  We prove that the boundary dynamics of the (semi)group generated by the
enriched dual transducer characterizes the algebraic property of being free for
an automaton group. We specialize this result to the class of bireversible
transducers and we show that the property of being not free is equivalent to
have a finite Schreier graph in the boundary of the enriched dual pointed on
some essentially non-trivial point. From these results we derive some
consequences from the dynamical, algorithmic and algebraic point of view. In
the last part of the paper we address the problem of finding examples of
non-bireversible transducers defining free groups, we show examples of
transducers with sink accessible from every state which generate free groups,
and, in general, we link this problem to the nonexistence of certain words with
interesting combinatorial and geometrical properties.

Group Theory;Formal Languages and Automata Theory