In this article we study domino snake problems on finitely generated groups.
We provide general properties of these problems and introduce new tools for
their study. The first is the use of symbolic dynamics to understand the set of
all possible snakes. Using this approach we solve many variations of the
infinite snake problem including the geodesic snake problem for certain classes
of groups. Next, we introduce a notion of embedding that allows us to reduce
the decidability of snake problems from one group to another. This notion
enable us to establish the undecidability of the infinite snake and ouroboros
problems on nilpotent groups for any generating set, given that we add a
well-chosen element. Finally, we make use of monadic second order logic to
prove that domino snake problems are decidable on virtually free groups for all
generating sets.Comment: Accepted to FCT 2023. Comments welcome