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
recognised 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.Comment: 18 page