The Schur multiplier of $\rm{SL}_2$, $K_2$, and Dedekind zeta-functions over $S$-integers

Abstract: In this paper, we obtain an exact sequence connecting $$H_2(\rm{SL}<em>2(\mathcal{O}</em>{K,S}), \mathbb{Z})$$ to $$H_2(\rm{SL}<em>2(\mathcal{O}</em>{K,T}), \mathbb{Z})$$, where $\mathcal{O}<em>{K,S}$ is a ring of $S$-integers and $T$ is a set of primes containing $S$. We apply this sequence to establish a relation between $H_2(\rm{SL}_2(\mathcal{O}</em>{K,S}), \mathbb{Z})$ with the second $K$-group $K_2(\mathcal{O}<em>{K,S})$, for $S$ large enough. This leads to a description of the rank and size of the torsion of $H_2(\rm{SL}_2(\mathcal{O}</em>{K,S}), \mathbb{Z})$. As an application, we propose a homological version of the Birch-Tate formula (conjecture) under these assumptions.