Suppose $[A,B]$ is elliptic for some $A,B \in PSL(2;\mathbb{R})$. Then $\langle A,B \rangle < PSL(2;\mathbb{R})$ is not free. Is there any nice way to see this? If $[A,B]$ is elliptic of finite order then it is clear, but what about elliptic of infinite order?
The claim is from: http://comet.lehman.cuny.edu/keenl/LiftingFreeSubgroups.pdf (the bottom paragraph of page 12).