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Possible Duplicate:
$\mathcal{B}_3$ 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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    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...