A geometric approach to (semi)-groups defined by automata via dual transducers

We give a geometric approach to groups defined by automata via the notion of enriched dual of an inverse transducer. Using this geometric correspondence we first provide some finiteness results, then we consider groups generated by the dual of Cayley type of machines. Lastly, we address the problem of the study of the action of these groups on the boundary. We show that examples of groups having essentially free actions without critical points lie in the class of groups defined by the transducers whose enriched duals generate torsion-free semigroup. Finally, we provide necessary and sufficient conditions to have finite Schreier graphs on the boundary yielding to the decidability of the algorithmic problem of the existence of Schreier graphs on the boundary whose cardinalities are bounded from above by some fixed integer.