Finiteness properties of torsion fields of abelian varieties

Let $A$ be an abelian variety defined over a field $K.$ We study finite generation properties of the profinite group $\mathrm{Gal}(\Omega/K)$ and of certain closed normal subgroups thereof, where $\Omega$ is the torsion field of $A$ over $K$. In fact, we establish more general finite generation properties for monodromy groups attached to smooth projective varieties via \'etale cohomology. We apply this in order to give an independent proof and generalizations of a recent result of Checcoli and Dill about small exponent subfields of $\Omega/K$ in the number field case. We also give an application of our finite generation results in the realm of permanence principles for varieties with the weak Hilbert property.