A congruence subgroup property for symmetric mapping class groups

TL;DR

This paper establishes the congruence subgroup property for certain subgroups of the mapping class group of hyperbolic surfaces, with implications for the conjugacy classification of torsion elements in low-genus cases.

Contribution

It proves the congruence subgroup property for centralizers of finite subgroups in mapping class groups of surfaces with genus constraints, extending understanding of their algebraic structure.

Findings

Proves the congruence subgroup property for specific subgroups

Shows torsion elements are conjugacy distinguished in genus ≤ 2

Provides new insights into the structure of mapping class groups

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.