Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like to find a quadrilateral whose sides are identified in pairs by $A$ and $B$, such that the sum of the internal angles equals the rotation angle of the commutator. In other words, I'd like to mimic what happens when one looks at the orbifold fundamental group of a once-coned torus, although in this case the group generated by $A$ and $B$ might be non-discrete.
Thank you!