Congruence kernels around affine curves

TL;DR

This paper investigates the congruence subgroup property of the mapping class group of affine curves, providing new proofs and insights into the structure of the congruence kernel and related complexes.

Contribution

It offers new proofs of existing theorems and establishes the behavior of the congruence kernel in affine curves, including its centralizer properties and dependence on genus.

Findings

Congruence kernel lies in the centralizer of every braid in Mod(S')

Proves simple-connectivity of Boggi's procongruence curve complex for punctured curves

Shows the congruence kernel depends only on genus in the affine case

Abstract

Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex for curves with at least two punctures, as well as a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.