Automorphisms of procongruence curve and pants complexes

Abstract

In this paper we study the automorphism group of the procongruence mapping class group through its action on the associated procongruence curve and pants complexes. Our main result is a rigidity theorem for the procongruence completion of the pants complex. As an application we prove that moduli stacks of smooth algebraic curves satisfy a weak anabelian property in the procongruence setting.

Notes on hyperelliptic mapping class groups

“…It is not restrictive to assume that ϒ λ = ϒ(S, P) and that σ is in the image of C(S, υ, P) • . The conclusion then follows from Corollary 7.8 and Theorem 4.9 in [12]. Corollary 7.9 implies, in particular, the claim we made at the beginning of this section: Corollary 7.10.…”

“…It is clear that we have Z ϒ(S,P) ( T(σ ) k ) = Z ˇ (S,P) ( T(σ ) k ) ∩ ϒ(S, P) and the same holds for the normalizers appearing in the statement of the corollary. But then the conclusion is an immediate consequence of Corollary 7.9 and Corollary 4.11 in [12] (see also Corollary 4.3 in [11]).…”

“…of an open subgroup is trivial (cf. (i) of Theorem 4.14 in [12]). This then implies the claim of the theorem for P = ∅.…”

“…Another interesting case which is not considered in Theorem 7.13 is when (S, υ, P) is a marked hyperelliptic closed surface of genus 1. In this case, ϒ(S, P) = (S, P) so that, by Theorem 4.14 in [12], we have that Z ϒ(S,P) ( ϒ λ ) is generated by the hyperelliptic involution for P ≤ 2 and is trivial for P > 2.…”