Given a principal GG-bundle P→MP→M and two C1C1 curves in MM with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on PP. The main result in this paper is that if GG is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, C1C1 homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.
