Decidability and Characterization of Expansivity for Group Cellular Automata

Group cellular automata are continuous, shift-commuting endomorphisms of $G^\mathbb{Z}$, where $G$ is a finite group. We provide an easy-to-check characterization of expansivity for group cellular automata on abelian groups and we prove that expansivity is a decidable property for general (non-abelian) groups. Moreover, we show that the class of expansive group cellular automata is strictly contained in that of topologically transitive injective group cellular automata.