Non-abelian big monodromy: NIF and BO for any non-isotrivial family

Establish that for any non-isotrivial smooth proper family of curves \mathscr{C}\to\mathscr{M} of genus g≥2, the induced \pi_1(\mathscr{M})-action on the Betti moduli M_B(Σ_g,r) has no invariant meromorphic functions (NIF) and possesses a Zariski-dense orbit (BO).

Background

This conjecture formalizes ‘big monodromy’ for the non-abelian character variety: the mapping class group (or base monodromy) action should be highly mixing, forbidding invariants and ensuring existence of dense orbits. It generalizes the phenomena proven for special families with large degree Lefschetz pencils and odd rank.