## Abstract

We consider an arbitrary topological group G definable in a structure M, such that some basis for the topology of G consists of sets definable in M. To each such group G we associate a compact G-space of partial types, S^{μ}_{G}.(M) = {pμ : p ϵ S_{G}.(M)}, which is the quotient of the usual type space S_{G}.(M) by the relation of two types being "infinitesimally close to each other". In the o-minimal setting, if p is a definable type then it has a corresponding definable subgroup Stab^{μ}.(p), which is the stabilizer of pμ. This group is nontrivial when p is unbounded; in fact, it is a torsion-free solvable group. Along the way, we analyze the general construction of S^{μ}_{G}.(M) and its connection to the Samuel compactification of topological groups.

Original language | English |
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Pages (from-to) | 2965-2995 |

Number of pages | 31 |

Journal | Journal of the European Mathematical Society |

Volume | 19 |

Issue number | 10 |

DOIs | |

State | Published - 2017 |

### Bibliographical note

Funding Information:The authors thank the US-Israel Binational Science Foundation for its support, and also thank the Mathematical Science Research Institute at Berkeley for its hospitality during Spring 2014. The second author thanks the National Science Foundation for support.

## Keywords

- Compactification
- Definable groups
- O-minimality

## ASJC Scopus subject areas

- Mathematics (all)
- Applied Mathematics