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.
Funding Statement
The authors were supported by NRF of Korea grants 2018R1D1A1A02085584 and 2021R1A2C1009639.
Citation
Byunghan KIM. Hyoyoon LEE. "Automorphism groups over a hyperimaginary." J. Math. Soc. Japan 75 (1) 21 - 49, January, 2023. https://doi.org/10.2969/jmsj/87138713
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