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Remarks on refined scissors congruence group and the third homology of $\mathrm{SL}_2$
This work revisits the key sufficient conditions on a ring $A$ that guarantee the existence of the exact sequence \[ H_3(\mathrm{SM}_2(A), \mathbb{Z})\to H_3(\mathrm{SL}_2(A), \mathbb{Z}) \to \mathcal{RB}(A) \to 0 \] and of the isomorphism \[ H_3(\mathrm{SL}_2(A), \mathrm{SM}_2(A); \mathbb{Z}) \
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