Automorphism groups over a hyperimaginary

Abstract

In this paper we study the Lascar group over a hyperimaginary e. We verify that various results about the group over a real set still hold when the set is replaced by e. First of all, there is no written proof in the available literature that the group over e is a topological group. We present an expository style proof of the fact, which even simplifies existing proofs for the real case. We further extend a result that the orbit equivalence relation under a closed subgroup of the Lascar group is type-definable. On the one hand, we correct errors appeared in the book written by the first author and produce a counterexample. On the other, we extend Newelski’s Theorem that ‘a G-compact theory over a set has a uniform bound for the Lascar distances’ to the hyperimaginary context. Lastly, we supply a partial positive answer to a question about the kernel of a canonical projection between relativized Lascar groups, which is even a new result in the real context. The Lascar (automorphism) group of a first-order complete theory and its quotient groups such as the Kim-Pillay group and the Shelah group have been central themes in contemporary model theory. The study on those groups enables us to develop Galois theoretic correspondence between the groups and their orbit-equivalence relations on a monster model such as Lascar types, Kim-Pillay types, and Shelah strong types. The notions of the Lascar group and its topology are introduced first by D. Lascar in [9] using ultraproducts. Later more favorable equivalent definition is suggested in [7] and [11], which is nowadays considered as a standard approach. However even a complete proof using the approach of the fundamental fact that the Lascar group is a topological group is not so well available. For example in [2], its proof is left to the readers, while the proof is not at all trivial. As far as we can see, only in [14], a detailed proof is written. Aforementioned results are for the Lascar group over ∅, or more generally over a real set A. In this paper we study the Lascar group over a hyperimaginary e and verify how results on the Lascar group over A can be extended to the case when the set is replaced by e. Indeed this attempt was made in [8] (and rewritten in [6, §5.1]). However those contain some errors, and moreover a proof of that the Lascar group over e is a topological group is also missing. In this paper we supply a proof of the fact in a detailed expository manner. Our proof is more direct and even simplifies that for the group over ∅ in [14]. We correct the mentioned errors in [6],[8], as well. In particular we correct the proof of that the orbit equivalence relation under a closed normal subgroup of the Lascar group over e is type-definable over e. Moreover we extend Newelski’s Theorem in [12] to the hyperimaginary context. Namely we show that if T is G-compact over e then there is 2020 Mathematics Subject Classification. Primary 03C60; Secondary 54H11.