Post-surjectivity and balancedness of cellular automata over groups

We discuss cellular automata over arbitrary finitely generated groups. Wecall a cellular automaton post-surjective if for any pair of asymptoticconfigurations, every pre-image of one is asymptotic to a pre-image of theother. The well known dual concept is pre-injectivity: a cellular automaton ispre-injective if distinct asymptotic configurations have distinct images. Weprove that pre-injective, post-surjective cellular automata are reversible.Moreover, on sofic groups, post-surjectivity alone implies reversibility. Wealso prove that reversible cellular automata over arbitrary groups arebalanced, that is, they preserve the uniform measure on the configurationspace.Comment: 16 pages, 3 figures, LaTeX "dmtcs-episciences" document class. Final version for Discrete Mathematics and Theoretical Computer Science. Prepared according to the editor's request