I'm reading this post by Charles Siegel on Monodromy Representations and there is a short remark on the proof of irreducibility of moduli space of genus $g$ curves ${\mathcal{M}_g}$ :
Just look at ${p:\tilde{X}\rightarrow X}$ a covering space of degree ${n}$. Then there’s a map ${\pi_1(X)\rightarrow S_n}$ given by the action of ${\pi_1(X)}$ by interchanging the sheets of the cover.
So this gave us a map from the fundamental group of the base into the automorphism group of the fibers. This actually happens in general, for any fiber bundle, just go around the loop and see how the transition functions change. But that’s not the most interesting case (though it can be adapted into a nice proof that ${\mathcal{M}_g}$ is irreducible). [...]
Could anybody elaborate the argument which Charles Siegel means that the fact that we have a map from the fundamental group of the base into the automorphism group of the fibers can be used to show that ${\mathcal{M}_g}$ is irreducible?
