Mu]). Let n ¥ 3. Then the torsion elements of Bn S are precisely powers of conjugates of the following three elements: ¤ ¤ ¤ ¡ (a) α0 σ1 σn 2σn ¡1 (which is of order 2n). 2 ¤ ¤ ¤ Ô ¡ Õ (b) α σ σ ¡ σ (of order 2 n 1 ). 1 1 n 2 n ¡1 2 ¤ ¤ ¤ Ô ¡ Õ (c) α σ σ ¡ σ (of order 2 n 2 ). 2 1 n 3 n ¡2 ¡ Õ Ô ¡ Õ Ô ∆ The three elements α0, α1 and α2 are respectively n, n 1 and n 2 roots of n, 2 n Õ Ô ¤ ¤ ¤ Õ ∆ Ô ∆ where n is the so-called ‘full twist’ braid of Bn S , defined by n σ1 σn ¡1 . So 2 Ô Õ ¡ Õ Ô ¡ Õ Bn S admits finite cyclic subgroups isomorphic to Z2n, Z2Ô n 1 and Z2 n 2 . In [GG2], 2 Õ ¥ Ô ∆ we showed that Bn S is generated by α0 and α1. If n 3, n is the unique element 2 2 Õ Ô Õ of Bn ÔS of order 2, and it generates the centre of Bn S . It is also the square of the Garside element (or ‘half twist’) defined by: Ô ¤ ¤ ¤ ÕÔ ¤ ¤ ¤ Õ ¤ ¤ ¤ Ô Õ ¡ Tn σ1 σn ¡1 σ1 σn 2 σ1σ2 σ1. 2 2 Ô Õ For n ¥ 4, Bn S is infinite. It is an interesting question as to which finite groups are 2 Õ Ü Ý realised as subgroups of Bn ÔS (apart of course from the cyclic groups αi and their subgroups given in Theorem 1). Another question is the following: how many conju- 2 Õ gacy classes are there in Bn ÔS of a given abstract finite group? As a partial answer to 2 Õ the first question, we proved in [GG2] that Bn ÔS contains an isomorphic copy of the 2 Õ finite group B3 ÔS of order 12 if and only if n 1 mod 3. While studying the lower central and derived series of the sphere braid groups, we 2 ¨ Õ Γ Ô showed that 2 B4 S is isomorphic to a semi-direct product of Q8 by a free group of rank 2 [GG3]. After having proved this result, we noticed that the question of the 2 Õ realisation of Q8 as a subgroup of Bn ÔS had been explicitly posed by R. Brown [ATD] in connection with the Dirac string trick [F, N] and the fact that the fundamental group Õ of SOÔ3 is isomorphic to Z2. The case n 4 was studied by J. G. Thompson [ThJ]. In a previous paper, we provided a complete answer to this question: ¥ THEOREM 2 ([GG4]). Let n È N, n 3. 2 Õ (a) Bn ÔS contains a subgroup isomorphic to Q if and only if n is even. ¨ 8 2 Õ Γ Ô (b) If n is divisible by 4 then 2 Bn S contains a subgroup isomorphic to Q8. As we also pointed out in [GG4], for all n ¥ 3, the construction of Q8 may be gener- 2 Ý Ô Õ alised in order to obtain a subgroup Ü α0, Tn of Bn S isomorphic to the dicyclic group Dic4n of order 4n. 2 Õ It is thus natural to ask which other finite groups are realised as subgroups of Bn ÔS . One common property of the above subgroups is that they are finite periodic groups 2 Õ of cohomological period 2 or 4. In fact, this is true for all finite subgroups of Bn ÔS . 2 Õ Indeed, by [GG2], the universal covering X of Fn ÔS is a finite-dimensional complex which has the homotopy type of S3 (we were recently informed by V. Lin that X is Õ biholomorphic to the direct product of SL Ô2, C by the Teichmüller space of the n- 2 Õ punctured Riemann sphere [Li]). Thus any finite subgroup of Bn ÔS acts freely on X, and so has period 2 or 4 by Proposition 10.2, Section 10, Chapter VII of [Br]. Since 2 2 Õ Ô Ô ÕÕ ∆n is the unique element of order 2 of Bn ÔS , and it generates the centre Z Bn S , 2 Õ the Milnor property must be satisfied for any finite subgroup of Bn ÔS . Recall also that a finite periodic group G satisfies the p2-condition (if p is prime and divides the order of G then G has no subgroup isomorphic to Zp ¢ Zp), which implies that a Sylow p-subgroup of G is cyclic or generalised quaternion, as well as the 2p-condition (each subgroup of order 2p is cyclic).
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