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$\mathcal{B}_3$ modulo the normal closure of $\mathbb{Z} \times \mathbb{Z} $

What is the index of $\langle (xy)^3,x \rangle$ in the braid group $\mathcal{B}_3 = \langle x,y | xyx = yxy \rangle$ ?

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    You had already asked this question as part 2 of your previous question; why not simply edit or add a comment in your original noting that this part has not been answered yet?2011-07-01
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    http://en.wikipedia.org/wiki/Todd%E2%80%93Coxeter_algorithm2011-07-01
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    I thought here the point was that the subgroup is *not* the normal closure...2011-07-01
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    I wasn't aware I could do that. And yes, I'm curious about the index of the subgroup, not the normal closure. The answer below is not help because, first, it is not a mathematical proof. Second, and more importantly, it does not enumerate all the cosets. So one cannot know for sure that the index is infinite. But thank you all for your help.2011-07-01
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    The central quotient is the free product of $C_2$ and $C_3$; the only free product that is virtually cyclic is the infinite dihedral group. So your subgroup is infinite index.2011-07-05

1 Answers 1

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I fire GAP and say

gap> f := FreeGroup(2);;
gap> x := f.1;;
gap> y := f.2;;
gap> g := f / [x*y*x*(y*x*y)^(-1)];;
gap> h := Subgroup(g, [(g.1*g.2)^3, g.1]);;
gap> Index(g, h);;
Error, the coset enumeration has defined more than 256000 cosets
 called from
TCENUM.CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
CosetTableFromGensAndRels( fgens, grels, fsgens ) called from
TryCosetTableInWholeGroup( H ) called from
CosetTableInWholeGroup( H ) called from
IndexInWholeGroup( H ) called from
...
Entering break read-eval-print loop ...
type 'return;' if you want to continue with a new limit of 512000 cosets,
type 'quit;' if you want to quit the coset enumeration,
type 'maxlimit := 0; return;' in order to continue without a limit
brk> 

This suggests that the index is infinite...