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Tomasz Rzepecki

Publications and source records attributed to Tomasz Rzepecki.

12 recordsLinked to original sources
arXiv2026

On idempotent measure conjecture and decomposition of invariant measures

We continue the study of the semigroup of global invariant types introduced by Gannon, Hoffmann, and Krupiński and the associated convolution semigroup of invariant Keisler measures. The first part of the paper concerns the Idempotent Measure Conjecture, studied in [CGK24] and [GHK25], which predicts that idempotent fim Keisler measures should be precisely the invariant Haar measures on relatively type-definable subgroups. We identify structural properties underlying all previously known positive instances of the conjecture and prove a conditional version under natural semigroup-theoretic hypotheses. In particular, we show that left simplicity of the support and a local invariance condition are sufficient for the conjectural characterization. The second part studies invariant Keisler measures in amenable NIP theories. We prove that the semigroup of global invariant types has a unique minimal left ideal consisting of f-generic types and identify its Ellis groups. Using normalized Haar measure on the Kim-Pillay group of T, we associate canonically an invariant Keisler measure to every Ellis group. We show that the resulting measures are supported on minimal subflows, prove that the flow (S_m(C), Aut(C)) is hereditarily amenable, establish unique ergodicity of every minimal subflow in the countable case, and characterize all ergodic Aut(C)-invariant Keisler measures as the measures arising from this construction.

math.LO↗
arXiv2025

Upside down and backwards

We investigate the semigroup of invariant types through the lens of Ellis theory; primarily focusing on definably amenable NIP groups. In this context, we observe that the collection of strong right $f$-generic types forms the unique minimal left ideal and thus, the Ellis subgroups are isomorphic to $G/G^{00}$ via the canonical quotient map. As consequence of the Newelski-Pillay conjecture, the Ellis subgroups of the semigroup of invariant types are abstractly isomorphic to the Ellis subgroups of the semigroup of finitely satisfiable types in the definable amenable NIP setting. We are interested in the existence of natural isomorphisms from invariant Ellis subgroups to finitely satisfiable Ellis subgroups and we determine when these isomorphisms can be witnessed by variants of the canonical NIP retraction map. Several limiting examples are provided. Outside of the NIP context, we provide an abelian group (and thus definably amenable) with an $\emptyset$-definable (dfg) type in which the invariant Ellis subgroups and finitely satisfiable Ellis subgroups not isomorphic.

math.LO↗
arXiv2024

Inner ultrahomogeneous groups

We define and study the class of inner ultrahomogeneous groups, which includes Hall's universal group and the universal locally recursively presentable group. We provide simple criteria for ample generic automorphisms, straight maximality, uniform simplicity and divisibility (all of which apply to both Hall's universal group and the universal locally recursively presentable group). We show that such groups of infinite exponent are not $\aleph_0$-saturated, their theories are not small, not rosy and have TP$_2$+SOP+IP$_n$ for all $n$. This strengthens and generalises known results about ample generic automorphisms and unstability of Hall's universal group. We also show that the exponents of finite exponent inner ultrahomogeneous groups are uniformly bounded, and we provide a series of examples of inner ultrahomogeneous groups.

math.LO↗
arXiv2022

Homogeneity of abstract linear spaces

We discuss homogeneity and universality issues in the theory of abstract linear spaces, namely, structures with points and lines satisfying natural axioms, as in Euclidean or projective geometry. We show that the two smallest projective planes (including the Fano plane) are homogeneous and, assuming the continuum hypothesis, there exists a universal projective plane of cardinality $\aleph_1$ that is homogeneous with respect to its countable and finite projective subplanes. We also show that the existence of a generic countable linear space is equivalent to an old conjecture asserting that every finite linear space embeds into a finite projective plane.

math.LO↗
arXiv2022

Generating ideals by additive subgroups of rings

We obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings. Let $R$ be any ring equipped with an arbitrary additional first order structure, and $A$ a set of parameters. We show that whenever $H$ is an $A$-definable, finite index subgroup of $(R,+)$, then $H+RH$ contains an $A$-definable, two-sided ideal of finite index. As a corollary, we positively answer Question 3.9 of [Bohr compactifications of groups and rings, J. Gismatullin, G. Jagiella and K. Krupiński]: if $R$ is unital, then $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$, where $\bar R \succ R$ is a sufficiently saturated elementary extension of $R$, and $(\bar R,+)^{00}_A$ [resp. $\bar R^{00}_A$] is the smallest $A$-type-definable, bounded index additive subgroup [resp. ideal] of $\bar R$. This implies that $\bar R^{00}_A=\bar R^{000}_A$, where $\bar R^{000}_A$ is the smallest invariant over $A$, bounded index ideal of $\bar R$. If $R$ is of finite characteristic (not necessarily unital), we get a sharper result: $(\bar R,+)^{00}_A + \bar R \cdot (\bar R,+)^{00}_A = \bar R^{00}_A$. We obtain similar results for finitely generated (not necessarily unital) rings and for topological rings. The above results imply that the simplified descriptions of the definable (so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of the aforementioned paper are valid for all unital rings. We analyze many examples, where we compute the number of steps needed to generate a group by $(\bar R \cup \{1\}) \cdot (\bar R,+)^{00}_A$ and study related aspects, showing "optimality" of some of our main results and answering some natural questions.

math.LO↗
arXiv2019

Hereditary G-compactness

We introduce the notion of hereditary G-compactness (with respect to interpretation). We provide a sufficient condition for a poset to not be hereditarily G-compact, which we use to show that any linear order is not hereditarily G-compact. Assuming that a long-standing conjecture about unstable NIP theories holds, this implies that an NIP theory is hereditarily G-compact if and only if it is stable (and by a result of Simon, this holds unconditionally for $\aleph_0$-categorical theories). We show that if $G$ is definable over $A$ in a hereditarily G-compact theory, then $G^{00}_A=G^{000}_A$. We also include a brief survey of sufficient conditions for G-compactness, with particular focus on those which can be used to prove or disprove hereditary G-compactness for some (classes of) theories.

math.LO↗
arXiv2018

Bounded Invariant Equivalence Relations

We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of problems related to cardinality and Borel cardinality of strong types, quotients of definable groups and related objecets, generalising (and often improving) essentially all hitherto known results in this area. In particular, we show that under reasonable assumptions, strong type spaces are "locally" quotients of compact Polish groups. It follows that they are smooth if and only if they are type-definable, and that a quotient of a type-definable group by an analytic subgroup is either finite or of cardinality at least continuum.

math.LO↗
arXiv2018

Galois groups as quotients of Polish groups

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an $F_σ$ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over $\emptyset$. As an easy conclusion of our main theorem, we get the main result from our recent paper joint with Andand Pillay, which says that for any strong type defined on a single complete type over $\emptyset$, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the aforementioned paper about bounded quotients of type-definable subgroups of definable groups.

math.LO↗
arXiv2018

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup. We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over $\emptyset$ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let $\mathfrak{C}$ be a monster model of a countable theory, $p \in S(\emptyset)$, and $E$ be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on $p(\mathfrak{C})$. Then, exactly one of the following holds: (1) $E$ is relatively definable (on $p(\mathfrak{C})$), smooth, and has finitely many classes, (2) $E$ is not relatively definable, but it is type-definable, smooth, and has $2^{\aleph_0}$ classes, (3) $E$ is not type definable and not smooth, and has $2^{\aleph_0}$ classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.

math.LO↗
arXiv2017

Equivalence relations invariant under group actions

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before. As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

math.LO↗
arXiv2015

Smoothness of bounded invariant equivalence relations

We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to get the conclusion.

math.LO↗