  We study the seeded domino problem, the recurring domino problem and the
$k$-SAT problem on finitely generated groups. These problems are generalization
of their original versions on $\mathbb{Z}^2$ that were shown to be undecidable
using the domino problem. We show that the seeded and recurring domino problems
on a group are invariant under changes in the generating set, are many-one
reduced from the respective problems on subgroups, and are positive equivalent
to the problems on finite index subgroups. This leads to showing that the
recurring domino problem is decidable for free groups. Coupled with the
invariance properties, we conjecture that the only groups in which the seeded
and recurring domino problems are decidable are virtually free groups. In the
case of the $k$-SAT problem, we introduce a new generalization that is
compatible with decision problems on finitely generated groups. We show that
the subgroup membership problem many-one reduces to the $2$-SAT problem, that
in certain cases the $k$-SAT problem many one reduces to the domino problem,
and finally that the domino problem reduces to $3$-SAT for the class of
scalable groups.
