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Abstract The rank of a tiling’s return module depends on the geometry of its tiles and is not a topological invariant. However, the rank of the first Čech cohomology$$\check H^1(\Omega )$$gives upper and lower bounds for the rank of the return module. For all sufficiently large patches, the rank of the return module is at most the rank of$$\check H^1(\Omega )$$. For a generic choice of tile shapes and an arbitrary reference patch, the rank of the return module is at least the rank of$$\check H^1(\Omega )$$. Therefore, for generic tile shapes and all sufficiently large patches, the rank of the return module is equal to the rank of$$\check H^1(\Omega )$$. 

Abstract Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an interesting way: for the Ising model on a free group, the free-boundary state typically fails to be equilibrium as long as it is not theonlyGibbs state. For every temperature between the uniqueness and reconstruction thresholds a typical sofic approximation gives this state finite but non-maximal pressure, and for every lower temperature the pressure is non-maximal overeverysofic approximation. We also show that, for more general interactions on sofic groups, the local on average limit of Gibbs states over a sofic approximation Σ, if it exists, is a mixture of Σ-equilibrium states. We use this to show that the plus- and minus-boundary-condition Ising states are Σ-equilibrium if Σ is any sofic approximation to a free group. Combined with a result of Dembo and Montanari, this implies that these states have the same entropy over every sofic approximation.