On compactifications and the topological dynamics of definable groups
Abstract
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).
Cite
@article{arxiv.1212.3176,
title = {On compactifications and the topological dynamics of definable groups},
author = {Jakub Gismatullin and Davide Penazzi and Anand Pillay},
journal= {arXiv preprint arXiv:1212.3176},
year = {2012}
}