Abelianization of ${\text{SL}}_2$ over Dedekind domains of arithmetic type

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.