Date: Mon, 20 May 1996 20:57:48 +1200
From: Marston Conder 
Subject: infinite simple quotients of (2,3,m) triangle groups



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