Date: Mon, 20 May 1996 18:08:19 -0600 (CST)
From: sjp 
Subject: RE: infinite simple quotients of (2,3,m) triangle groups

In message Mon, 20 May 1996 20:57:48 +1200,
  Marston Conder   writes:

>
>
> Alex Mason (Glasgow) asked me the following question a few weeks ago:
>
>
>> I wonder if you can tell me if anything is known about INFINITE simple
>>   quotients of the (2,3,7)-group?
>>   More generally, is anything known about infinite simple quotients of
>>   (2,3,m)-groups?
>>
>
>
> For positive integers p,q,r, the (p,q,r) triangle group is the group with
> presentation  < x,y,z | x^p = y^q = z^r = xyz = 1 >,  more commonly
> presented  in the equivalent form  < x,y | x^p = y^q = (xy)^r = 1 >.
>
> As some of you possibly know, there certainly do exist infinite simple
> quotients of the modular group  C2*C3 = < x,y | x^2 = y^3 = 1 >.
>
> For example, this is proved by Schupp in J London Math Soc 13 (1976)
> using  small cancellation theory.   Now I don't think the proof can be
> modified to  give infinite simple quotients of (2,3,7), but I believe it
> can for simple  quotients of (2,3,m) for suitably large m.
>
> What about (2,3,7)?   Does anyone have any other ideas?
>
> Marston Conder
>
You are certainly right about simple quotients of (2,3,m) for large m.
Alex Mason also asked me about this a while ago. Using small
cancellation type arguments I mapped out a proof that for m>=36,
and where the smallest prime divisor of m is at least 7, then every
countable group can be embedded in a simple quotient of the
(2,3,m) group. This seemed to be adequate for Alex' purpose. I did
think a bit about lowering m. From a geometric point of view it
would involve probably fairly intricate curvature arguments on
"pictures"  (or van Kampen diagrams), but I didn't feel very
motivated to pursue this.     Steve Pride (Glasgow)