On Monodromically Full Points of Configuration Spaces of Hyperbolic Curves

In the present paper, we introduce and discuss the notion of monodromically full points of configuration spaces of hyperbolic curves. This notion leads to complements to M. Matsumoto’s result concerning the difference between the kernels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to the automorphism and outer automorphism groups of the geometric fundamental group of the hyperbolic curve. More concretely, we prove that any hyperbolic curve over a number field has many “nonexceptional” closed points, i.e., closed points which do not satisfy a condition considered by Matsumoto, but that there exist infinitely many hyperbolic curves which admit many “exceptional” closed points, i.e., closed points which do satisfy the condition considered by Matsumoto. Moreover, we prove a Galois-theoretic characterization of equivalence classes of monodromically full points of configuration spaces, as well as a Galois-theoretic characterization of equivalence classes of quasimonodromically full points of cores. In a similar vein, we also prove a necessary and sufficient condition for quasi-monodromically full Galois sections of hyperbolic curves to be geometric.