Abelianization of SL2 over Dedekind domains of arithmetic type

Abstract

We determine the exact group structure of the abelianization of ${\text{SL}}_2\left(A\right)$, in which A is a Dedekind domain of arithmetic type with infinitely many units. In particular, our results show that ${\text{SL}}_2{\left(A\right)}^{\text{ab}}$ is finite, with exponent dividing 12 when $\text{char}\left(A\right)=0$, and dividing 6 when $\text{char}\left(A\right)>0$. As illustrative examples, we compute ${\text{SL}}_2{\left(A\right)}^{\text{ab}}$ explicitly for instances where A is the ring of integers of a real quadratic field or a cyclotomic extension.