One-relator groups with at least three generators are never periodically rigid, and quasi-planar groups are rigid exactly when virtually cyclic or torsion-free virtually Z^2.
Aperiodic Subshifts on Polycyclic Groups
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We prove that every polycyclic group of nonlinear growth admits a strongly aperiodic SFT and has an undecidable domino problem. This answers a question of [4] and generalizes the result of [2].
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Period-rigidity of one-relator groups
One-relator groups with at least three generators are never periodically rigid, and quasi-planar groups are rigid exactly when virtually cyclic or torsion-free virtually Z^2.