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Consider the triangle group $\Delta(a, b, c)$ for some $a, b, c > 1$, given by three generators $x$, $y$, and $z$, such that $x^2 = y^2 = z^2 = (xy)^a = (yz)^b = (zx)^c = 1$.

Can anyone provide an example of a non-normal subgroup of some $\Delta(a, b, c)$ of finite index, containing only products of even numbers of generators (i.e., a subgroup of the corresponding von Dyck group), and containing no conjugate of any nontrivial power of $xy$, $yz$, or $zx$?

Furthermore, I would particularly be interested in examples where $a = 2$.

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  • $\begingroup$ At least you know that there are such subgroups (unless $(a,b,c)=(2,2,2)$ as $\Delta$ is then abelian). Indeed otherwise we can assume $c>2$ and then you can choose the subgroup $\{1,z\}$ if $\Delta$ is finite. If $\Delta$ is infinite, there is a normal subgroup of finite index $N$ such that in $\Delta/N$, the orders of $xy,yz,zx$ are $a,b,c$ respectively (it exists because $\Delta$ is residually finite). Then choose any non-normal subgroup of $N$, since you want a non-normal subgroup. $\endgroup$
    – YCor
    Commented Dec 13, 2016 at 2:32
  • $\begingroup$ Ah, $\{1, z\}$ doesn't work; I want subgroups which contain only products of even numbers of generators, while $z$ is a product of an odd number of generators. (As for the latter case, I have to think about the residually finite reasoning some more.) $\endgroup$
    – Sridhar Ramesh
    Commented Dec 13, 2016 at 2:47
  • $\begingroup$ Oh yes I forgot. In the infinite case it's no problem, just intersect with the von Dyck group first if necessary. $\endgroup$
    – YCor
    Commented Dec 13, 2016 at 3:00
  • $\begingroup$ Then in the finite case it seems not to exist at all: if $a=b=2$, you have a dihedral group (times one of order 2). Then in the dihedral group, all subgroup of the von Dyck group will be normal. So the only remaining cases are the Coxeter groups of type $A_3,B_3, H_3$. In $A_3$ and $H_3$, the von Dyck subgroup has index 2, and index 4 in $B_3$. In $A_3=Sym_4$, it does not exist either, and also in the binary icosahedral (of order 120) $H_3$. I haven't checked $B_3$ (of order 48). $\endgroup$
    – YCor
    Commented Dec 13, 2016 at 3:08
  • $\begingroup$ In the infinite case, pardon my inability to see it right away, but why is it clear the triangle group is residually finite? And, supposing we take N as above (intersected with the von Dyck group if necessary), how do we know it has a non-normal subgroup? $\endgroup$
    – Sridhar Ramesh
    Commented Dec 13, 2016 at 3:34

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