Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!

share|improve this question
1  
$B_3$ is a central extension of $PSL_2(\mathbb Z)$. That's a fairly standard one. –  Ryan Budney Jul 12 '12 at 19:29
2  
Every braid group splits as a semidirect product: simply map each (standard) generator to $1\in\mathbb{Z}$. –  Steve D Jul 12 '12 at 21:29
    
To be precise, a homomorphism to $\mathbf{Z}$ yields not one semidirect product decomposition, but plenty of them. –  YCor Jun 28 at 13:30

1 Answer 1

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

share|improve this answer
1  
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) –  YCor Jun 28 at 13:31

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.