Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups

Automorphisms of procongruence curve and pants complexes

“…However, thanks to a recent result by Hoshi, Minamide and Mochizuki (cf. [11]), we are able to fill this gap in genus 0. Since, in this case, we also know that the congruence subgroup property holds, we get: Theorem 1.3.…”

“…Let us consider the short exact sequence 1 → P Γ(S) → Γ(S) → Σ n → 1, where we put n := n(S). From Corollary C in [11], it follows that the outer representation associated to this short exact sequence identifies Σ n with a normal subgroup of Out I (P Γ(S)). Since P Γ(S) is center free, from Lemma 2.11, it follows that the given element f ∈ Aut I (P Γ(S)) extends to an automorphism of Γ(S), which we also denote by f , so that we have:…”

“…The conclusion then follows from the identity Aut(P Γ(S)) = Aut I (P Γ(S)) (cf. Lemma 3.13 in [7] and Corollary 2.8 in [11]). □…”

“…and (cf. Corollary C in [11]) 1 → Σ 5 → Out I (P Γ(S 0,5 )) → Out I (P Γ(S 0,5 )) /Σ 5 → 1 Since Σ 4 and Σ 5 are complete groups, the above short exact sequences split and there are natural isomorphisms as stated in the lemma (cf. Theorem 7.15 in [23]).…”

Galois-theoretic characterization of geometric isomorphism classes of quasi-monodromically full hyperbolic curves with small numerical invariants

“…Proof. Assertion (i) follows immediately from a similar argument to the argument applied in the proof of [13,Lemma 1.5]. Assertion (ii) follows immediately, by considering a point of M g,[r] that classifies a "totally degenerate pointed stable curve of type (g, r)", from [12, Lemma 5.4, (ii)] and [12,Proposition 5.6,(ii)].…”