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We present a construction that yields infinite families of non-isomorphic semidirect products N ⋊ F m sharing a specified profinite completion.Within each family, m ≥ 2 is constant and N is a fixed group.For m = 2 we can take N to be free of rank ≥ 10, free abelian of rank ≥ 12, or a surface group of genus ≥ 5.
Paweł Piwek (Fri,) studied this question.
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