Krzysztof Krupinski

Superrosy dependent groups having finitely satisfiable generics

Abstract We develop a basic theory of rosy groups and we study groups of small U þ -rank satisfying NIP and having finitely satisfiable generics: U þ -rank 1 implies that the group is abelian-by-finite, U þ -rank 2 implies that the group is solvable-by-finite, U þ -rank 2, and not being nilpotent-by-finite implies the existence of an interpretable algebraically closed field.

SOME MODEL THEORY OF POLISH STRUCTURES

We introduce a notion of Polish structure and, in doing so, provide a setting which allows the application of ideas and techniques from model theory, descriptive set theory, topology and the theory of profinite groups. We define a topological notion of independence in Polish structures and prove that it has some nice properties. Using this notion, we prove counterparts of some basic results from geometric stability theory in the context of small Polish structures. Then, we prove some structural theorems about compact groups regarded as Polish structures: each small, nm-stable compact G-group is solvable-by-finite; each small compact G-group of finite N M-rank is nilpotent-by-finite. Examples of small Polish structures and groups are also given.

On -categorical groups and rings with NIP

We show that ω-categorical rings with NIP are nilpotent-by-finite. We prove that an ω-categorical group with NIP and fsg is nilpotent-by-finite. We also notice that an ω-categorical group with at least one strongly regular type is abelian. Moreover, we get that each ω-categorical, characteristically simple p-group with NIP has an infinite, definable abelian subgroup. Assuming additionally the existence of a non-algebraic, generically stable over ∅ type, such a group is abelian.

On Bounded Type-Definable Equivalence Relations

We investigate some topological properties of the spaces of classes of bounded type-definable equivalence relations.

On model-theoretic connected components in some group extensions

We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of SL2(ℝ) (interpreted in a monster model), as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus ≥ 2, expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for SL2(ℤ) instead of SL2(ℝ), which (as most of our examples) was not accessible by the previously known methods.

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewskis conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group

On stable fields and weight

(M,<,+,0)

Topological dynamics and the complexity of strong types

of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group

Superrosy fields and valuations

(M,<,+,0)

Profinite structures interpretable in fields

of exponent 2, and we show that each almost linear, minimal group is torsion.