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 S-G(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). (C) 2013 Elsevier B.V. All rights reserved.