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A congruence subgroup property for symmetric mapping class groups

Marco Boggi

TL;DR

This work proves a congruence subgroup property for the centralizer of a finite group action on a hyperbolic surface, reducing to the quotient surface $S_{/G}^{\circ}$ and obtaining the property for ${\Gamma}(S)^G$ when $g(S_{/G})\le 2$. The authors develop a procongruence framework, establish key centralizer identities in profinite completions for good groups, and handle both free and nonfree actions. They extend the results to relative mapping class groups and derive conjugacy-separability conclusions: in genus up to $2$, torsion elements and solvable finite subgroups are conjugacy distinguished. The methods combine profinite rigidity, normalizer-centralizer exact sequences, and cohomological tools (Lyndon–Hochschild–Serre, Shapiro) to connect classical mapping class group properties with their profinite analogues, yielding strong separability results with potential applications in arithmetic and geometric topology.

Abstract

We prove the congruence subgroup property for the centralizer of a finite subgroup $G$ in the mapping class group of a hyperbolic oriented and connected surface of finite topological type $S$ such that the genus of the quotient surface $S/G$ is at most $2$. As an application, we show that torsion elements in the mapping class group of a surface of genus $\leq 2$ are conjugacy distinguished.

A congruence subgroup property for symmetric mapping class groups

TL;DR

This work proves a congruence subgroup property for the centralizer of a finite group action on a hyperbolic surface, reducing to the quotient surface and obtaining the property for when . The authors develop a procongruence framework, establish key centralizer identities in profinite completions for good groups, and handle both free and nonfree actions. They extend the results to relative mapping class groups and derive conjugacy-separability conclusions: in genus up to , torsion elements and solvable finite subgroups are conjugacy distinguished. The methods combine profinite rigidity, normalizer-centralizer exact sequences, and cohomological tools (Lyndon–Hochschild–Serre, Shapiro) to connect classical mapping class group properties with their profinite analogues, yielding strong separability results with potential applications in arithmetic and geometric topology.

Abstract

We prove the congruence subgroup property for the centralizer of a finite subgroup in the mapping class group of a hyperbolic oriented and connected surface of finite topological type such that the genus of the quotient surface is at most . As an application, we show that torsion elements in the mapping class group of a surface of genus are conjugacy distinguished.
Paper Structure (12 sections, 7 theorems, 14 equations)

This paper contains 12 sections, 7 theorems, 14 equations.

Table of Contents

  1. Introduction
  2. The congruence subgroup property
  3. Conjugacy separability of torsion elements in mapping class groups
  4. Preliminary results
  5. Normalizers and centralizers of finite subgroups of the mapping class group of a surface
  6. The congruence topology
  7. The centralizer of a finite subgroup in the profinite completion of a good group
  8. Proof of Theorem A
  9. The case of a free action
  10. The general case
  11. The congruence subgroup property for relative mapping class groups
  12. Conjugacy separability

Key Result

Proposition 2.1

The natural epimorphism of mapping class groups $q^\circ\colon{\Gamma}(S^\circ)\to{\Gamma}(S)$, associated to the embedding of $S^\circ$ in $S$, induces the isomorphisms:

Theorems & Definitions (15)

  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 5 more