Domino Snake Problems on Groups

TL;DR

This paper explores domino snake problems on finitely generated groups, introducing symbolic dynamics, embedding techniques, and logical methods to analyze their properties, decidability, and undecidability across various group classes.

Contribution

It introduces new tools like symbolic dynamics and embedding notions to study domino snake problems, establishing results on their decidability and undecidability in different group contexts.

Findings

Solved many variations of the infinite snake problem.

Proved undecidability of snake problems on nilpotent groups.

Established decidability of domino snake problems on virtually free groups.

Abstract

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.