On the abelianization of congruence subgroups of ${\mathrm{SL}}_2$ over $S$-integers

Abstract

In this work, we compute the first integral homology, or abelianization, of the congruence subgroups $\Gamma (A,{\mathfrak{m}}_A),{\Gamma }_1(A,{\mathfrak{m}}_A)$, and ${\Gamma }_0(A,{\mathfrak{m}}_A)$ for a local ring $A$ with maximal ideal ${\mathfrak{m}}_A$, showing that $H_1(\Gamma (A,{\mathfrak{m}}_A),\mathbb{Z})$ is isomorphic to the additive group of ${\mathfrak{sl}}_2({\mathfrak{m}}_A/{\mathfrak{m}}_A^2)$. We then use these results to determine the structure of the groups $H_1(\Gamma ({\mathcal{O}}_{K,S},\mathfrak{p}),\mathbb{Z})$, $H_1({\Gamma }_1({\mathcal{O}}_{K,S},\mathfrak{p}),\mathbb{Z})$ and $H_1({\Gamma }_0({\mathcal{O}}_{K,S},\mathfrak{p}),\mathbb{Z})$, where ${\mathcal{O}}_{K,S}$ is a Dedekind domain of arithmetic type, not totally imaginary, $|S|\ge 2$, and $\mathfrak{p}$ is a nonzero prime ideal. The computations are given in terms of the residue field $\kappa (\mathfrak{p})$ and the known $H_1({\mathrm{SL}}_2({\mathcal{O}}_{K,S}),\mathbb{Z})$. As a consequence, we also obtain the torsion subgroup of their second integral cohomology. These results will be of paramount importance for a forthcoming work concerning $H_2({\mathrm{SL}}_2({\mathcal{O}}_{K,S}),\mathbb{Z})$.

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