Infinite Simple (2, 3, n )-Groups and Congruence Hulls in the Modular Group

Abstract

We prove that if n > 66 and (n, 30) = 1, then there exist uncountably many infinite simple (2, 3, n)groups, that is, groups generated by a pair of elements x, y, say, where the orders of x, y and xy are 2, 3 and n, respectively. This extends previous results of Schupp and the authors.These results are used to prove the existence of subgroups of the modular group with special arithmetic properties.

Infinite quotients of Fuchsian groups

“…Perhaps the most important Fuchsian group is the modular group G PSL2Y Z, and in this case Schupp [19] used small cancellation theory to show that G has 2 d 0 non-isomorphic in®nite simple quotients. His result was developed further by Mason and Pride [13], [14], whose work implies that a triangle group h2Y 3Y n also has 2 d 0 non-isomorphic in®nite simple quotients if n b 66 and nY 30 1. In this paper we extend this result to all but thirty non-elementary ®nitely generated Fuchsian groups.…”