Pseudo-elementary generalized triangle groups

Abstract

Abstract.A generalized triangle group is a group that can be presented in the form G ¼ hx; y j x l ¼ y m ¼ wðx; yÞ n ¼ 1i where l; m; n A f2; 3; 4; . . .g U fyg and wðx; yÞ is an element of the free product hx; y j x l ¼ y m ¼ 1i involving both x and y. A homomorphism f : G ! G is said to be essential if fðxÞ, fð yÞ, fðwðx; yÞÞ have orders l, m, n respectively. Every generalized triangle group G admits an essential representation to PSLð2; CÞ. In most cases there will be such a representation with infinite or non-elementary image.Vinberg and Kaplinsky say that G is pseudo-finite if the image of any essential representation G ! PSLð2; CÞ is finite and they have obtained a partial classification of such groups. Extending this concept, we call G pseudo-elementary if the image of any essential representation G ! PSLð2; CÞ is elementary. In this paper we classify the pseudo-elementary generalized triangle groups G with n d 3 and obtain partial results in the case n ¼ 2.