On groups and counter automata

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller–Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognized by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata.