Contractible subshifts are introduced as block-map gluings strengthening strong irreducibility; they coincide with retracts of full shifts precisely when they are SFTs with fixed points, and imply dense periodic points on virtually polycyclic, metabelian Baumslag-Solitar, and lamplighter groups.
When are group shifts of finite type?
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abstract
It is known that a group shift on a polycyclic group is necessarily of finite type. We show that, for trivial reasons, if a group does not satisfy the maximal condition on subgroups, then it admits non-SFT abelian group shifts. In particular, we show that if group is elementarily amenable or satisfies the Tits alternative, then it is virtually polycyclic if and only if all its group shifts are of finite type. Our theorems are minor elaborations of results of Schmidt and Osin.
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Contractible subshifts
Contractible subshifts are introduced as block-map gluings strengthening strong irreducibility; they coincide with retracts of full shifts precisely when they are SFTs with fixed points, and imply dense periodic points on virtually polycyclic, metabelian Baumslag-Solitar, and lamplighter groups.