Definable homomorphisms of abelian groups in o-minimal structures
The setting is a definable structure \({\mathcal M}\) (which might be stable) in an o-minimal structure \({\mathcal N}\), with a pair of abelian groups \(A\), \(B\) definable in \({\mathcal M}\). The main results are that if there is in \({\mathcal M}\) an infinite definable set of definable homomorphisms \(A\rightarrow B\), then in \({\mathcal M}\) there is a definable infinite field, whose additive group is definably isomorphic in \({\mathcal M}\) both to a definable subgroup of \(B\) and to a definable section of \(A\). Furthermore, the group \({\mathcal H}\) of all \({\mathcal M}\)-definable homomorphisms \(A\rightarrow B\) has a definable subgroup \(H\) such that every \({\mathcal N}\)-definable subgroup of \({\mathcal H}\) lies in the union of finitely many cosets of \(H\). Among various applications, it is shown that any \({\mathcal N}\)-definable group which is not abelian-by-finite interprets an infinite field. There are also applications to definably compact groups: for example, if \(A,B\) are definable abelian groups (in \({\mathcal N}\)) with \(A\) definably compact, then there is no infinite definable family of homomorphisms \(A\rightarrow B\), or \(B\rightarrow A\). The methods are model-theoretic. The authors work with an infinitary notion of definability for groups and rings, and work with infinitesimals (with respect to a large elementary extension) and dimension arguments.
scientific article; zbMATH DE number 1398636