Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T04:37:03Z http://mathoverflow.net/feeds/question/88704 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/88704/number-of-connected-components-of-the-hurwitz-space-h-no-and-subgroups-of-the Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group Harized 2012-02-17T10:29:08Z 2012-08-24T21:24:01Z <p>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 <code>$\mathbf{P}^1_{\overline{\mathbf Q}}-\{0,1,\infty,\lambda\}$</code>, where $\lambda$ is allowed to vary in <code>$\overline{\mathbf Q}-\{0,1\}$</code>. </p> <p><strong>Q1.</strong> How many connected components does $H_n^o$ have? (I'm looking for a formula in terms of $n$.)</p> <p>There is a finite etale morphism <code>$\pi_n^0:H_n^o \to \mathbf{P}^1_{\overline{\mathbf{Q}}} \backslash \{0,1,\infty\}$</code> 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 $\infty$.</p> <p><strong>Q2.</strong> The degree of $\pi_n^o$ is the number of subgroups of the fundamental group of <code>$\mathbf{P}^1(\mathbf{C})-\{0,1,\infty,\lambda\}$</code> of index $n$. Is there a nice formula for this?</p> http://mathoverflow.net/questions/88704/number-of-connected-components-of-the-hurwitz-space-h-no-and-subgroups-of-the/105409#105409 Answer by JSE for Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group JSE 2012-08-24T18:40:15Z 2012-08-24T18:40:15Z <p>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.")</p> <p>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. </p> http://mathoverflow.net/questions/88704/number-of-connected-components-of-the-hurwitz-space-h-no-and-subgroups-of-the/105423#105423 Answer by Igor Rivin for Number of connected components of the Hurwitz space $H_n^o$ and subgroups of the fundamental group Igor Rivin 2012-08-24T21:24:01Z 2012-08-24T21:24:01Z <p>Question 2 is answered in more detail than you could possibly want in <a href="http://www.math.ou.edu/~nbrady/teaching/s11-5863/lubotzky.pdf" rel="nofollow">Lubotzky's paper.</a> (since your fundamental group is just a free group on three generators)</p>