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$$ \langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=1\rangle =\Delta(p,q,r) $$ This is a presentation of a triangle group $\Delta(p,q,r)$, a special kind of Coxeter group.

EDIT In fact, these are called extended triangle groups, by G. Jones and D. Singerman in Maps, hypermaps and triangle groups...

What about the following presentation: $$ \langle a,b,c\mid a^2=b^2=c^2=(ab)^p=(bc)^q=(ca)^r=(abc)^s=1\rangle $$ Do these groups have a name and where are they treated?

The presentation in question are motivated by this and that...

ANOTHER EDIT if $p=q=r$ is prime and $s=1$ this is called triangular Fuchsian group here...

draks ...
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  • Have you come across generalised triangle groups? What I call a triangle group is the group of transformations of the tiled plane (no reflections); Wikipedia calls these von Dyck groups. Such groups have presentation $\langle x, y, x; x^p, y^q, z^r, xyz\rangle$, and here $ab=x$, $bc=y$ and $ca=z$. A generalised triangle group is a group with presentation $\langle x, y, x; x^p, y^q, z^r, W(x, y, z)\rangle$. Fine and Rosenberger wrote a whole book motivated by these groups and their generalisations (one-relator products). Jim Howie has also written a lot about them. – user1729 Jul 08 '16 at 10:36
  • @user1729 you mean $<x,y,{\bf z};\dots>$? Do you have some explicite references or links to them? – draks ... Jul 08 '16 at 13:17
  • Yes, $x, y, z$. When I google the first link is to a paper of Jim Howie (http://www.macs.hw.ac.uk/~jim/preprint27.pdf). The references look extensive. I am pretty sure he gave a talk on these at a conference in 2012 I was at. The book of Fine and Rosenberger is "Algebraic generalizations of discrete groups: a path to combinatorial group theory through one-relator products". – user1729 Jul 08 '16 at 14:02
  • @user1729 why not posting this as an answer... – draks ... Jul 12 '16 at 05:11
  • Instead of editing this old post again, you might be better asking a new question about your new edit. If you think Derek Holt's answer is insufficient then you could consider asking on MathOverflow. – user1729 Sep 08 '20 at 08:42
  • @user1729, sorry, and I just realised that Derek points out exactly what I edited... – draks ... Sep 08 '20 at 08:45
  • @user1729 but you're right. I did: https://math.stackexchange.com/q/3819674/19341 – draks ... Sep 09 '20 at 09:30

1 Answers1

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I haven't come across a name for this family in full generality, but the special case in which $p=2$ was defined and studied by Coxeter in his paper

H. S. M. Coxeter, The abstract groups $G^{ m, n, p}$, Trans. Amer. Math. Soc. 45 (1939), 73-150.

where (in your notation) the group is called $G^{q,r,s}$.

Also, when $s$ is even, your group has a subgroup of index $2$ with presentation $\langle x,y \mid x^p=y^q=(xy)^r=[x,y]^{s/2} \rangle$.

These groups were studied in the same paper by Coxeter, and denoted $(p,q,r;s/2)$.

Both of these families have been extensively studied since then, in particular concerning their finiteness. They are generally infinite for sufficiently large values of the parameters, and there is just a handful of remaining cases for which their finiteness is still unknown.

A few years ago Havas and I showed, using a big computer calculation, that $(2,3,13;4)$ is finite of order $358\,848\,921\,600$. So your group with $(p,q,r,s) = (2,3,13,8)$ has twice that order.

Derek Holt
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