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Let $B_n$ be the braid group on $n$ strands, $n \geq 3$. Consider a subgroup $H$ generated by $\sigma_i$ and $\sigma_{i+1}^2$. Is $H$ a free group of rank $2$?

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No. Let $a=\sigma_i$ and $b=\sigma_{i+1}$. Then $b^2ab^2a^{-1}b^{-2}a^{-1} b^{-2}a=1$.

The group generated by the squares $\sigma_{i}^2$ and $\sigma_{i+1}^2$ is known to be free.

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  • $\begingroup$ May I know how you come up with this solution? I mean about the strategy to solve these kinds of questions. $\endgroup$
    – eyp
    Oct 6, 2019 at 23:51
  • $\begingroup$ I suspected it was not free (I might have come across a similar problem before) and then just did a brute force computer search for a relator. So I don't really have a helpful strategy! $\endgroup$
    – Derek Holt
    Oct 7, 2019 at 7:04

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