Finite index subgroups of surface groups

Question

It is not hard to see (say from the perspective of covering spaces) that there are a finite number of subgroups of a fixed finite index $n$ in a finitely generated free group $F_n$. Given a closed oriented surface $F$, are there a finite number of subgroups of $\pi_1(F)$ of a given index? If so is there a formula for this number?

Answer 1

@Eric Wofsey has already explained why the number of subgroups of a given index is finite in a finitely generated group, so I'll just attempt to reproduce the formula for counting them.

There is a kind of recursive formula, depending on the genus $g$ of $F$. This comes from Chapter 14 of the book "Subgroup Growth", by Lubotzky and Segal. who attribute the result to A.D. Mednykh.

Let $a_n$ denote the number of subgroups of index $n$ in $\pi_1(F)$. Then $$a_n = \frac{1}{(n-1)!}h_n - \sum_{k=1}^{n-1}\frac{1}{(n-k)!}h_{n-k}a_k,$$ where $h_n$ is the number of homomorphisms from $\pi_1(F)$ to the symmetric group $S_n$. (In fact, this much is independent of $g$, and holds for any group in place of $\pi_1(F)$.)

But, for surface groups $F$ of genus $g$ (either orientable or non-orientable), there is a formula for $h_n$ that relies on the representation theory of the symmetric group: $$h_n = (n!)^{2g-1}\sum_{\chi\in\operatorname{Irr}(S_n)}\chi(1)^{2-2g}.$$ Here, $\operatorname{Irr}(S_n)$ denotes the set of irreducible characters of the symmetric group $S_n$.

Assuming I didn't make any mistake in my code, here are expressions in $g$ for the first few values of $n$ (of course $a_1 =1$ for any $g$):

> a := proc(n,g)
>   local   k;
>   h(n,g)/(n-1)! - add( h(n-k,g)*thisproc(k,g),k=1..n-1)
> end proc:
> 
> h := proc( n, g )
>   uses    GroupTheory;
>   (n!)^(2*g-1) * `+`( op( map( `$` @ op, CharacterDegrees( CharacterTable( Symm( n ) ) ) ) ^~ (2-2*g) ) )
> end proc:

> for i from 2 to 5 do print(simplify(a(i,g))) end:       
                                                          g
                                                         4  - 1

                                                    g     g
                                                  36     9        g
                                                  --- + ---- - 2 4  + 1
                                                   6     3

                                         g      g     g     g
                                      576    144    64    36     g     g      g
                                      ---- + ---- + --- - --- - 9  - 16  + 3 4  - 1
                                       72     36     8     2

                      g      g         g      g         g       g     g      g
                 14400    900    23 576    400    25 144    7 64    36    5 9        g    (2 + 2 g)
                 ------ + ---- - ------- + ---- - ------- - ----- - --- + ---- + 3 16  - 2          + 1
                  1440     90      288      80      36        8      6     3

(Even the simplified expressions get a bit messy after that.)

Note that it is fairly easy to see directly why $a_2 = 4^g - 1$. Any subgroup of index $2$ is normal, so $a_2$ just counts the number of homomorphisms from $\pi_1(F)$ onto a cyclic group $C_2$ of order $2$. It is clear from the defining relation for $\pi_1(F)$ that sending any non-empty subset of the $2g$ generators to the nontrivial element of $C_2$ (and the others to $1$) gives rise to a (surjective) homomorphism, which yields the value $a_2 = 2^{2g} - 1 = 4^g - 1$.

Answer 2

More generally, if $G$ is any finitely generated group, then it has only finitely many subgroups of any given index. Indeed, if $H\subseteq G$ is a subgroup of index $n$, then it is the stabilizer of a point in an action of $G$ on a set with $n$ elements (namely, the cosets of $H$). But there are only finitely many such actions since $G$ is finitely generated, so there can only be finitely many such subgroups $H$.

I don't know how to count how many subgroups there are of a given index when $G=\pi_1(F)$.