Are the braid groups $\mathcal{B}_n$ virtually abelian ? virtually free ?
Subgroups of the braid groups $\mathcal{B}_n$
4
$\begingroup$
group-theory
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1See http://math.stackexchange.com/questions/48780/mathcalb-3-modulo-the-normal-closure-of-mathbbz-times-mathbbz/48950#48950 for virtual abelianness when $n=3$. – 2011-07-23
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1Braid groups are torsion free, so they can't be virtually free (or they would be free themselves). They can't be virtually abelian because then they would have to be nilpotent (but they contain free subgroups). – 2011-07-23
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0But free groups are virtually free! – 2011-07-23
1 Answers
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Well the 2-string braid group is infinite cyclic which is both virtually free and virtually abelian! As Mariano commented, it was pointed out in this earlier discussion
$\mathcal{B}_3$ modulo the normal closure of $\mathbb{Z} \times \mathbb{Z} $
that the 3-string braid group has a subgroup of index 6 which is the direct product of an infinite cyclic group and the free group of rank 2, so the answer is no and no for braid groups on 3 or more strings.