On the abelianization of congruence subgroups of $\mathrm{SL}_2$ over $S$-integers
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, $|S| \geq 2$, 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 $H_2(\mathrm{SL}_2(\mathcal{O}_{K,S}), \mathbb{Z})$.