Math. Proc. Camb. Phil. Soc. (1987), 102, 25 25 Printed in Great Britain Generalized triangle groups BY GILBERT BAUMSLAG Department of Mathematics, City College, New York, NY 10031, U.S.A. JOHN W. MORGAN Department of Mathematics, Columbia University, New York, NY 10027, U.S.A. AND PETER B. SHALEN Department of Mathematics, University of Illinois at Chicago, Chicago, IL 60680, U.S.A. (Received 6 August 1986; revised 4 November 1986) 1. Introduction A group G is called a triangle group if it can be presented in the form G = (a,b;a l = b m = (ab) n = 1> (l,m,n>l). It is well-known that G is isomorphic to a subgroup of PSL 2 (C), that a is of order I, b is of order m and ab is of order n. If s(G) = l/l+l/m+l/n (1) then G contains the fundamental group of a positive genus orientable surface as a subgroup of finite index whenever s(G) ^ 1; in particular G is infinite. Furthermore, if s(G) < 1, the genus of the surface is greater than 1 and consequently G contains a free group of rank 2. We are interested in the family of groups that can be presented in the form G= (a,b;a l = b m = w n = 1> (l,m,n> I), where w is any element in the free group freely generated by a and b. Since w can be replaced by any one of its conjugates in such a presentation of G, we can always assume that w i s a cyclically reduced word in a and b. If w is either a power of a or a power of 6, G is a free product of two cyclic groups and so its structure is completely understood. Thus we concentrate on the cases where w is a more interesting word. Here we term a group G a generalized triangle group if it can be presented in the form G = (a,b;a l = b m = w n = 1> (l,m,n>l) (2) where w = a r >6 s i...a r *6 8 * (k ^ 1,0 < r i < 1,0 < s t < m). (3) Furthermore, we call a representation <r of G in a group X special if o-(a), o~(b), a(w) are of orders I, m and n respectively; if, in addition, <r(G) is cyclic, <r will be called a special cyclic representation and if er(6?) is a dihedral group, then o~ will be called a special dihedral representation. The following two theorems are analogues for generalized triangle groups of the results stated above for triangle groups.