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Preprint Number 544
544. Jakub Gismatullin, Davide Penazzi, Anand Pillay
On compactifications and the topological dynamics of definable groups
Submission date: 14 December 2012.
We discuss definable compactifications and topological dynamics. 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*00_M and the universal definable G-ambit as the type space S_G(M). We also prove 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).
Mathematics Subject Classification: 03C45, 03C64, 37B99
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