Self-simulable groups
Sebastián Barbieri, Mathieu Sablik, Ville Salo
Source abstract
We say that a finitely generated group Γ \Gamma is self-simulable if every effectively closed action of Γ \Gamma on a closed subset of { 0 , 1 } N \{\mathtt {0},\mathtt {1}\}^{\mathbb {N}} is the topological factor of a Γ \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 × F k F_k \times F_k of two free groups of rank k ≥ 2 k \geq 2 , non-amenable finitely generated branch groups, the simple groups of Burger and Mozes, Thompson’s V V , the groups GL n ( Z ) \operatorname {GL}_n(\mathbb {Z}) , SL n ( Z ) \operatorname {SL}_n(\mathbb {Z}) , A u t ( F n ) Aut(F_n) and O u t ( F n ) Out(F_n) for n ≥ 5 n \geq 5 ; The braid groups B m B_m for m ≥ 7 m \geq 7 , and certain classes of RAAGs. We also show that Thompson’s F F is self-simulable if and only if F 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.
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