A note on Lascar strong types in simple theories

Let T be a countable, small simple theory. In this paper, we prove for such T, the notion of Lascar Strong type coincides with the notion of a strong type,over an arbitrary set

The Lascar groups and the 1st homology groups in model theory

Let $p$ be a strong type of an algebraically closed tuple over B=\acl^{\eq}(B) in any theory $T$. Depending on a ternary relation \indo^* satisfying some basic axioms (there is at least one such, namely the trivial independence in $T$), the first homology group $H^*_1(p)$ can be introduced, similarly to \cite{GKK1}. We show that there is a canonical surjective homomorphism from the Lascar group over $B$ to $H^*_1(p)$. We also notice that the map factors naturally via a surjection from the `relativised' Lascar group of the type (which we define in analogy with the Lascar group of the theory) onto the homology group, and we give an explicit description of its kernel. Due to this characterization, it follows that the first homology group of $p$ is independent from the choice of \indo^*, and can be written simply as $H_1(p)$. As consequences, in any $T$, we show that $|H_1(p)|\geq 2^{\aleph_0}$ unless $H_1(p)$ is trivial, and we give a criterion for the equality of stp and Lstp of algebraically closed tuples using the notions of the first homology group and a relativised Lascar group. We also argue how any abelian connected compact group can appear as the first homology group of the type of a model.Comment: 30 pages, no figures, this merged with the article arXiv:1504.0772

Automorphism groups over a hyperimaginary

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

Transitivity, lowness, and ranks in NSOP1_1 theories

We develop the theory of Kim-independence in the context of NSOP$_{1}$ theories satsifying the existence axiom. We show that, in such theories, Kim-independence is transitive and that \ind^{K}-Morley sequences witness Kim-dividing. As applications, we show that, under the assumption of existence, in a low NSOP$_{1}$ theory, Shelah strong types and Lascar strong types coincide and, additionally, we introduce a notion of rank for NSOP$_{1}$ theories