(This question was discussed by people at the PCMI workshop on moduli spaces, without any clear resolution, so I thought I'd throw it open to MO.)

The hyperelliptic mapping class group is (by definition) the subgroup of mapping classes commuting with an involution. Not all mapping classes are hyperelliptic. In algebraic geometry, we would say that the map $H_g \to M_g$ (where $H_g$ is the moduli space of hyperelliptic genus $g$ curves) does not induce a surjection on fundamental groups.

What about the trigonal locus $T_g$, parametrizing genus-g curves endowed with a degree-3 map to $\mathbb{P}^1$? Does the map $T_g \to M_g$ induce a surjection, or a finite-index inclusion on fundamental groups? (We do know that $\pi_1(T_g)$, like $\pi_1(H_g)$, surjects onto $Sp_{2g}(\mathbb{Z})$, or at least we know its image is Zariski dense; I'm not sure whether we know its image is finite-index, now that I think of it.)

In topology, we would ask the following question: (equivalent? if not, close to it) Let $\phi$ be a surjection from the free group on $k$ generators to $S_3$, sending each generator to the class of a transposition, and let $\Gamma$ be the (finite-index) subgroup of the $k$-strand braid group consisting of elements which stabilize $\phi$ under left composition. The realization of $\phi$ as a degree-$3$ simply branched cover of a sphere yields a map from $\Gamma$ to some mapping class group $\Gamma_g$, whose image is what we might call the trigonal genus-g mapping class group; the question is whether this is a *proper* subgroup.

More generally, one could define the gonality of a mapping class $f$ to be the minimal $d$ such that $[f]$ lies in the image of the fundamental group of the space of $d$-gonal curves of genus $g$. Is this an interesting invariant? (i.e. if it is always 2 or 3 it is not so interesting.)

`$\mathbb{C}^*\times \mathbb{C}^*$`

and also of an elliptic curve. – algori Jul 28 '11 at 3:49