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Type-definable and invariant groups in o-minimal structures. (English) Zbl 1118.03028
A 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 \(M^n\).

MSC:
03C64 Model theory of ordered structures; o-minimality
20A15 Applications of logic to group theory
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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)
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