Ballier–Stein conjecture on the domino problem and virtually free groups

Prove that the domino problem on any finitely generated group G is decidable if and only if G is virtually free, thereby establishing the Ballier–Stein characterization of decidability for the domino problem.

Background

The domino problem asks whether, for a given finite tileset (C,D) on a finitely generated group G=⟨S⟩, there exists a vertex-coloring of the Cayley graph by C such that each labeled edge obeys D. While the problem is solvable on Z and undecidable on Z^2, its decidability across groups is conjecturally characterized by virtual freeness via monadic second-order logic decidability.

The paper recalls the Ballier–Stein conjecture and uses it as a central context: their main theorem constructs a non–virtually free group with decidable snake tiling, implying either a refutation of the Ballier–Stein conjecture or a separation between domino and snake problems.