On Monodromically Full Points of Configuration Spaces of Hyperbolic Curves

Abstract

Abstract. 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.

Congruence topologies on the mapping class group

“…Theorem 2.2 in [18]) and then extended by Hoshi and Mochizuki to the closed surface case (cf. Corollary 6.2 in[15] and Lemma 20 in[13]).…”

Conditional results on the birational section conjecture over small number fields

“…This follows immediately from a similar argument to the argument applied in the proof of Lemma 1.5 [cf. also the proof of [7], Lemma 54 (respectively, [15], Proposition 2.8, (iv)), concerning the proof of the equivalence (1) ⇔ (2) (respectively, (1) ⇔ (3))].…”

“…Moreover, observe that it follows immediately from the equivalence (1) ⇔ (4) of Proposition A.6 that, to verify the equivalence (1) ⇔ (2), by considering the pro-l Galois section of X/k naturally determined by swhere l ranges over elements of Σ -we may assume without loss of generality that Σ is of cardinality 1. On the other hand, since Σ is of cardinality 1, it follows immediately from [7], Proposition 19, (ii), that the kernel of the composite G k s → Π Σ X/k → Aut(∆ Σ X/k ) coincides with the kernel of the pro-Σ outer Galois representation associated to the hyperbolic curve X \ {x} over k. Thus, the equivalence (1) ⇔ (2) follows immediately from [15] …”

“…On the other hand, since H ⊆ ∆ Σ X/k is characteristic, and ∆ Σ X/k is slim [where we refer to the discussion entitled "Profinite groups" in §0 concerning the term "slim"], it follows from [7], Lemma 5, that we have a natural injection Aut(∆ Σ X/k ) → Aut(H). Thus, to verify Proposition A.2, by replacing Π Σ X/k by the open subgroup H · Im(s), we may assume without loss of generality that X is of genus ≥ 1.…”

“…First, we verify assertion (i). Now since, as is well-known, the profinite group ∆ Σ Y /k is slim [where we refer to the discussion entitled "Profinite groups" in §0 concerning the term "slim"], for any open subgroup H ⊆ ∆ Σ Y /k of ∆ Σ Y /k , it follows immediately from [7], Lemma 5, …”