Type-definable and invariant groups in o-minimal structures.

*(English)*Zbl 1118.03028A famous theorem of Anand Pillay says that, if \(M\) is an o-minimal structure and \(G\) is a group definable in \(M\), then there is a unique manifold definable in \(M\) with underlying set \(G\) and the same dimension as \(G\) such that the manifold topology makes \(G\) into a topological group. Further investigations of Berarducci and Otero on this matter imply that, if \(M\) expands a real closed field, then this manifold is even definably isomorphic to a definable subset of \(M^k\) for some \(k\).

The paper under review extends these results to groups \(G\) type-definable in some big o-minimal structure \(M\). It is shown that such a group \(G\) is a type-definable subset of a definable manifold inducing on it a group topology. Furthermore, if \(M\) expands a real closed field, then \(G\) with this group topology is even definably isomorphic to a type-definable group in some \(M^k\) with the topology induced by \(M^k\).

Actually, part of these results holds in a wider setting, i.e., for invariant groups. A subset \(G \subset M^n\) is said to be invariant if there is some small subset \(A\) of \(M\) such that every automorphism of \(M\) acting identically on \(A\) fixes \(G\) setwise. An invariant group is a group \(G\) such that both \(G\) as a set and its group operation are invariant. Invariant groups properly extend type-definable groups. In this enlarged framework a dimension theory is developed, and it is proved that each invariant group \(G \subseteq M^n\) has a unique topology making it a topological group and inducing on some large invariant \(U \subseteq G\) the same topology as \(M^n\).

The paper under review extends these results to groups \(G\) type-definable in some big o-minimal structure \(M\). It is shown that such a group \(G\) is a type-definable subset of a definable manifold inducing on it a group topology. Furthermore, if \(M\) expands a real closed field, then \(G\) with this group topology is even definably isomorphic to a type-definable group in some \(M^k\) with the topology induced by \(M^k\).

Actually, part of these results holds in a wider setting, i.e., for invariant groups. A subset \(G \subset M^n\) is said to be invariant if there is some small subset \(A\) of \(M\) such that every automorphism of \(M\) acting identically on \(A\) fixes \(G\) setwise. An invariant group is a group \(G\) such that both \(G\) as a set and its group operation are invariant. Invariant groups properly extend type-definable groups. In this enlarged framework a dimension theory is developed, and it is proved that each invariant group \(G \subseteq M^n\) has a unique topology making it a topological group and inducing on some large invariant \(U \subseteq G\) the same topology as \(M^n\).

Reviewer: Carlo Toffalori (Camerino)

##### MSC:

03C64 | Model theory of ordered structures; o-minimality |

20A15 | Applications of logic to group theory |

Full Text:
DOI

**OpenURL**

##### References:

[1] | Tame topology and o-minimal structures 248 (1998) |

[2] | Journal of Pure and Applied Algebra 53 pp 239–255– (1988) |

[3] | Annals of Pure and Applied Logic 107 pp 87–119– (2001) |

[4] | Elements of mathematics, general topology 293 |

[5] | Annals of Pure and Applied Logic 101 pp 1–27– (2000) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.