The following was inspired by Jesse Peterson's answer.
Let $M^n$ be a closed hyperbolic manifold, eg a surface, and consider the action of $\Gamma=\pi_1M$ on the boundary at infinity of hyperbolic $n$-space. Then each $\gamma\in\Gamma\smallsetminus 1$ has precisely two fixed points, namely the end points of its axis of translation.
Any non-commuting pair of elements whose axes share an end point now provide an example of the kind required. It isn't too difficult to construct explicit examples.