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The large scale geometry of strongly aperiodic subshifts of finite type. (English) Zbl 1400.20034
Summary: A subshift on a group \(G\) is a closed, \(G\)-invariant subset of \(A^G\), for some finite set \(A\). It is said to be a subshift of finite type (SFT) if it is defined by a finite collection of “forbidden patterns”, to be strongly aperiodic if all point stabilizers are trivial, and weakly aperiodic if all point stabilizers are infinite index in \(G\). We show that groups with at least 2 ends have a strongly aperiodic SFT, and that having such an SFT is a QI invariant for finitely presented groups. We show that a finitely presented torsion free group with no weakly aperiodic SFT must be QI-rigid. The domino problem on \(G\) asks whether the SFT specified by a given set of forbidden patterns is empty. We show that decidability of the domino problem is a QI invariant.

MSC:
20F65 Geometric group theory
37B10 Symbolic dynamics
37B50 Multi-dimensional shifts of finite type, tiling dynamics (MSC2010)
20F05 Generators, relations, and presentations of groups
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