Bruno R. Ramos

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 isomorph...