We are inspired by recent efforts to generalize the classical embedding theorem of Krieger for $\mathbb{Z}$ subshifts, which states that if $X$ is an SFT and $Y$ is a mixing SFT, then $X$ embeds into $Y$ if certain necessary conditions on the periodic points and entropy are satisfied. Moving to subshifts over groups $G$ beyond $\mathbb{Z}$, an extra essential hypothesis emerges: that there is a homomorphism (a continuous and shift-commuting map, not necessarily injective) from $X$ to $Y$ at all. This is trivially satisfied if, e.g., $Y$ contains a fixed point, but necessary and sufficient conditions for the existence of a homomorphism are not known in general.

In this talk, we present joint work with K. McGoff that constructs a homomorphism $\phi : X \to Y$ in the case that $X$ is aperiodic, $Y$ has the finite extension property, and the underlying group $G$ has the property that every finitely generated subgroup of $G$ has polynomial growth (i.e., $G$ is locally virtually nilpotent by Gromov’s theorem). The finite extension property (FEP) can be seen as a very strong mixing-like condition which has been considered before for subshifts over $\mathbb{Z}^d$ [Briceño, McGoff, Pavlov 2016].