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, $\mathrm{Mod}\_g$?



Fundamental group of $M_g^\circ$
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.


