On compactifications and the topological dynamics of definable groups

Gismatullin, Jakub, Penazzi, Davide orcid iconORCID: 0000-0002-9732-1577 and Pillay, Anand (2014) On compactifications and the topological dynamics of definable groups. Annals of Pure and Applied Logic, 165 (2). pp. 552-562. ISSN 01680072

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Official URL: http://dx.doi.org/10.1016/j.apal.2013.07.020


For G a group definable in some structure M , we define notions of “definable” compactification of G and “definable” action of G on a compact space X (definable G -flow), where the latter is under a definability of types assumption on M . We describe the universal definable compactification of G as G⁎/(G⁎)M00 and the universal definable G -ambit as the type space SG(M)SG(M). We also point out the existence and uniqueness of “universal minimal definable G-flows”, and discuss issues of amenability and extreme amenability in this definable category, with a characterization of the latter. For the sake of completeness we also describe the universal (Bohr) compactification and universal G-ambit in model-theoretic terms, when G is a topological group (although it is essentially well-known).

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