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A Tits alternative for topological full groups. (English) Zbl 07277639
Summary: We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for \(\mathbf{Z}\)-actions. On the other hand, when a finitely generated group \(G\) is not virtually cyclic, then we construct a minimal free action of \(G\) on a Cantor space such that the topological full group contains a non-abelian free group.
MSC:
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
43A07 Means on groups, semigroups, etc.; amenable groups
20F65 Geometric group theory
37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\)
22F05 General theory of group and pseudogroup actions
22F10 Measurable group actions
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