Date: Mon, 20 May 1996 20:57:48 +1200 From: Marston ConderSubject: 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