Incomplete: 01/29/08 Here is what C-F-P-V says applied just to pure-cycle classes. We can label Nielsen classes as follows. Put in the distinct integers d_1^*< ...< d_s^* that appear as disjoint cycle lengths. If, however, all these lengths are odd, and d_s^*=n, then you might have two conjugacy classes to consider, and they would need to be considered. (My survey takes this seriously, but I suppress this here.) Now you consider just Nielsen classes with support in d_1^*< ...< d_s^*: The corresponding conjugacy classes appear with multiplicity, say respectively, (m_1,...,m_s). I'll call this Nielsen class \ni_{\bd^*,\bm}. What C-F-P-V says is that if EACH of the m_i's is suitably large, then there is a precise number of braid orbits. I gave you the easiest case to state, the one that appears in today's serious applications, when all d_i^*s are odd. Then there are exactly two braid orbits. What my survey will contain is this. For a given \bd^* I will give an algorithm to compute the relation between \ni_{\bd^*,\bm} and for the same set, but where the group is Spin_n. I will also do this for the case where the d_i^*s include some even integers, but then you replace Spin_n by the representation cover of S_n. If we knew that this analog determined the components (braid orbits) exactly, that would be the exact analog of Thms 1.2 and 1.3, and it would given an analog of my Figure 1 for any given d_i^*s. Instead for given d_i^*s it gives something like Figure 1, though there is a region where we don't know the orbits precisely. Still, we have increasing reason to believe that the analog of Figure 1 is exactly correct. Notice, however, in any case, the analog of Figure 1 depends on knowing which lifting invariant values occur for which m_is. I can give a serious result when all the d_i^*s are odd precisely because I can use a trick to apply my Thm. 1.3. That is the one original result I'm putting in this survey. This is what I meant when I said I was using the Fried-Serre formula. This is one kind of generalization of the Fried-Serre formula (something I flirted with in the Bailey-Fried paper).

It is this Fried-Voelklein Addendum that I am using to give a plausibility reason for your conjecture/result. That is, the proof of the result says that as you increase the use of the conjugacy classes in \bfC^*, there are two regions of multiplicity of these conjugacy classes. There is an early region where if all appear often enough the number of orbits can only go UP. So, once you get to that region, you know that the limit number of orbits is given by a bound on the order of a quotient of the Schur multiplier of G. In the case of G=A_n or S_n, that is enough. Incidentally, somewhere, from this view you have to explain why in your genus 0 case there can only be one possible value of the lifting invariant -- rather than two, which would give two braid orbits. That follows from Invariance Corollary 2.3 in the paper I sent you. The archetype of the theorem you want is stated at the bottom of p. 2 (or better yet the Constellation diagram on p. 5). It has two parts. Theorem 1.3, for r\ge n 3-cycles: both the inner and absolute Hurwitz spaces of covers of the sphere with 3-cycle branching have exactly two components.