Automorphism groups over a hyperimaginary

TL;DR

This paper extends fundamental properties of the Lascar group to hyperimaginaries, providing new proofs, correcting errors, and generalizing key theorems in model theory related to automorphism groups.

Contribution

It offers the first proof that the Lascar group over a hyperimaginary is a topological group, corrects previous errors, and generalizes important theorems to the hyperimaginary setting.

Findings

Lascar group over hyperimaginary is a topological group

Orbit equivalence relation under closed subgroup is type-definable

Extended bounds on Lascar distances to hyperimaginaries

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, "Simplicity Theory" [6, 5.1.14-15] and produce a counterexample. On the other, we extend Newelski's Theorem in "The diameter of a Lascar strong type" [12] 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 raised in "The relativized Lascar groups, type-amalgamations, and algebraicity" [4, 2.11], which is even a new result in the real context.