Abstract. We prove that if $\{x,y\},\{u,v\} $ are two sets of generating pairs for a free group $F$ satisfying the equation $ [x,y^n] = [u,v^m]$ then $n = m$. Further if $n = m \ge2$ then $y$ is conjugate in $F$ to $v^{\pm1}$. This theorem rose out of a question concerning Schottky groups. The method of proof is used to consider certain related equations in free groups and generalizations to genus one Fuchsian groups.

AMS Subject Classification (1991): 20E05

Keyword(s): Free Groups, Equations, Test Elements, Scottky Groups

Received January 19, 2001, and in revised form July 12, 2001. (Registered under 2874/2009.)