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I need the following information about the quotients of infinite triangle (or von Dyck) groups.

(1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l+1/m+1/n<1$) triangle group. What are the additional conditions $S$ and $T$ should satisfy in order to ensure that the given quotients of this group is finite?

(2) Is there exist any reference where such quotient groups have been identified as isomorphic to other famous group?

I am a student of Physics and have very limited knowledge of the group theory, so simple answer will be very helpful.

Thanks in advance.

Ketan Patel

Ketan Patel
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  • Although your question is slightly more general, I think it was probably answered here: http://mathoverflow.net/questions/22459/x-y-xp-yp-xyp-1/22463#22463 . – HJRW May 15 '13 at 11:16
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    In particular, your group is finite precisely when $1/l+1/m+1/n>1$. – HJRW May 15 '13 at 11:22
  • Since, I am talking about the hyperbolic group, I have edited the question. – Ketan Patel May 15 '13 at 11:25
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    Ketan: Now it is no longer clear what your question is. Are you interested in finite subgroups of these groups? If so, they are all cyclic, whose generators are conjugate to powers of $S$ or $T$ or $ST$. What else do you need to know? (I do not know what "famous group" means.) – Misha May 15 '13 at 12:02
  • Ketan - in the question I linked to above, it is explained that in the hyperbolic case these groups are always infinite. These groups are pretty famous in their own right. – HJRW May 15 '13 at 12:17
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    Ketan: are you perhaps interested in finite quotients rather than subgroups. So your question becomes: what relations must I add to obtain a finite group? (It's a question with no very straight-forward answer by the way.) – Nick Gill May 15 '13 at 12:34
  • Misha and HW, generally the hyperbolic groups of these kind are infinite. However there exist some finite subgroups of them in certain cases. G(2,3,7) with an additional defining relation $(ST^{-1}ST)^{4}=E$ is such an example which is same as the group $PSL_2(7)$. My question is: Is there exist a generalization of the above example so we can find finite subgroups of other $G(l,m,n)$. – Ketan Patel May 15 '13 at 12:42
  • Thanks Nick. The rephrasing of my question by Nick sounds more correct and appropriate. I will look forward to some answer or hint in this direction. – Ketan Patel May 15 '13 at 12:46
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    Ketan - it appears that Nick is exactly right. You are interested in finite quotients, not finite subgroups. – HJRW May 15 '13 at 13:02
  • Perhaps this needs emphasis. Subgroups and quotients are very different! If you want a good answer to your question, you should edit the title and the body to make this change. – HJRW May 15 '13 at 13:51
  • Thanks HW. I have modified it according to your suggestion. Sorry for my ignorance of correct vocabulary. – Ketan Patel May 15 '13 at 14:18
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    That's an improvement. Now I have a further question. You ask for conditions to ensure that 'the given quotients' are finite, but you haven't given us any quotients! In general, these groups have many finite quotients (because they are residually finite) and also many infinite quotients. Which quotients are you specifically interested in? – HJRW May 15 '13 at 14:35
  • Here's a paper addressing finite quotients of the $(2,3,7)$ triangle group, but the techniques should apply to any hyperbolic triangle group. http://link.springer.com/article/10.1007%2FBF02784148 – Ian Agol May 15 '13 at 14:49
  • Finite quotients of $G(2,3,7)$ are known as Hurwitz groups and have been much studied. It is known, for example, that the alternating group $A_n$ is a Hurwitz group for all sufficiently large $n$. – Derek Holt May 15 '13 at 14:54
  • You may start by reading this paper: http://pages.uoregon.edu/kantor/PAPERS/GKKL3.pdf, where the authors show that "most" simple groups are 2-generated. Thus, such groups are quotients of von Dyck groups. – Misha May 15 '13 at 23:50
  • Try adding the order of [a,b], the commutator of a and b. It should give some finite quotients. Be careful though, sometimes the relations are incompatible, and the group collapses to the trivial group. – Thomas Aug 21 '13 at 04:27

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