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International audienceThere is a natural inclusion of SL_2(Z) into SL_2(Z[i]), but it does not induce an injection of commutator factor groups (Abelianizations). In order to see where and how the 3-torsion of the Abelianization of SL_2(Z) disappears, we study a double cover of the amalgamated product decomposition of SL_2(Z) as Z/(4Z) times Z/(6Z) amalgamated over Z/(2Z) inside SL_2(Z[i]); and then compute the homology of the covering amalgam

The subgroup measuring the defect of the abelianization of $$\mathrm{SL}_2(\mathbb{Z }[i])$$