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Sep 1 at 9:26 comment added YCor Note: the proposition is highly specific to $G$ being finitely presented (otherwise take $A=B=\mathbf{Z}$, $G=\mathbf{Z}\wr\mathbf{Z}$). Suitably reformulated (as in your paper) it indeed implies as a corollary Baumslag's result that wreath products are not fp.
Sep 1 at 6:20 history answered AGenevois CC BY-SA 4.0