Are the braid groups $\mathcal{B}_n$ virtually abelian ? virtually free ?
Subgroups of the braid groups $\mathcal{B}_n$
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$\begingroup$
group-theory
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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.