The strongly aperiodic SFT conjecture

Let $G$ be a finitely generated group. A strongly aperiodic subshift of finite type (SFT) is an SFT with no periodic configurations. The strongly aperiodic SFT conjecture. $G$ admits a strongly aperiodic SFT if and only if $G$ is one-ended and has decidable word problem. The general case remains widely open, although the claim is known for virtually polycyclic, solvable Baumslag–Solitar, hyperbolic, and lamplighter groups.