Automorphism groups over a hyperimaginary

Abstract

In this paper we study the Lascar group over a hyperimaginary $\boldsymbol{e}$. We verify that various results about the group over a real set still hold when the set is replaced by $\boldsymbol{e}$. First of all, there is no written proof in the available literature that the group over $\boldsymbol{e}$ 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 hand, 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.