  Let $G$ be a group and let $V$ be an algebraic group over an algebraically
closed field. We introduce algebraic group subshifts $\Sigma \subset V^G$ which
generalize both the class of algebraic sofic subshifts of $V^G$ and the class
of closed group subshifts over finite group alphabets. When $G$ is a
polycyclic-by-finite group, we show that $V^G$ satisfies the descending chain
condition and that the notion of algebraic group subshifts, the notion of
algebraic group sofic subshifts, and that of algebraic group subshifts of
finite type are all equivalent. Thus, we obtain extensions of well-known
results of Kitchens and Schmidt to cover the case of many non-compact group
alphabets.
