Determine the homology groups of arithmetic special linear groups in general

Determine the structure of the homology groups H_d(SL_n(\mathcal{O}_{K,S}),\mathbb{Z}) for rings of S-integers \mathcal{O}_{K,S} in the unstable range and in general arithmetic settings.

Background

The paper studies the second integral homology of SL_2 over rings of S-integers and develops exact sequences relating these groups under localization at additional primes. It emphasizes that homological stability results address the stable range, whereas the unstable range remains substantially less understood.

The authors explicitly identify the general determination of these homology groups as unresolved, providing the broad context for their focus on H_2(SL_2(\mathcal{O}_{K,S}),\mathbb{Z}).

References

In fact, finding the structure of the groups $H_d(n(O{K,S}), Z)$, where $O_{K,S}$ is a ring of $S$-integers, is still an open problem in general.

— The Schur multiplier of $\rm{SL}_2$, $K_2$, and Dedekind zeta-functions over $S$-integers  (2609.28677 - Amorim et al., 23 Sep 2026) in Introduction

We suspect that the condition $2 \nmid |H_1(2(O{K,S}), Z)|$ from Theorem \ref{Main} could be improved or replaced by $3 \nmid |H_1(2(O{K,S}), Z)|$, but this depends on a version of Theorem \ref{Gg_0Glob} (the abelianization of $0(O{K,S}, p)$), for $|\kappa(p)|=2$.

— The Schur multiplier of $\rm{SL}_2$, $K_2$, and Dedekind zeta-functions over $S$-integers  (2609.28677 - Amorim et al., 23 Sep 2026) in Remark following Corollary 3.4

Let $O_{K,S}$ be a ring of $S$-integers of a totally real extension $K$, then there is an $S_0 \supseteq \mathcal{S}_2$, such that for all $S \supseteq S_0$, we have, up to a power of $2$,

|\zeta_KS(-1)|=\frac{2r \cdot |H_2(2(O{K,S}), Z)_{\operatorname{tor}|}{|w_2(K)|},

where $r=[K: Q]$.

— The Schur multiplier of $\rm{SL}_2$, $K_2$, and Dedekind zeta-functions over $S$-integers  (2609.28677 - Amorim et al., 23 Sep 2026) in Conjecture 4.4, Section 4