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 Z action as a factor of a subaction of a Z-SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with Z. Let H be a finitely generated group and G = Z o 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,σ) such that the H-subaction of (X,σ) is an extension of (Y, f). In the case where f is an expansive action, 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.