Bi-invariant types, reliably invariant types, and the comb tree property

Abstract

We introduce and examine some special classes of invariant types$\unicode{x2014}$bi-invariant, strongly bi-invariant, extendibly invariant, and reliably invariant types$\unicode{x2014}$and show that they are related to certain model-theoretic tree properties. We show that the comb tree property (recently introduced by Mutchnik) is equivalent to the failure of Kim's lemma for bi-invariant types and is implied by the failure of Kim's lemma for reliably invariant types over invariance bases. We show that every type over an invariance base extends to a reliably invariant type$\unicode{x2014}$generalizing an unpublished result of Kruckman and Ramsey$\unicode{x2014}$and use this to show that, under a reasonable definition of Kim-dividing, Kim-forking coincides with Kim-dividing over invariance bases in theories without the comb tree property. Assuming a measurable cardinal, we characterize the comb tree property in terms of a form of dual local character. We also show that the antichain tree property (introduced by Ahn and Kim) seems to have a somewhat similar relationship to strong bi-invariance. In particular, we show that NATP theories satisfy Kim's lemma for strongly bi-invariant types and (assuming a measurable cardinal) satisfy a different form of dual local character. Furthermore, we examine a mutual generalization of the local character properties satisfied by NTP$_2$ and NSOP$_1$ theories and show that it is satisfied by all NATP theories. Finally, we give some related minor results$\unicode{x2014}$a strengthened local character characterization of NSOP$_1$ and a characterization of coheirs in terms of invariant extensions in expansions$\unicode{x2014}$as well as a pathological example of Kim-dividing.