Congruence kernels around affine curves

Abstract

Abstract Let S be a smooth affine algebraic curve, and let S ˚ ${\text{{$\mathring{S}$}}}$ be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of mapping class groups by showing that the congruence kernel ker Mod ( S ˚ ) ^ → Out ( π 1 ( S ˚ ) ^ ) $ \ker \Bigl (\widehat{\mathrm {Mod}(\mathring{S})} \rightarrow \mathrm {Out}(\widehat{\pi _1(\mathring{S})})\Bigr ) $ lies in the centralizer of every braid in Mod ( S ˚ ) ${{\mathrm {Mod}(\text{{$\mathring{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 𝒞 ˇ ( S ˚ ) ${\check{\mathcal {C}}(\text{{$\mathring{S}$}})}$ when S is affine, and a new proof of Matsumoto's theorem that the congruence kernel depends only on the genus in the affine case.