Geodesic intersections and isoxial Fuchsian groups

The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified in for groups of the second kind by G. Mess and for arithemetic groups by by D. Long and A. Reid. Here we show that the conjecture holds for almost all Fuchsian groups and explain why our method fails for arithemetic groups.