Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk.  show that, in general, the set of singular points has measure zero. Then  focus our attention on several classes of automata.  characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group.  prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. Then  provide some necessary conditions for such examples to exist, and study some dynamical properties of their Schreier graphs in the boundary. Finally  relate some of those automata with aperiodic tilings of the discrete plane via Wang tilings. This has a series of consequences from the algorithmic and dynamical points of view, and is related to a problem of Gromov regarding the searching for examples of CAT(0) complexes whose fundamental groups are not hyperbolic and contain no subgroup isomorphic to $\mathbb{Z}^{2}$.
["Daniele D'Angeli", 'Thibault Godin', 'Ines Klimann', 'Matthieu Picantin', 'Emanuele Rodaro']
I am writing a research paper on a idea Boundary action of automaton groups without singular points and Wang tilings
[{"from": "human", "value": "I am writing a research paper on a idea Boundary action of automaton groups without singular points and Wang tilings"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  study automaton groups without singular points, that is, points in the boundary for which the map that associates to each point its stabilizer, is not continuous. This is motivated by the problem of finding examples of infinite bireversible automaton groups with all trivial stabilizers in the boundary, raised by Grigorchuk and Savchuk.  show that, in general, the set of singular points has measure zero. Then  focus our attention on several classes of automata.  characterize those contracting automata generating groups without singular points, and apply this characterization to the Basilica group.  prove that potential examples of reversible automata generating infinite groups without singular points are necessarily bireversible. Then  provide some necessary conditions for such examples to exist, and study some dynamical properties of their Schreier graphs in the boundary. Finally  relate some of those automata with aperiodic tilings of the discrete plane via Wang tilings. This has a series of consequences from the algorithmic and dynamical points of view, and is related to a problem of Gromov regarding the searching for examples of CAT(0) complexes whose fundamental groups are not hyperbolic and contain no subgroup isomorphic to $\\mathbb{Z}^{2}$."}]