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In the article The third homology of $SL_{2}(\mathbb{Q})$, Hutchinsondetermined 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 theproperties of the refined scissors congruence group in order to extend thisresult to the case of imaginary quadratic number fields whose ring of integersis a Euclidean domain with respect to the norm.

Third homology of $\mathrm{SL}_{2}$ over Number fields: The norm-Euclidean quadratic imaginary case