On dynamical finiteness properties of algebraic group shifts

Abstract

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.