Abstract
In this paper we show that membership in finitely generated submonoids is undecidable for the free metabelian group of rank 2 and for the wreath product ℤ≀(ℤ×ℤ). We also show that subsemimodule membership is undecidable for finite rank free (ℤ×ℤ)-modules. The proof involves an encoding of Turing machines via tilings. We also show that rational subset membership is undecidable for two-dimensional lamplighter groups.
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