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We generalize the classical definition of effectively closed subshift tofinitely generated groups. We study classical stability properties of thisclass and then extend this notion by allowing the usage of an oracle to theword problem of a group. This new class of subshifts forms a conjugacy classthat contains all sofic subshifts. Motivated by the question of whether thereexists a group where the class of sofic subshifts coincides with that ofeffective subshifts, we show that the inclusion is strict for several groups,including recursively presented groups with undecidable word problem, amenablegroups and groups with more than two ends. We also provide an extended model ofTuring machine which uses the group itself as a tape and characterizes ourextended notion of effectiveness. As applications of these machines we provethat the origin constrained domino problem is undecidable for any group of theform $G \times \mathbb{Z}$ subject to a technical condition on $G$ and wepresent a simulation theorem which is valid in any finitely generated group.

A notion of effectiveness for subshifts on finitely generated groups