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.
KeywordsAutomata groups Stallings construction Schreier graphs Dynamics on the boundary
Mathematics Subject Classification20E08 20M05 20M35 68R15 68R99
The authors acknowledge the anonymous referee for his/her precious remarks that have certainly improved the quality of the paper.
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