TITLE: Fundamental group of $M_g^\circ$
QUESTION [7 upvotes]: Let's work over the complex numbers $\mathbb{C}$. Let $g\geq3$ be an integer. Let $\mathcal{M}_g$ be moduli stack of smooth genus $g$ curves. Let $M_g$ be the corresponding coarse moduli scheme. They share an open subscheme $M_g^\circ$ parametrizing  automorphism-free smooth genus $g$ curves. 
What is the fundamental group $\pi_1(M_g^\circ)$?
How does it compare to the following notions:
(1) The orbifold mapping class group $\pi_1(\mathcal{M}_g)$, 
(2) The topological fundamental group  $\pi_1(M_g)$, 
(3) The mapping class group of a genus $g$ surface, $\mathrm{Mod}_g$?

REPLY [12 votes]: Except in a few trivial cases, the locus of curves which have an extra automorphism will have codimension greater than one in $\mathcal M_g$. When that happens, the fundamental group of $\mathcal M_g$ must equal the fundamental group of $\mathcal M_g^{\circ}$. (To see this, one can pass to the $3$-torsion cover of $\mathcal M_g$, which is a smooth scheme, and use the fact that removing a codimension two subscheme of a smooth scheme does not affect $\pi_1$.) So it equals cases 1 and 3.