Date: Mon, 20 May 1996 18:08:19 -0600 (CST) From: sjpSubject: 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)