Groups, measures, and the NIP. (English) Zbl 1134.03024
The paper deals with a well-known conjecture of the third author linking groups definable in an o-minimal structure and Lie groups. Several papers (by Berarducci, Otero, Edmundo and Peterzil, and Pillay himself) contributed to this matter in the past. Now a complete proof of the full conjecture is provided.
The statement of the main theorem is the following: Let \(G\) be a definable compact group definable in a saturated o-minimal expansion of a real closed field, and let \(G^{00}\) be its smallest type-definable subgroup of bounded index. Then the quotient group \(G/G^{00}\), equipped with the logic topology, is a compact Lie group whose dimension (as a Lie group) equals the dimension of \(G\) (as a definable set in an o-minimal structure).
The proof proceeds by induction on the dimension of \(G\). The key cases are when (a) \(G\) is commutative and (b) \(G\) is definably simple. In the commutative case the ingredients are the failure of the Independence Property (NIP for short) and the existence on an invariant finitely additive measure on all subsets of \(G\). The authors refer here to Keisler’s theory of measure and forking, and discuss in particular smooth, definable and finitely satisfiable measures. They also point out various consequences of NIP, mainly in the presence of measures. Case (b) was already considered by Peterzil and Pillay under weaker hypotheses.
In the induction step, one assumes that \(G\) has a normal commutative definable subgroup \(N\) such that both \(N\) and \(G/N\) satisfy the claim of the theorem. Indeed a stronger assumption (the “finitely satisfiable generics” property) is necessary on \(N\) and \(G/N\), singling out some suitable features of stable groups. This clearly requires to show this property in cases (a) and (b).
The final section of the paper proposes the new notion of “compact domination” (analogous to the “stable domination” introduced by the first author in his analysis of algebraically closed valued fields). It is conjectured that a definably compact group \(G\) in an o-minimal structure is definably dominated by the quotient \(G/G^{00}\). This is proved in several particular cases.
The statement of the main theorem is the following: Let \(G\) be a definable compact group definable in a saturated o-minimal expansion of a real closed field, and let \(G^{00}\) be its smallest type-definable subgroup of bounded index. Then the quotient group \(G/G^{00}\), equipped with the logic topology, is a compact Lie group whose dimension (as a Lie group) equals the dimension of \(G\) (as a definable set in an o-minimal structure).
The proof proceeds by induction on the dimension of \(G\). The key cases are when (a) \(G\) is commutative and (b) \(G\) is definably simple. In the commutative case the ingredients are the failure of the Independence Property (NIP for short) and the existence on an invariant finitely additive measure on all subsets of \(G\). The authors refer here to Keisler’s theory of measure and forking, and discuss in particular smooth, definable and finitely satisfiable measures. They also point out various consequences of NIP, mainly in the presence of measures. Case (b) was already considered by Peterzil and Pillay under weaker hypotheses.
In the induction step, one assumes that \(G\) has a normal commutative definable subgroup \(N\) such that both \(N\) and \(G/N\) satisfy the claim of the theorem. Indeed a stronger assumption (the “finitely satisfiable generics” property) is necessary on \(N\) and \(G/N\), singling out some suitable features of stable groups. This clearly requires to show this property in cases (a) and (b).
The final section of the paper proposes the new notion of “compact domination” (analogous to the “stable domination” introduced by the first author in his analysis of algebraically closed valued fields). It is conjectured that a definably compact group \(G\) in an o-minimal structure is definably dominated by the quotient \(G/G^{00}\). This is proved in several particular cases.
Reviewer: Carlo Toffalori (Camerino)
MSC:
| 03C64 | Model theory of ordered structures; o-minimality |
| 03C45 | Classification theory, stability, and related concepts in model theory |
| 22C05 | Compact groups |
| 28E05 | Nonstandard measure theory |
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