SELF-SIMULABLE GROUPS - (:unkn) unknown (American Mathematical Society, 2026)
We say that a finitely generated group Gamma is self-simulable if every effectively closed action of Gamma on a closed subset of {0, 1}N is the topological factor of a Gamma-subshift of finite type. We show that self-simulable groups exist, that any direct product of non-amenable finitely generated groups is self-simulable, that under technical conditions self-simulability is inherited from subgroups, and that the subclass of self-simulable groups is stable under commensurability and quasi-isometries of finitely presented groups. Some notable examples of self-simulable groups obtained are the direct product F-k x F-k of two free groups of rank k > 2, non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson s V, the groups GL(n)(Z), SLn(Z), Aut(F-n) and Out(F-n) for n > 5; The braid groups B-m for m > 7, and certain classes of RAAGs. We also show that Thompson s F is self-simulable if and only if F is non-amenable, thus giving a computability characterization of this well-known open problem. We also exhibit a few applications of self-simulability on the dynamics of these groups, notably, that every self-simulable group with decidable word problem admits a non-empty strongly aperiodic subshift of finite type.