Cantor systems, piecewise translations and simple amenable groups. (English) Zbl 1283.37011

A Cantor dynamical system comprises a homeomorphism \(T:C\to C\) of the Cantor set \(C\), and its associated topological full group \([[T]]\) is the group of homeomorphisms of \(C\) that are piecewise given by powers of \(T\), with the pieces being open subsets of \(C\). This group is countable, and work of T. Giordano, I. F. Putnam and C. F. Skau [Isr. J. Math. 111, 285–320 (1999; Zbl 0942.46040)] shows that the topological full group is a complete invariant for flip-conjugacy. Among several results showing that this construction yields interesting groups, H. Matui [Int. J. Math. 17, No. 2, 231–251 (2006; Zbl 1109.37008)] showed that the topological full group is simple for any minimal Cantor system, and is finitely-generated if and only if the system is conjugate to a minimal subshift. R. Grigorchuk and K. Medynets [“On algebraic properties of topological full groups ”, Preprint, arXiv:1105.0719] conjectured that \([[T]]\) is amenable if \(T\) is minimal, and this is the first theorem here. As a result the authors deduce that there exist finitely-generated simple groups that are infinite and amenable, and moreover that there are \(2^{\aleph_0}\) mutually non-isomorphic examples. None of these examples are finitely presented.


37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
37B10 Symbolic dynamics
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