Discrete Free Products of Two Complex Cyclic Matrix Groups

Observe that the elements of ftxUf^ have infinite order, while those of ft2Uft3 have finite order. We will prove that whenever A e ft and B e ft do not both have a fixed point on L, then the group (A,B') is the discrete free product (A)*(B'), where B' denotes the transpose of B. The case where both A and B have a fixed point on L is also discussed. We show in addition that if A eft and Beft are real, then (A,B') is the discrete free product (A)*(B') if and only if for every real u,