A Refined scissors congruence group and the third homology of ${\text{SL}}_2$

There is a natural connection between the third homology of ${\text{SL}}_2(A)$ and the refined Bloch group $\mathcal{R}\mathcal{B}(A)$ of a commutative ring $A$. In this article we investigate this connection and as the main result we show that if $A$ is a universal ${\text{GE}}_2$-domain such that $-1\in A^{\times 2}$, then we have the exact sequence $H_3({\text{SM}}_2(A),\mathbb{Z})\to H_3({\text{SL}}_2(A),\mathbb{Z})\to \mathcal{R}\mathcal{B}(A)\to 0$, where ${\text{SM}}_2(A)$ is the group of monomial matrices in ${\text{SL}}_2(A)$. Moreover we show that ${\mathcal{R}\mathcal{P}}_1(A)$, the refined scissors congruence group of $A$, naturally is isomorph with the relative homology group $H_3({\text{SL}}_2(A),{\text{SM}}_2(A),\mathbb{Z})$.