A generalization of the simulation theorem for semidirect products

We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed $\mathbb{Z}^d$ action as a factor of a subaction of a $\mathbb{Z}^{d+2}$-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with $\mathbb{Z}^2$. Let $H$ be a finitely generated group and $G = \mathbb{Z}^2 \rtimes H$ a semidirect product. We show that for any effectively closed $H$-dynamical system $(Y,f)$ where $Y$ is a Cantor set, there exists a $G$-subshift of finite type $(X,\sigma)$ such that the $H$-subaction of $(X,\sigma)$ is an extension of $(Y,f)$. In the case where $f$ is an expansive action of a recursively presented group $H$, a subshift conjugated to $(Y,f)$ can be obtained as the $H$-projective subdynamics of a $G$-sofic subshift. As a corollary, we obtain that $G$ admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of $H$ is decidable.