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

TL;DR

This paper precisely characterizes the abelianization of the special linear group over Dedekind domains of arithmetic type, revealing finiteness and explicit bounds on its exponent, with concrete computations for specific number rings.

Contribution

It provides the first exact descriptions of the abelianization of SL_2 over Dedekind domains of arithmetic type, including explicit calculations for rings of integers in quadratic and cyclotomic fields.

Findings

SL_2(A)^ab is finite with exponent dividing 12 in characteristic zero

SL_2(A)^ab has exponent dividing 6 in positive characteristic

Explicit computations for rings of integers in real quadratic and cyclotomic fields

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.