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A subshift on a group G is a closed, G-invariant subset of A^G, for somefinite set A. It is said to be a subshift of finite type (SFT) if it is definedby a finite collection of 'forbidden patterns', to be strongly aperiodic if allpoint stabilizers are trivial, and weakly aperiodic if all point stabilizersare infinite index in G. We show that groups with at least 2 ends have astrongly aperiodic SFT, and that having such an SFT is a QI invariant forfinitely presented torsion free groups. We show that a finitely presentedtorsion free group with no weakly aperiodic SFT must be QI-rigid. The dominoproblem on G asks whether the SFT specified by a given set of forbiddenpatterns is empty. We show that decidability of the domino problem is a QIinvariant.

The large scale geometry of strongly aperiodic subshifts of finite type