Counting surface branched covers

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compat- ibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum D there ex- ists a branched cover f such that D(f) = D. One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f's, and choose an equivalence relation up to which one regards them. As it turns out, quite a few natural choices for this relation are possible. In this short note we carefully an- alyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different from each other we employ Grothendieck's dessins d'enfant.