Abelianization of $\text{SL}_2$ over Dedekind domains of arithmetic type
Behrooz Mirzaii, Bruno R. Ramos, Thiago Verissimo
TL;DR
The paper determines the exact abelianization ${SL_2(A)}^{ab}$ for Dedekind domains of arithmetic type $A$, proving finiteness with exponent dividing $12$ in characteristic $0$ and $6$ in positive characteristic, and providing explicit prime-by-prime decompositions. The main method combines the GE$_2$-property for these rings (via Vaserstein–Liehl) with the congruence subgroup property to reduce to finite local factors, yielding a direct-sum description in terms of local data at primes in distinguished sets $S_2$, $S_2'$, and $S_3$. Special cases are worked out for rings of integers of real quadratic fields and cyclotomic fields, including a complete modulo-$24$ classification and cyclotomic formulas. The results generalize and unify prior computations, enabling explicit abelianizations for a broad class of arithmetic rings and guiding computations in concrete number-theoretic settings.
Abstract
We determine the exact group structure of the abelianization of $\text{SL}_2(A)$, where $A$ is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that $\text{SL}_2(A)^\text{ab}$ is finite, with exponent dividing $12$ when $\text{char}(A)=0$, and dividing $6$ when $\text{char}(A)>0$. As illustrative cases, we compute $\text{SL}_2(A)^\text{ab}$ explicitly for instances where $A$ is the ring of integers of a real quadratic field or a cyclotomic extension.