  We say that a finitely generated group $\Gamma$ is (dynamically)
self-simulable if every effectively closed action of $\Gamma$ on a closed
subset of $\{\mathtt{0},\mathtt{1}\}^{\mathbb{N}}$ 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 $F_k \times F_k$ of two free groups of rank $k \geq 2$, non-amenable
finitely generated branch groups, the simple groups of Burger and Mozes,
Thompson's $V$, the groups $\operatorname{GL}_n(\mathbb{Z})$,
$\operatorname{SL}_n(\mathbb{Z})$, $\operatorname{Aut}(F_n)$ and
$\operatorname{Out}(F_n)$ for $n \geq 5$; The Braid groups $B_m$ for $m \geq
7$, 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 nonempty
strongly aperiodic subshift of finite type.
