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Krzysztof Krupinski

Publications and source records attributed to Krzysztof Krupinski.

At least 19 recordsLinked to original sources
arXiv2022

On the topological dynamics of automorphism groups; a model-theoretic perspective

We give a model-theoretic treatment of the fundamental results of Kechris-Pestov-Todorčević theory in the more general context of automorphism groups of not necessarily countable structures. One of the main points is a description of the universal ambit as a certain space of types in an expanded language. Using this, we recover various results of Kechris-Pestov-Todorčević, Moore, Ngyuen Van Thé, in the context of automorphism groups of not necessarily countable structures, as well as Zucker.

math.LO↗
arXiv2022

Invariant measures in simple and in small theories

We give examples of (i) a simple theory with a formula (with parameters) which does not fork over the empty set but has mu measure 0 for every automorphism invariant Keisler measure mu, and (ii) a definable group G in a simple theory such that G is not definably amenable, i.e. there is no translation invariant Keisler measure on G We also discuss paradoxical decompositions both in the setting of discrete groups and of definable groups, and prove some positive results about small theories, including the definable amenability of definable groups, and nontriviality of the graded Grothendieck ring.

math.LO↗
arXiv2022

Bohr compactifications of groups and rings

We introduce and study model-theoretic connected components of rings as an analogue of model-theoretic connected components of definable groups. We develop their basic theory and use them to describe both the definable and classical Bohr compactifications of rings. We then use model-theoretic connected components to explicitly calculate Bohr compactifications of some classical matrix groups, such as the discrete Heisenberg group $UT_3(Z)$, the continuous Heisenberg group $UT_3(R)$, and, more generally, groups of upper unitriangular and invertible upper triangular matrices over unital rings.

math.LO↗
arXiv2021

Ramsey theory and topological dynamics for first order theories

We investigate interactions between Ramsey theory, topological dynamics, and model theory. We introduce various Ramsey-like properties for first order theories and characterize them in terms of the appropriate dynamical properties of the theories in question (such as [extreme] amenability of a theory or some properties of the associated Ellis semigroups). Then we relate them to profiniteness and triviality of the Ellis groups of first order theories. In particular, we find various criteria for [pro]finiteness and for triviality of the Ellis group of a given theory from which we obtain wide classes of examples of theories with [pro]finite or trivial Ellis groups. As an initial motivation, we note that profiniteness of the Ellis group of a theory implies that the Kim-Pillay Galois group of this theory is also profinite, which in turn is equivalent to the equality of the Shelah and Kim-Pillay strong types. We also find several concrete examples illustrating the lack of implications between some fundamental properties. In the appendix, we give a full computation of the Ellis group of the theory of the random hypergraph with one binary and one 4-ary relation. This example shows that the assumption of NIP in the version of Newelski's conjecture for amenable theories (proved in [15]) cannot be dropped.

math.LO↗
arXiv2019

Boundedness and absoluteness of some dynamical invariants in model theory

Let ${\mathfrak C}$ be a monster model of an arbitrary theory $T$, $\bar α$ any tuple of bounded length of elements of ${\mathfrak C}$, and $\bar c$ an enumeration of all elements of ${\mathfrak C}$. By $S_{\bar α}({\mathfrak C})$ denote the compact space of all complete types over ${\mathfrak C}$ extending $tp(\bar α/\emptyset)$, and $S_{\bar c}({\mathfrak C})$ is defined analogously. Then $S_{\bar α}({\mathfrak C})$ and $S_{\bar c}({\mathfrak C})$ are naturally $Aut({\mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${\mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${\mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{\bar α}({\mathfrak C})$ and $S_{\bar c}({\mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${\mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{\bar c}({\mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{\bar c}({\mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{\bar α}({\mathfrak C})$ in place of $S_{\bar c}({\mathfrak C})$.

math.LO↗
arXiv2019

Amenability, definable groups, and automorphism groups

We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if $M$ is a countable, $ω$-categorical structure and $Aut(M)$ is amenable, as a topological group, then the Lascar Galois group $Gal_{L}(T)$ of the theory $T$ of $M$ is compact, Hausdorff (also over any finite set of parameters), that is $T$ is G-compact. An essentially special case is that if $Aut(M)$ is extremely amenable, then $Gal_{L}(T)$ is trivial, so, by a theorem of Lascar, the theory $T$ can be recovered from its category $Mod(T)$ of models. On the side of definable groups, we prove for example that if $G$ is definable in a model $M$, and $G$ is definably amenable, then the connected components ${G^{*}}^{00}_{M}$ and ${G^{*}}^{000}_{M}$ coincide, answering positively a question from an earlier paper of the authors. We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.

math.LO↗
arXiv2017

Definable topological dynamics

For a group $G$ definable in a first order structure $M$ we develop basic topological dynamics in the category of definable $G$-flows. In particular, we give a description of the universal definable $G$-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of $G$, that is to the quotient $G^*/{G^*}^{00}_M$ (where $G^*$ is the interpretation of $G$ in a monster model). More generally, we obtain these results locally, i.e. in the category of $Δ$-definable $G$-flows for any fixed set $Δ$ of formulas of an appropriate form. In particular, we define local connected components ${G^*}^{00}_{Δ,M}$ and ${G^*}^{000}_{Δ,M}$, and show that $G^*/{G^*}^{00}_{Δ,M}$ is the $Δ$-definable Bohr compactification of $G$. We also note that some deeper arguments from the topological dynamics in the category of externally definable $G$-flows can be adapted to the definable context, showing for example that our epimorphism from the Ellis group to the $Δ$-definable Bohr compactification factors naturally yielding a continuous epimorphism from the $Δ$-definable generalized Bohr compactification to the $Δ$-definable Bohr compactification of $G$. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

math.LO↗
arXiv2016

Generalized Bohr compactification and model-theoretic connected components

For a group $G$ first order definable in a structure $M$, we continue the study of the "definable topological dynamics" of $G$. The special case when all subsets of $G$ are definable in the given structure $M$ is simply the usual topological dynamics of the discrete group $G$; in particular, in this case, the words "externally definable" and "definable" can be removed in the results described below. Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant $G^{*}/(G^{*})^{000}_{M}$ of $G$, which appears to be "new" in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalized Bohr compactification of $G$; [externally definable] strong amenability. Among other things, we essentially prove: (i) The "new" invariant $G^{*}/(G^{*})^{000}_{M}$ lies in between the externally definable generalized Bohr compactification and the definable Bohr compactification, and these all coincide when $G$ is definably strongly amenable and all types in $S_G(M)$ are definable, (ii) the kernel of the surjective homomorphism from $G^*/(G^*)^{000}_M$ to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup, and (iii) when $Th(M)$ is NIP, then $G$ is [externally] definably amenable iff it is externally definably strongly amenable. In the situation when all types in $S_G(M)$ are definable, one can just work with the definable (instead of externally definable) objects in the above results.

math.LO↗
arXiv2013

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) (intepreted 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. 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.

math.LO↗
arXiv2013

Superrosy fields and valuations

We prove that every non-trivial valuation on an infinite superrosy field of positive characteristic has divisible value group and algebraically closed residue field. In fact, we prove the following more general result. Let $K$ be a field such that for every finite extension $L$ of $K$ and for every natural number $n>0$ the index $[L^*:(L^*)^n]$ is finite and, if $char(K)=p>0$ and $f: L \to L$ is given by $f(x)=x^p-x$, the index $[L^+:f[L]]$ is also finite. Then either there is a non-trivial definable valuation on $K$, or every non-trivial valuation on $K$ has divisible value group and, if $char(K)>0$, it has algebraically closed residue field. In the zero characteristic case, we get some partial results of this kind. We also notice that minimal fields have the property that every non-trivial valuation has divisible value group and algebraically closed residue field.

math.LO↗
arXiv2012

On regular groups and fields

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $ω_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.

math.LO↗
arXiv2012

Borel equivalence relations and Lascar strong types

The space of Lascar strong types, on some sort and relative to a given first order theory T, is in general not a compact Hausdorff space. This paper has at least three aims. First to show that spaces of Lascar strong types and other related spaces such as the Lascar group, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well as some new examples. The third is to explore notions of definable map, embedding and isomorphism between these and related quotient objects. The motivation for writing this paper is the recent discovery, via definable groups, of new examples of non G-compact first order theories.

math.LO↗
arXiv2011

Small, $nm$-stable compact $G$-groups

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq ω$, then $H$ is abelian-by-finite. Both results are significant steps towards the proof of the conjecture that each small, $nm$-stable compact $G$-group is abelian-by-finite. We give examples of small, $nm$-stable compact $G$-groups of infinite ordinal $\NM$-rank, providing counter-examples to the $\NM$-gap conjecture.

math.LO↗
arXiv2010

On relationships between algebraic properties of groups and rings in some model-theoretic contexts

We study relationships between certain algebraic properties of groups and rings definable in a first order structure or $*$-closed in a compact $G$-space. As a consequence, we obtain a few structural results about $ω$-categorical rings as well as about small, $nm$-stable compact $G$-rings, and we also obtain surprising relationships between some conjectures concerning small profinite groups.

math.LO↗
arXiv2010

On $ω$-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 $\emptyset$ type, such a group is abelian.

math.LO↗
arXiv2009

On stable fields and weight

We prove that if K is an infinite stable field whose generic type has weight 1 then K is separably closed. We also obtain partial results when the generic of K has finite weight and when K is strongly stable.

math.LO↗