Third homology of ${\mathrm{SL}}_2$ over Number fields: The norm-Euclidean quadratic imaginary case

Abstract

In the article The third homology of $S{L}_2(\mathbb{Q})$, Hutchinson determined the structure of $H_3\left({\mathrm{SL}}_2(\mathbb{Q}),\mathbb{Z}\left[\frac{1}{2}\right]\right)$ by expressing it in terms of $K_3^{\mathrm{ind}}(\mathbb{Q})\cong \mathbb{Z}/24$ and the scissor congruence group of the residue field ${\mathbb{F}}_p$ with $p$ a prime number. In this paper, we develop further the properties of the refined scissors congruence group in order to extend this result to the case of imaginary quadratic number fields whose ring of integers is a Euclidean domain with respect to the norm.