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Are the braid groups $\mathcal{B}_n$ virtually abelian ? virtually free ?

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    But free groups are virtually free!2011-07-23

1 Answers 1

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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.