A refined scissors congruence group and the third homology of SL2

There is a natural connection between the third homology of SL2(A) and the refined Bloch group 7Z3(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 GE2-domain such that -1 E Ax2, then we have the exact sequence H3(SM2(A),Z) -+ H3(SL2(A), Z) -+7Z3(A) -+ 0, where SM2(A) is the group of monomial matrices in SL2(A). Moreover, we show that 7ZP1(A), the refined scissors congruence group of A, is naturally isomorphic to the relative homology group H3(SL2(A), SM2(A); Z). (c) 2024 Elsevier B.V. All rights reserved.