Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group

Question

A cover (of $\mathbf{P}^1_{\overline{\mathbf{Q}}}$) is a finite morphism $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$, where $X$ is a smooth projective connected curve over $\overline{\mathbf{Q}}$. For any positive integer $n$, let $H_n^o$ be the moduli space of covers $X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ of degree $n$ which are etale over $\mathbf{P}^1_{\overline{\mathbf Q}}-\{0,1,\infty,\lambda\}$, where $\lambda$ is allowed to vary in $\overline{\mathbf Q}-\{0,1\}$.

Q1. How many connected components does $H_n^o$ have? (I'm looking for a formula in terms of $n$.)

There is a finite etale morphism $\pi_n^0:H_n^o \to \mathbf{P}^1_{\overline{\mathbf{Q}}} \backslash \{0,1,\infty\}$ which sends the class of a cover $f:X\to \mathbf{P}^1_{\overline{\mathbf{Q}}}$ to the unique element $\lambda$ in its branch locus different from $0$, $1$ and $\infty$.

Q2. The degree of $\pi_n^o$ is the number of subgroups of the fundamental group of $\mathbf{P}^1(\mathbf{C})-\{0,1,\infty,\lambda\}$ of index $n$. Is there a nice formula for this?

Answer 1

There is not going to be a nice answer to your Q1, because the decomposition of that Hurwitz space into components can be quite wild. In fact, any algebraic curve over Qbar is birational to a connnected component of one of these spaces! (See Diaz-Donagi-Harbater, "Every curve is a Hurwitz space.")

As for your Q2, these numbers are called "Hurwitz numbers" and a lot is known about them. Often one splits things up by specifying the conjugacy class of monodromy around the branch points. But given that you are NOT doing that, I think you are essentially just asking for the number of triples of elements of S_n which generate a transitive permutation group. I imagine you could do that by some kind of inclusion-exclusion argument. Or look up "subgroup zeta function" for the very rich general story about counting subgroups of index n in finitely generated groups.

Answer 2

Question 2 is answered in more detail than you could possibly want in Lubotzky's paper. (since your fundamental group is just a free group on three generators)