Frank O. Wagner

• D.Phil.
• Professor at Claude Bernard University Lyon 1

Given a $T$-rough definably amenable $T$-rough approximate subgroup $A$ of a group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ of bounded index in $\langle A\rangle$.

A sharply 2-transitive permutation group of characteristic 0 whose point stabiliser has an abelian subgroup of finite index splits. More generally, a near-domain of characteristic 0 with a multiplicative subgroup of finite index avoiding all multipliers $d_{a,b}$ must be a near-field. In particular this answers question 12.48 b) of the Kourovka Not...

We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimension. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that a pseudofinite $\widetilde {\mathfrak M}_c$ -group of finite positive dimension contains a finite-by-abelian subgr...

We define k-genericity and k-largeness for a subset of a group, and determine the value of k for which a k-large subset of G^n is already the whole of G^n , for various equationally defined subsets. We link this with the inner measure of the set of solutions of an equation in a group, leading to new results and/or proofs in equational probabilistic...

An ω-categorical group of finite burden is virtually finite-by-abelian; an ω-categorical ring of finite burden is virtually finite-by-null; an ω-categorical NTP2 ring is nilpotent-by-finite.

A sharply 2-transitive permutation group of finite Morley rank and characteristic 2 splits; a split sharply 2-transitive permutation group of finite Morley rank and characteristic different from 2 is the group of affine transformations of an algebraically closed field. In particular, a sharply 2-transitive permutation group of finite Morley rank of...

An $\omega$-categorical inp-minimal group is virtually finite-by-abelian; an $\omega$-categorical inp-minimal ring is virtually finite-by-null; an $\omega$-categorical NTP 2 ring is virtually nilpotent.

If F is a type-definable family of commensurable subsets, subgroups or sub-vector spaces in a metric structure, then there is an invariant subset, subgroup or sub-vector space commensurable with F. This in particular applies to type-definable or hyper-definable objects in a classical first-order structure.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the twentieth publication in the Lecture Notes in Logic series, contains the proceedings of...

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the twenty-sixth publication in the Lecture Notes in Logic series, contains the proceedings...

There is no bad group of Morley rank 2n+1 with an abelian Borel subgroup of Morley rank n. In particular, there is no bad group of Morley rank 3 (O. Fr{\'e}con).

There is no sad group of Morley rank 2n + 1 with an abelian Borel subgroup of rank n. In particular, Fr{\'e}con's Theorem follows: There is no bad group of Morely rank 3.

Résumé Un groupe interprétable dans le mauvais corps vert est isogène à un quotient d’un sous-groupe définissable d’un groupe algébrique par une puissance du groupe vert. Un sous-groupe définissable d’un groupe algébrique dans un corps vert ou rouge est une extension des points colorés d’un groupe algébrique multiplicatif ou additif par un groupe a...

Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular , it is shown that all notions coincide for non-multidimensional theories where the dimensions...

Unimodularity is localized to a complete stationary type, and its properties are analysed. Some variants of unimodularity for definable and type-definable sets are introduced, and the relationship between these different notions is studied. In particular, it is shown that all notions coincide for non-multidimensional theories where the dimensions a...

A pseudofinite group satisfying the uniform chain condition on centralizers up to finite index has a big finite-by-abelian subgroup.

A bounded automorphism of a field or a group with trivial approximate centre is definable.

In this paper, we shall study type-definable groups in a simple theory with respect to one or several stable reducts. While the original motivation came from the analysis of definable groups in structures obtained by Hrushovski's amalgamation method, the notions introduced are in fact more general, and in particular can be applied to certain expans...

We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of the colour...

The Fitting subgroup of a type-definable group in a simple theory is relatively definable and nilpotent. Moreover, the Fitting subgroup of a supersimple hyperdefinable group has a normal hyperdefinable nilpotent subgroup of bounded index, and is itself of bounded index in a hyperdefinable subgroup.

Given a definably amenable approximate subgroup $A$ of a (local) group in some first-order structure, there is a type-definable subgroup $H$ normalised by $A$ and contained in $A^4$ such that every definable superset of $H$ has positive measure.

We prove a version of Hrushovski's socle lemma for rigid groups in an arbitrary simple theory.

If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make sense of this statement, local simplicity theory for hyperdefinable sets is developped.

Non-n-ampleness as defined by Pillay and Evans is preserved under analysability. Generalizing this to a more general notion of Sigma-ampleness, we obtain an immediate proof for all simple theories of CHatzidakis weak Canonical Base Property (CBP) for types of finite SU-rank. This is then applied to the special case of 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) < ω or NM(H) = ω α for some ordinal α, 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 provide counter-example...

We generalize Frécon’s construction of the inevitable radical to groups in stable and even simple theories.

We classify the groups definable in the coloured fields obtained by Hrushovski amalgamation. A group definable in the bad green field is isogenous to the quotient of a subgroup of an algebraic group by a Cartesian power of the group of green elements. A definable subgroup of an algebraic group in the green or red field is an extension of a Cartesia...

We introduce a generalization of CM-triviality relative to a fixed invariant collection of partial types, in analogy to the Canonical Base Property defined by Pillay, Ziegler and Chatzidakis which generalizes one-basedness. We show that, under this condition, a stable field is internal to the family, and a group of finite Lascar rank has a normal n...

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 Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an in...

We define a reasonably well-behaved class of ultraimaginaries, i.e. classes modulo (Formula presented.)-invariant equivalence relations, called tame, and establish some basic simplicity-theoretic facts. We also show feeble elimination of supersimple ultraimaginaries: If (Formula presented.) is an ultraimaginary definable over a tuple (Formula prese...

This paper completes the proof of the group configuration theorem for simple theories started in [1]. We introduce the notion of an almost hyperdefinable (poly-)structure, and show that it has a reasonable model theory. We then construct an almost hyperdefinable group from a polygroup chunk.

We reconstruct the group action in the group configuration theorem. We apply it to show that in an ω-categorical theory a finitely based pseudolinear regular type is locally modular, and the geometry associated to a finitely based locally modular regular type is projective geometry over a finite field.

If 𝒞 is a pseudo-variety, then a supersimple residually 𝒞 group is nilpotent-by-poly-𝒞.

We generalize Fr\'econ's construction of the inevitable radical to groups in stable and even simple theories.

We show that dependent elds have no Artin-Schreier extension, and that simple elds have only a nite number of them.

We show that NIP fields have no Artin-Schreier extension, and that simple fields have only a finite number of them.

We start an analysis of geometric properties of a structure relative to a reduct. In particular, we look at definability of groups and fields in this context. In the relatively one-based case, every definable group is isogenous to a subgroup of a product of groups definable in the reducts. In the relatively CM-trivial case, which contains certain H...

An overview is given of the various expansions of fields and fusions of strongly minimal sets obtained by means of Hrushovski's amalgamation method, as well as a characterization of the groups definable in these structures.

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

We prove that if $(H,G)$ is a small, $nm$-stable compact $G$-group, then $H$ is nilpotent-by-finite, and if additionally $\NM(H) \leq \omega$, 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, $nm$-stab...

The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world...

Zusammenfassung Wir konstruieren einen schlechten Körper der Charakteristik Null. Mit anderen Worten, wir konstruieren einen algebraisch abgeschlossenen Körper mit einem Dimensionsbegriff analog der Zariski-Dimension, zusammen mit einer unendlichen echten multiplikativen Untergruppe der Dimension Eins, so daβ der Körper selbst Dimension Zwei hat. D...

Elliptic curves over a supersimple,eld with exactly one extension of degree 2 have s-generic rational points.

Logicum Lugdunensis - Volume 7 Issue 4 - Tuna Altinel, Thomas Blossier, Eric Jaligot, Amador Martín Pizarro, Abderezak Ould Houcine, Frank Wagner

The second of a two volume set showcasing current research in model theory and its connections with number theory, algebraic geometry, real analytic geometry and differential algebra. Each volume contains a series of expository essays and research papers around the subject matter of a Newton Institute Semester on Model Theory and Applications to Al...

We prove a theorem comparing a well-behaved dimen- sion notion to a second, more rudimentary dimension. Specialising to a non-standard counting measure, this generalizes a theorem of Larsen and Pink on an asymptotic upper bound for the inter- section of a variety with a general finite subgroup of an algebraic group. As a second application we apply...

A small profinite ring has an open nil ideal of finite nil exponent. We also discuss various conjectures about small profinite rings, and give examples illustrating them.

A unipotent linear group over a skew field of characteristic 2 is nilpotent.

We define a notion of genericity for arbitrary subgroups of groups interpretable in a simple theory, and show that a type generic for such a group is generic for the minimal hyperdefinable supergroup (the definable hull). In particular, at least one generic type of the definable hull is finitely satisfiable in the original subgroup. If the subgroup...

1. We show that if p is a real type which is internal in a set Σ of partial types in a simple theory, then there is a type p′ interbounded with p, which is finitely generated over Σ, and possesses a fundamental system of solutions relative to Σ.2. If p is a possibly hyperimaginary Lascar strong type, almost Σ-internal, but almost orthogonal to Σ ω...

A type analysable in one-based types in a simple theory is itself one-based.

There is a model-completion T-n of the theory of a (reflexive) n-coloured graph <X, R-1,..., R-n> such that R-n is total, and R-i omicron R-j subset of or equal to Ri+j for all i, j. For n > 2, the theory T-n is not simple, and does not have the strict order property. The theories T-n combine to yield a non-simple theory T-infinity without the stri...

There is a model-completion Tn of the theory of a (reflexive) n-coloured graph $\langle X, R_{1}, ..., R_{n}\rangle$ such that Rn is total, and $R_{i} \circ R_{j} \subseteq R_{i+j}$ for all i, j. For n > 2, the theory Tn is not simple, and does not have the strict order property. The theories Tn combine to yield a non-simple theory T∞ without the s...

If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p.

If there are infinitely many p-Mersenne prime numbers, there is no bad field of positive characteristic p.

A totally ordered group G (possibly with extra structure) is called coset-minimal if every definable subset of G is a finite union of cosets of definable subgroups intersected with intervals with endpoints in G∪{±∞}. Continuing work in Belegradek et al. (J. Symbolic Logic 65(3) (2000) 1115) and Point and Wagner (Ann. Pure Appl. Logic 105(1–3) (2000...

A small profinite m-stable group has an open abelian subgroup of finite M-rank and finite exponent.

We show that if p is a real type which is almost internal in a formula φ in a simple theory, then there is a type p′ interalgebraic with a finite tuple of realizations of p, which is generated over φ. Moreover, the group of elementary permutations of p′ over all realizations of φ is type-definable.

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as pro...

We study local strengthenings of the simplicity condition. In particular, we define and study a local Lascar rank, as well as short, low, supershort and superlow theories. An example of a low, non supershort theory is given.

An $\omega$-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl($\emptyset$)-definable subgroup. Every finitely based regular type in a CM-trivial $\omega$-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple $\omega$-...

If K is a field of finite Morley rank, then for any parameter set A ⊆ Keq the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl(∅).

We study hyperdefinable groups, the most general kind of groups interpretable in a simple theory. After developing their basic theory, we prove the appropriate versions of Hrushovski's group quotient theorem and the Weil–Hrushovski group chunk theorem. We also study locally modular hyperdefinable groups and prove that they are bounded-by-Abelian-by...

This chapter presents a tool for the model-theoretic analysis of arbitrary structures through the group configuration and the binding group. Stability theory or classification theory is a tool for the classification of the models of a complete first-order theory. The stable group forms an interesting class of infinite groups which is amenable to gr...

A minimal field of non-zero characteristic is algebraically closed.

A totally ordered group G is essentially periodic if for every definable non-trivial convex subgroup H of G every definable subset of G is equal to a finite union of cosets of subgroups of G on some interval containing an end segment of H; it is coset-minimal if all definable subsets are equal to a finite union of cosets, intersected with intervals...

A structure (M, $<$,...) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a t...

It is proved that any supersimple eld has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and elds whose theory is simple. 1 Introduction Simple theories were introduced by Shelah in [12]. In [3], Kim, continuing Shelah's work, showed...

this paper assumes some acquaintance with model theory and its language. In section 2 we give definitions and background results on simple theories and hyperimaginaries, as well as additional machinery required. The proof of the main result (elimination of hyperimaginaries for supersimple theories) is given in section 4. The proof is quite technica...

An !-categorical supersimple group is nite-b y-abelian-by-nite, and has nite SU-rank. Every denable subgroup is commensurable with an acl(;)- denable subgroup. Every nitely based regular type in a CM-trivial !-categorical simple theory is non-orthogonal to a type ofSU-rank 1. In particular, a supersimple !-categorical CM-trivial theory has nite SU-...

If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups G/H are the images of the Sylow-2-subgroups of G. Zusammenfassung. Sei G eine omega-stabile Gruppe und H ein definierbarer Normalteiler von G. Dann sind die Sylow-2-Untergruppen von G/H Bilder der Sylow-2-Untergruppen von G.

If G is an omega-stable group with a normal definable subgroup H, then the Sylow-2-subgroups of G|H are the images of the Sylow-2-subgroups of G.

The notion of a large set in an arbitrary group is introduced in analogy to the generic sets in an algebraic or stable group. The question is studied which properties “satisfied largely” by a group hold on the entire group. ZUSAMMENFASSUNG. Wir definieren den Begriff einer groβen Teilmenge einer be-liebigen Gruppe in Analogie zu den generischen Men...

The Sylow-2-subgroups of a periodic group with minimal condition on centralizers are locally finite and conjugate. The same holds for the Sylow-p-subgroups for any prime p, provided the subgroups generated by any two p-elements of the group are finite. In the non-periodic context, the bounded left Engel elements of a group with minimal condition on...

We define a generalized version of CM-triviality, and show that in the presence of enough regular types, or solubility, a stable CM-trivial group is nilpotent-by-finite. A torsion-free small CM-trivial stable group is abelian and connected. The first result makes use of a generalized version of the analysis of bad groups.

It is proved that any supersimple field has trivial Brauer group, and more generally that any supersimple division ring is commutative. As prerequisites we prove several results about generic types in groups and fields whose theory is simple.

. We construct an example of a quasi-o-minimal group without the Exchange Property. A structure hM; !; : : : i is called quasi-o-minimal [1] if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0definable subsets and intervals. A complete theory T is said to have the Exchange Property if, f...

An infinite field with only countably many pure types is algebraically closed.

. An !-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl(;)- definable subgroup. It is well-known that an !-categorical superstable group is abelian-by-finite [1] and has finite rank (in fact by [3] every !-categorical superstable theory is even one-based of f...

We prove for a wide class of structures that if [script F] is a family of substructures which are pairwise uniformly commensurable, then there is a commensurable substructure invariant under automorphisms stabilizing [script F] setwise.

If $G$ is an omega-stable group with a normal definable subgroup $H$, then the Sylow-$2$-subgroups of $G/H$ are the images of the Sylow-$2$-subgroups of $G$.

A supersimple theory eliminates hyperimaginaries. In particular, in a supersimple theory Lascar strong type is the same as strong type, and every strong type has a canonical base in C eq . It follows that the Amalgamation Theorem (Independence Theorem) holds for types over algebraically closed sets.