On compactifications and the topological dynamics of definable groups

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 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).