Self-simulable groups

We say that a finitely generated group Γ Γ is self-simulable if every effectively closed action of Γ Γ on a closed subset of { 0 , 1 } N \{0,1\}N is the topological factor of a Γ Γ -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 × F k F_k × F_k of two free groups of rank k ≥ 2 k ≥ 2 , non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson’s V V , the groups GL n ⁡ ( Z ) GL_n(Z) , SL n ⁡ ( Z ) SL_n(Z) , A u t ( F n ) Aut(F_n) and <inline-formula content-type="math/ma