On the domino problem of the Baumslag-Solitar groups

In [1] we construct aperiodic tile sets on the Baumslag-Solitar groups BS(m, n). Aperiodicity plays a central role in the undecidability of the classical domino problem on Z(2), and analogously to this we state as a corollary of the main construction that the Domino problem is undecidable on all Baumslag-Solitar groups. In the present work we elaborate on the claim and provide a full proof of this fact. We also provide details of another result reported in [1]: there are tiles that tile the Baumslag-Solitar group BS(m,n) but none of the valid tilings is recursive. The proofs are based on simulating piecewise affine functions by tiles on BS(m, n). (C) 2021 The Author(s). Published by Elsevier B.V.