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I've been trying to google this question, but to no avail. The question sounds elementary but I hope it's suitable for the experts at MO!

Let $B_n$ be the braid group on $n$-strands and $P_n$ be the pure braid group on $n$-strands. We have a well know decomposition of $P_n$ as $F_n \rtimes P_{n-1}$. In this light, my question is simply:

Are there known decompositions of $B_n$ in terms of direct/semidirect products of $B_{k}$ for $1\leq k < n$ and (possibly) other subgroups of $B_n$

I'd even be interested for special cases of $n$

Thanks!

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    $\begingroup$ $B_3$ is a central extension of $PSL_2(\mathbb Z)$. That's a fairly standard one. $\endgroup$
    – Ryan Budney
    Commented Jul 12, 2012 at 19:29
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    $\begingroup$ Every braid group splits as a semidirect product: simply map each (standard) generator to $1\in\mathbb{Z}$. $\endgroup$
    – Steve D
    Commented Jul 12, 2012 at 21:29
  • $\begingroup$ To be precise, a homomorphism to $\mathbf{Z}$ yields not one semidirect product decomposition, but plenty of them. $\endgroup$
    – YCor
    Commented Jun 28, 2015 at 13:30

1 Answer 1

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The answer is known for $B_3$ and $B_4$:

$B_3=\langle \sigma_1,\sigma_2\mid \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2\rangle$ decomposes as $B_3=F_2\rtimes \mathbb Z$, where $F_2=B_3'=\langle\sigma_2\sigma_1^{-1},\sigma_1\sigma_2\sigma_1^{-2}\rangle$, and $\mathbb Z=\langle \sigma_1\rangle$.

Similarly,

$B_4=\langle \sigma_1,\sigma_2,\sigma_3\mid \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2, \sigma_1\sigma_3=\sigma_3\sigma_1,\sigma_2\sigma_3\sigma_2=\sigma_3\sigma_2\sigma_3 \rangle$ decomposes as $B_4=F_2\rtimes B_3=F_2\rtimes (F_2\rtimes \mathbb Z)$, where the outer $F_2$ is generated by $\sigma_3\sigma_1^{-1}$ and $\sigma_2\sigma_3\sigma_1^{-1}\sigma_2^{-1}$, and $B_3$ has generators as above.

See Leonid Bokut, Andrei Vesnin, Grobner–Shirshov bases for some braid groups, Journal of Symbolic Computation 41 (2006) 357–371 http://math.nsc.ru/~vesnin/papers/bokut-vesnin2006.pdf

or Theorem 2.1 (presentation for $B_n'$ for all $n$) here:

E. A. Gorin, V. Ya. Lin, “Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids”, Mat. Sb. (N.S.), 78(120):4 (1969), 579–610 http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=3572&option_lang=eng

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    $\begingroup$ What do you mean by "the answer is known"? the question is too vague to have a definite answer (unless you describe all semidirect product decompositions) $\endgroup$
    – YCor
    Commented Jun 28, 2015 at 13:31
  • $\begingroup$ @YCor Sorry, did not notice your comment earlier. What I meant, that for $B_4$ this is a known decomposition which is more interesting than splitting off a copy of $\mathbb Z$. $\endgroup$
    – mathreader
    Commented Aug 14, 2017 at 7:23

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