Connected components of definable groups and o-minimality. I
The authors consider the ``connected components'' \(G^{00}\) and \(G^{000}\) of groups \(G\) definable in o-minimal expansions of real closed fields, or structures closely related to o-minimal structures. Most of the results in this paper follow from structural results on definable groups from the first author's doctoral thesis. These structural results are reviewed in the second section, with notable decompositions involving the various simple factors, compact or not, of the semisimple part of the group. Everything starts in Proposition 2.1 with a proof of the existence of a maximal normal definable torsion-free (and automatically solvable) subgroup, which essentially relies on a lemma on lifting of torsion. In the paper that lemma is proved by invoking very specific machinery on Euler characteristic in o-minimal structures, but it is worth mentioning that it is valid in any group with the descending chain condition on definable subgroups (by arguments similar to the usual proof in the finite Morley rank case).NEWLINENEWLINEIn the third section the authors prove that if \(G\) is a saturated model of the universal cover of \(SL_2(R)\) (interpretable in the two-sorted structure consisting of the field \(\mathbb R\) of reals and the infinite cyclic group), then \(G^{000}\) is a proper subgroup of \(G^{00}=G\). They then give a group definable in the reals and with a similar property (and expect that this property is indeed typical). These groups also provide new ``natural'' examples of theories where the group of Lascar strong automorphisms is properly contained in the group of Kim-Pillay strong automorphisms. In the fourth section the authors raise questions about definable amenability and the nature of types with bounded orbits in the o-minimal and, more generally, NIP environment. In particular they obtain that for a group definable in an o-minimal expansion of a real closed field, the existence of a type with a bounded orbit is equivalent to the definable amenability of the group. They conclude on links between types with bounded orbits and genericity.
scientific article; zbMATH DE number 6084075
scientific article