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.

翻译：我们研究了有限生成群上的种子多米诺问题、递归多米诺问题及$k$-SAT问题。这些问题是对经典$\mathbb{Z}^2$上通过多米诺问题证明不可判定的原始版本的推广。我们证明了群上的种子多米诺问题和递归多米诺问题在生成集变换下保持不变性，可从子群的对应问题通过多一归约得到，并且与有限指数子群上的问题具有正等价性。由此推导出自由群上的递归多米诺问题是可判定的。结合不变性性质，我们猜想种子和递归多米诺问题可判定的群仅限近乎自由群。在$k$-SAT问题方面，我们提出了一种与有限生成群上判定问题兼容的新推广形式。研究表明子群成员问题可通过多一归约转化为2-SAT问题，特定情况下$k$-SAT问题可多一归约为多米诺问题，最后证明对于可伸缩群类，多米诺问题可归约为3-SAT问题。