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

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.