A class of 2-generator subgroups of PSL(2,\({\mathbb{C}})\)
Let G be the subgroup of PSL(2,\({\mathbb{C}})\) generated by the elements A and B, where A has two fixed points, and B maps one fixed point of A onto the other. The following results were obtained by \textit{B. Maskit} [Proc. Am. Math. Soc. 106, No.1, 175-186 (1989; Zbl 0676.30028)]: (i) if G is discrete, then either A is elliptic of order 2, 3, 4, or 6, or B is elliptic of order 2; (ii) if B is elliptic of order 2, then G is discrete if and only if A is a loxodromic or elliptic element of finite order; (iii) if A is elliptic of order 2, 3, 4, or 6, and B is parabolic hyperbolic or primitive elliptic then G is discrete. In the present paper the author shows that if in the case (iii) B is a non-primitive elliptic element, then the group G is not discrete. The question of discreteness of G in the case (iii) if B is loxodromic is still open.
scientific article; zbMATH DE number 4156785