The Quaternion Group As a Subgroup of the Sphere Braid Groups Daciberg Gonçalves, John Guaschi

The quaternion group as a subgroup of the sphere braid groups Daciberg Gonçalves, John Guaschi

To cite this version:

Daciberg Gonçalves, John Guaschi. The quaternion group as a subgroup of the sphere braid groups. Bulletin of the London Mathematical Society / The Bulletin of the London Mathematical Society, 2007, 39 (2), pp.232-234. 10.1112/blms/bdl041. hal-00020804

HAL Id: hal-00020804 https://hal.archives-ouvertes.fr/hal-00020804 Submitted on 15 Mar 2006

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The Abelianisation of Bn(S ) is isomorphic to the cyclic group Z2(n−1). 2 The kernel of the associated projection ξ : Bn(S ) → Z2(n−1) (which is defined by ξ(σi)= 1 2 2 for all 1 ≤ i ≤ n − 1) is the commutator subgroup Γ2 (Bn(S )). If w ∈ Bn(S ) then ξ(w) is the exponent sum (relative to the σi) of w modulo 2(n − 1). The torsion elements of the braid groups of S2 and RP 2 were classified by Murasugi [M]: 2 if M = S and n ≥ 3, they are all conjugates of powers of the three elements α0 = 2 σ1 ··· σn−2σn−1 (which is of order 2n), α1 = σ1 ··· σn−2σn−1 (of order 2(n − 1)) and α2 = 2 th th th σ1 ··· σn−3σn−2 (of order 2(n − 2)) which are respectively n ,(n − 1) and (n − 2) roots 2 n of Tn, where Tn is the so-called ‘full twist’ of Bn(S ), defined by Tn = (σ1 ··· σn−1) . If 2 2 n ≥ 3, Tn is the unique element of Bn(S ) of order 2 and generates the centre of Bn(S ). 2 In [GG2], we showed that Bn(S ) is generated by α0 and α1. 2 For n ≥ 4, Bn(S ) is infinite. It is an interesting question as to which finite groups are 2 realised as subgroups of Bn(S ) (apart of course from the cyclic groups hαii). In [GG2], 2 2 we proved that Bn(S ) contains an isomorphic copy of the finite group B3(S ) of order 12 if and only if n 6≡ 1 mod 3. The quaternion group Q8 of order 8 appears in the study of braid groups of non-orientable surfaces, being isomorphic to the 2-string pure braid group 2 2 2 P2(RP ). Further, since the projection F3(RP ) → F2(RP ) of configuration spaces of RP 2 onto the first two coordinates admits a section [VB], it follows using the Fadell- 2 Neuwirth short exact sequence that P3(RP ) is a semi-direct product of a free group of rank 2 by Q8 [GG1]. While studying the lower central and derived series of the sphere braid groups, we 2 showed that Γ2 (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 realisation of Q8 as a subgroup of Bn(S ) was explicitly posed by R. Brown in connection with the fact that the fundamental group of SO(3) is isomorphic to Z2 [ATD]. In this paper, we give a complete answer to this question: Theorem. Let n ∈ N, n ≥ 3. 2 (a) If n is a multiple of 4 then Γ2 (Bn(S )) contains a subgroup isomorphic to Q8. 2 (b) If n is an odd multiple of 2 then Bn(S ) contains a subgroup isomorphic to Q8. 2 (c) If n is odd then Bn(S ) contains no subgroup isomorphic to Q8. Proof. We first suppose that n is even, so that n = 2m with m ∈ N. Let H be the 2 subgroup of B2m(S ) generated by x and y, where

x =(σ1 ··· σ2m−1)(σ1 ··· σ2m−2) ··· (σ1σ2)σ1, −1 −1 −1 y =(σ1 ··· σm−1)(σ1 ··· σm−2) ··· (σ1σ2)σ1 · σ2m−1(σ2m−2σ2m−1) ··· −1 −1 −1 −1 ··· (σm+2 ··· σ2m−1)(σm+1 ··· σ2m−1).

Geometrically, x may be interpreted as the ‘half twist’ or Garside element of B2m [Bi]. Further, y may be considered as the commuting product of the positive half twist of the 2 first m strings with the negative half twist of the last m strings. Then x = T2m and 2 m −1 −1 m S2 y = (σ1 ...σm−1) (σm+1 ...σ2m−1) = T2m in B2m( ) (cf. [FVB, GVB]). It is well −1 2 known that xσix = σ2m−i in B2m [Bi], and thus in B2m(S ), from which we obtain −1 −1 xyx = y . Hence H is isomorphic to a quotient of Q8. But x is of order 4, and the induced permutation of y on the symmetric group S2m is different from that of the elements of hxi. It follows that H contains the five distinct elements of hxi∪{y}, and

2 ∼ 2 so H = Q8. If m is even then x, y ∈ Ker(ξ), and thus H ⊆ Γ2 (Bn(S )). This proves parts (a) and (b). 2 To prove part (c), suppose that n is odd, and suppose that x, y ∈ Bn(S ) generate a 2 2 −1 −1 subgroup H isomorphic to Q8, so x = y and xyx = y . In particular, x and y are of ±(n−1)/2 order 4, and thus are conjugates of α1 by Murasugi’s classiﬁcation. By considering ε1(n−1)/2 ε2(n−1)/2 −1 a conjugate of H if necessary, we may suppose that x = α1 and y = wα1 w , 2 −1 where w ∈ Bn(S ) and ε1,ε2 ∈{1, −1}. Replacing x by x if necessary, we may suppose ε1(n−1)/2 further that ε1 = −ε2. Thus xy = hα1 ,wi, and is of exponent sum zero. On the ±(n−1)/2 other hand, xy is an element of H of order 4, and so is conjugate to α1 by Murasugi’s ± − − ξ α (n 1)/2 n(n 1) n classiﬁcation. But 1 = ± 2 , which is non zero modulo 2( −1). This yields a contradiction, and proves part (c).

Remark. Let n ≥ 3. Using techniques similar to those of the proof of the Theorem, 2 one may show that the subgroup of Bn(S ) generated by σ1 ··· σn−1 and the half twist x is isomorphic to the dicyclic group of order 4n. In particular, if n is a power of two 2 then Bn(S ) contains a subgroup isomorphic to the generalised quaternion group of order 2 2 4n. Further investigation into the ﬁnite subgroups of Bn(S ) and Bn(RP ) will appear elsewhere.

References

[ATD] Algebraic topology discussion list, January 2004, http://www.lehigh.edu/~dmd1/pz119.txt.

[A1] E. Artin, Theorie der Z¨opfe, Abh. Math. Sem. Univ. Hamburg 4 (1925), 47–72.

[A2] E. Artin, Theory of braids, Ann. Math. 48 (1947), 101–126.

[Bi] J. S. Birman, Braids, links and mapping class groups, Ann. Math. Stud. 82, Princeton University Press, 1974.

[FVB] E. Fadell and J. Van Buskirk, The braid groups of E2 and S2, Duke Math. Journal 29 (1962), 243–257.

[FoN] R. H. Fox and L. Neuwirth, The braid groups, Math. Scandinavica 10 (1962), 119–126.

[GVB] R. Gillette and J. Van Buskirk, The word problem and consequences for the braid groups and mapping class groups of the 2-sphere, Trans. Amer. Math. Soc. 131 (1968), 277–296.

[GG1] D. L. Gon¸calves and J. Guaschi, The braid groups of the projective plane, Algebraic and Geo- metric Topology 4 (2004), 757–780.

2 [GG2] D. L. Gon¸calves and J. Guaschi, The braid group Bn,m(S ) and the generalised Fadell-Neuwirth short exact sequence, J. Knot Theory and its Ramiﬁcations 14 (2005), 375–403.

2 [GG3] D. L. Gon¸calves and J. Guaschi, The lower central and derived series of the braid groups Bn(S ) 2 and Bm(S \{x1,...,xn}), preprint, March 2006.

[M] K. Murasugi, Seifert ﬁbre spaces and braid groups, Proc. London Math. Soc. 44 (1982), 71–84.

[VB] J. Van Buskirk, Braid groups of compact 2-manifolds with elements of ﬁnite order, Trans. Amer. Math. Soc. 122 (1966), 81–97.