  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
$\mathbb Z\wr (\mathbb Z\times \mathbb Z)$. We also show that subsemimodule
membership is undecidable for finite rank free $(\mathbb Z\times \mathbb
Z)$-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.
