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Russian Academy of Sciences

St.Petersburg Department of Steklov Mathematical Institute

Euler International Mathematical Institute

Logic, Algebra and Geometry June 1-7, 2004

Program

Abstracts

Russian Academy of Sciences

St.Petersburg Department of Steklov Mathematical Institute

Euler International Mathematical Institute

Logic, Algebra and Geometry June 1-7, 2004

Program

Abstracts

Program Committee Marat Arslanov Andreas Blass Andrey Bovykin Yuri Ershov Su Gao Edward Griffor Alexander Kechris Juris Steprans Anatoly Vershik

— co-chair Yi Zhang

— co-chair Boris Zilber Organizing Committee Andrey Bovykin Elena Novikova Anatoly Vershik Yi Zhang

Logic, Algebra and Geometry. Short abstracts of the international meeting held on June 1–7, 2004.– St. Petersburg, 2004.– 33pp.

An international conference “Logic, Algebra and Geometry” is organized by the St.Petersburg Department of Steklov Institute of Mathematics. Included are 22 short abstracts of the talks accepted for presentation at the meeting; More information about the conference can be

found on the meeting’s site http://logic.pdmi.ras.ru/june2004.

The conference “Logic, Algebra and Geometry” is supported by the Russian Foundation for Fundamental Research (grant no. 04-01-10119g).

We express our sincere gratitude towards the State Hermitage Museum for their generous help in organizing the cultural program of our conference.

\begin{matrix}c \\ ◯ \end{matrix}POMI RAN, 2004

Contents

Conference Program . . . . . . . . . . . . . . . . . . . . . . . . 1 Petr Andreev . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 John T. Baldwin . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Andreas Blass . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Jörg Brendle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Peter J. Cameron . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Mauro Di Nasso . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Yuri Ershov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Su Gao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Misha Gavrilovich . . . . . . . . . . . . . . . . . . . . . . . . . 14 Evgeniy Gordon . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Greg Hjorth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 Tapani Hyttinen . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Vladimir Kanovei, Michael Reeken, Saharon Shelah . . . . . . . 16 Jaroslav Nesetril . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Nicolai Reshetikhin . . . . . . . . . . . . . . . . . . . . . . . . . 18 Juris Steprans . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Carlo Toffalori . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Monica VanDieren . . . . . . . . . . . . . . . . . . . . . . . . . 19 Boban Velickovic . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Anatoly Vershik . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Roman Wencel . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Sergey Zyubin . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Conference Program June 1, Tuesday

11:00–14:00 Registration, coffee
14:00–14:10 Welcoming address by Anatoly Vershik

14:10–15:00 Yuri Ershov Model theory of multi valued fields

15:10–16:00 Greg Hjorth An anti-Ramsey theorem

16:00–16:30 Coffee break

16:30–17:20 Andreas Blass Powers of the Group of Integers

June 2, Wednesday

10:00 - 10:50 Su Gao Classification problems of Polish metric spaces 1

11:10 - 12:00 Boban Velickovic Set theoretic aspects of the Control Measure Problem

12:00 - 14:00 LUNCH

14:10 - 15:00 Yuri Ershov Nice extensions and global class field theory

15:10 - 16:00 Juris Steprans Geometric cardinal invariants, maximal functions and a measure theoretic pigeonhole principle

16:00 - 16:30 Coffee

16:30 - 17:20 Jörg Brendle Van Douwen’s diagram and its relatives

June 3, Thursday

10:00 - 10:50 Su Gao Classification problems of Polish metric spaces 2

11:10 - 12:00 Anatoly Vershik Universality, randomness, complexity 1

12:00 - 14:00 LUNCH

14:10 - 15:00 Jaroslav Nesetril Ramsey classes and homogeneous structures

15:10 - 16:00 Vladimir Kanovei Fully saturated extensions of standard universe

16:00 - 16:30 Coffee

16:30 - 17:20 Mauro Di Nasso A purely algebraic characterization of the hyperreals of nonstandard analysis

June 4, Friday

10:00 - 10:50 Su Gao Classification problems of Polish metric spaces 3

11:10 - 12:00 Anatoly Vershik Universality, randomness, complexity 2

12:00 - 13:30 LUNCH
13:30  A bus takes us to Steklov Institute

14:10 - 15:00 Peter Cameron Isometries of Urysohn space

15:10 - 16:00 Nicolai Reshetikhin The random geometry of the 6-vertex model

18:00  A banquette in the Marble Hall of Steklov Institute

Saturday, June 5

The guided tour of the Hermitage Museum starts at 10:30. We meet at the group entrance of Hermitage, near the Neva river. (Do not confuse: there is another entrance to Hermitage, from the Palace Square, we DO NOT meet there.) We arranged two simultaneous English-language tours and one Russian-language tour.

June 6, Sunday

10:00 - 10:25 Petr Andreev An approximation of the hyperfinite world

10:30 - 10:55 Roman Wencel A generalization of o-minimality for expansions of Boolean algebras

11:15 - 11:40 Carlo Toffalori Minimalities and modules over group rings

11:45 - 12:10 Sergey Zyubin

Conjugately dense subgroups of free products of groups

12:10 - 14:00 LUNCH

14:10 - 15:00 Evgeniy Gordon A model-theoretic approach to approximation of topological algebraic systems by finite ones

15:10 - 16:00 Misha Gavrilovich ℵ_{1}-categoricity of universal covering spaces of abelian varieties and Galois groups of their torsion points  16:00 - 16:30 Coffee

16:30 - 17:20 Tapani Hyttinen Local modularity in homogeneous structures

June 7, Monday

10:00 - 10:50 Monica VanDieren Shelah’s Categoricity Conjecture for tame abstract elementary classes

11:10 - 12:00 John Baldwin 0–1 laws for probability n^{−α}

Abstracts

An axiomatization of the hyperfinite world Petr Andreev

(joint work with E.I. Gordon, Eastern Illinois University)

                      petr@sherwood.ru

After the invention of nonstandard analysis there can be no doubt that “finite” and “continuous” are not opposites. Moreover, the universe of sets can even be thought of as genuinely hyperfinite which means that hyperfinite structures could be taken as objects of the primary interest while continuous ones serve as their approximations. The theory of hyperfinite sets (THS) is a theory in the ∈–language obtained by introducing, for the universe of hereditarily finite sets, some additional principles which, in particular, imply the existence of relevant structures behaving as usual infinite ones. Therefore it is possible in THS to use continuous mathematics for investigating hyperfinite structures. We will present the axioms of THS and briefly discuss ways of doing mathematics in it.

0-1 laws for probability n^{−α} John T. Bladwin

                   jbaldwin@math.uic.edu

A seminal 1987 paper by Shelah and Spencer proved that if α is an irrational 0 < α < 1 and graphs on n vertices are given edge probability n^{−α} then for any first order sentence ϕ in the language of graphs lim_{n→∞} P_{n}(ϕ) is 0 or 1. We will begin with a short account of 0-1 laws, separating a model theoretic and combinatorial component of the argument for proving them. We will discuss the method of determined theories, introduced by Baldwin and Shelah, which fills a gap on the model theoretic side of the Shelah-Spencer argument and gets the additional property that the almost sure theory is stable (behaves well from a model theorists viewpoint). Then we will discuss our proof that the 0-1 law remains true if we consider graphs not just on n points but on the structure n with successor. (It fails on (n, <).) We will focus on a new combinatorial argument to compute the basic estimates which also yields a new proof of the original Shelah-Spencer result. Finally, we will discuss the logical differences between these almost sure theories and apparently analogous theories arising by the construction of expansions of algebraically closed fields.

Powers of the Group of Integers Andreas Blass

                      ablass@umich.edu

This will be a survey talk on some application of set theory in the theory of abelian groups. A central role will be played by powers of the additive group of integers, i.e., groups of sequences (possibly of transfinite length) of integers with the operation of componentwise addition. Some important subgroups of these powers will also be discussed. I plan to begin by reviewing classical results (from the 1950’s and even earlier) and then to present more recent results (including some that are not yet published). I hope to emphasize not only the applicability of set-theoretic methods in this part of algebra but also the use of such applications in motivating new results in set theory.

Van Douwen’s diagram and its relatives Jörg Brendle

              brendle@kurt.scitec.kobe-u.ac.jp

Consider the structure P(ω)/fin, the power set of the natural numbers ω modulo the finite sets. For A, B ⊆ ω, say A is eventually contained in B (and write A ⊆^{∗} B) if A \ B is finite. Denoting equivalence classes of sets A ⊆ ω by [A], P(ω)/fin is a Boolean algebra with ordering relation [A] ≤ [B] if A ⊆^{∗} B. A number of cardinal invariants of the continuum which describe the combinatorial structure of (P(ω)/fin, ≤) like, for example, the almost-disjointness number 𝔞 or the distributivity number 𝔥, have been investigated and their interrelationship is usually displayed in van Douwen’s diagram.

In recent years research has increasingly focused on cardinals which are defined in a similar way for structures other than P(ω)/fin, the analogue of van Douwen’s diagram has been redrawn, and the connection of the new cardinals with the classical ones has been studied. The purpose of my talk is to give a survey on results in this field.

For example, consider Dense(ℚ)/nwd, the dense subsets of the rational numbers ℚ modulo the nowhere dense sets. This is a lattice under the relation [A] ≤ [B] if A ⊆_{nwd} B which means that A\B is nowhere dense. Cardinal invariants of this structure have been investigated by Balcar and Hrušák. They established, for example, that 𝔥_{ℚ} ≤ add(meager) where 𝔥_{ℚ} is the distributivity number of Dense(ℚ)/nwd and, thus, that 𝔥_{ℚ} < 𝔥 is consistent. Answering one of their questions we proved the consistency of 𝔥 < 𝔥_{ℚ} in joint work with M. Hrušák.

On another track, Cichoń, Krawczyk, Majcher-Iwanow and Weglorz have looked at the collection (ω) of partitions of the natural numbers ordered by ≤^{∗} where A ≤^{∗} B if by gluing finitely many blocks of A we can get a partition which is coarser than B. The resulting cardinal invariants, called dual cardinals, are fairly well-known by now: for example, 𝔱_{d} = ℵ_{1} (Carlson), 𝔞_{d} = C (Krawczyk), etc. However, while ((ω), ≤^{∗}) is a lattice it is not a Boolean algebra and thus not isomorphic to the converse structure ((ω), ≥^{∗}) the investigation of which becomes an interesting topic by itself. Building on earlier work of B. Majcher-Iwanow, we proved in joint work with S. Zhang that most cardinals of the latter structure agree with the corresponding cardinals of P(ω)/fin and that 𝔞_{c} = 𝔞_{𝔰} where 𝔞_{c} is the smallest size of a maximal family A of nontrivial partitions such that the common refinement of any two members of A is trivial, that is, almost equal (=^{∗}) to the partition into singletons.

Isometries of Urysohn space Peter J. Cameron

                   p.j.cameron@qmul.ac.uk

In a posthumous paper in 1927, P. S. Urysohn constructed the unique complete separable metric space which is universal and homogeneous. I will report on joint work with A. M. Vershik in which we construct various interesting groups of isometries of Urysohn space, including transitive abelian groups and a free group which is dense in the full isometry group. The methods used are similar to those which have been used to investigate automorphisms of the countable random graph or Rado graph. Some background on Urysohn space will be given.

A purely algebraic characterization of the hyperreals of nonstandard analysis Mauro Di Nasso

                    dinasso@dm.unipi.it

I will show that the hyperreal numbers of nonstandard analysis can be characterized in purely algebraic terms as homomorphic images of a suitable class of rings of real-valued functions (this is joint work with V. Benci).

Time permitting, I will also present a characterization of those subspaces of the Stone-Cech compactification βN that are sets of hypernatural numbers in the sense of nonstandard analysis. The existence of such subspaces is equivalent to the existence of a special class of ultrafilters which are independent of ZFC.

Model theory of multi valued fields Yuri Ershov

                     ershov@math.nsc.ru

Multi valued field is a pair (F, R), F is a field, R ≤ F is a Pruefer ring (with F as the quotient field). R is viewed as a “code” of the family W (R) = {R_{m}|m is a maximal ideal of R} of the valuation rings of F (and R is the holomorphy ring of the family W (R): R = ⋃{R_{m}|R_{m} ∈ W (R)}). An important example of a multi valued field is (Q, Z), Q is the field of rational numbers and Z is the ring of integers.

For valued fields (F, R) (then R is a valuation ring of F) the notion of henselian valuation ring (a henselian valued field) is very important. Does a right generalization of this notion exist for multi valued fields?

Model theory suggests as a satisfactory answer to distinguish the multi valued fields (F, R) satisfying following two properties:

LG_{A} (Arithmetical local-global principle) An absolutely irreducible affine variety V defined over F has a nonsingular R-point if V has the nonsingular H(R_{m})-points for every R_{m} in W (R).

M (Maximality) There if no proper separable algebraic extention F_{0} of F such that F_{0} is embeddable in every field H_{R_{m}}(F) for R_{m} in W (R).

Here: H(R_{m}) is a henselization of R_{m} and H_{R_{m}}(F) is the quotient field of H(R_{m}).

Many important algebraic and model theoretical results for such multi valued field were found (see [1]).

References

[1] Yu. L. Ershov, Multi valued fields, Novosibirsk, 2000, Scientific Books (English translation of the book appeared in 2001, Kluwer Academic/Plenum Publishers).

[2] Yu. L. Ershov, On wonderful extensions of the field of rational numbers, Doklady Mathematics, 2000.vol 62, #1, 12-13.

Nice extensions and global class field theory Yuri Ershov

                     ershov@math.nsc.ru

Nice extension of an algebraic number field F is a subfield N of the adeles ring A(F) of F, containing F and satisfying the local global principle LG.

For a nice extension N of F let N̂ (F̂) be the algebraic closer of N (F). Then G(F)-module N̂ ∖ {0}/F̂ ∖ {0} can be used for a definition of global class field theory of F instead the ideles class group of F.

Using an existence of a decidable nice extensions it is possible to present an effective (algorithmic) version of global class field theory.

References

[1] Yu.L.Ershov, Nice extensions and global class field theory. Doklady Mathematics, 2003, vol 67, #1, 21-23.

Classification Problems of Polish Metric Spaces Su Gao

                        sgao@unt.edu

I will give an introduction to the descriptive set theoric framework in which the classification problems can be investigated and their complexity measured. Then I will turn to the isometric classification problem of Polish metric spaces. For various important classes of Polish metric spaces I will give complete characterizations of the complexity of their isometric classification problems. Among these classes I will elaborate more on the class of all locally compact separable metric spaces. I will also discuss other notions of equivalence and open problems in the field.

ℵ_{1}-categoricity of universal covering spaces of abelian varieties

and Galois groups of their torsion points Misha Gavrilovich

                  gavrilov@maths.ox.ac.uk

We consider a structure consisting of the universal covering space U together with the covering map to a commutative algebraic group A, and its non-first order theory T in an appropriate infinitary language. We relate the stability and categoricity of the structure to arithmetic and geometric properties of the algebraic group A. We show that omega-stability of the structure is equivalent to a question about the geometry and fundamental group of A and that ℵ_{1}-categoricity is equivalent to an arithmetical question about Galois groups of torsion and division points.

For elliptic curves the positive answers to both questions are known. Kummer theory provides answers to the arithmetical questions; the geometrical question is known as the question about the existence of Shafarevich morphisms, which is also known for elliptic curves.

A model-theoretic approach to approximation of topological

algebraic systems by finite ones Evgeniy Gordon

                   cfyig@ux1.cts.eiu.edu

We discuss a notion of approximation of a topological algebraic system A by finite algebraic systems of some class K. For the case of a discrete algebraic system the definition of approximability is equivalent to the well-known definition of a local embedding of an algebraic system A in a class K of algebraic systems. The approximations of discrete groups by finite ones and their applications were studied by A.M. Vershik and the author.

It is known that all locally compact abelian groups and a huge class of nilpotent Lie groups are approximable by finite groups. On the other hand a compact Lie group G is approximable by finite groups iff for any positive ε there exists a finite subgroup H ⊆ G that is an ε-grid in G. This condition does not hold for the most important compact Lie groups, e.g. for SO(3). The non-approximability of such groups makes it of interest to investigate more general classes of finite algebraic systems that approximate locally compact groups non-approximable by finite groups. We show that the approximability of a locally compact group G by finite semigroups implies the approximability of G by finite groups. We prove also that G is approximable by finite quasigroups iff it is unimodular. This is the first known approximation of unimodularity in terms of algebra and topology.

Using the result that all locally compact groups approximable by finite semigroups are unimodular we prove that a locally compact field cannot be approximated by finite associative rings (not necessary commutative). Finite approximations of the field R can be considered as computer systems for reals. Thus, it is impossible to construct a computer arithmetic for reals that is an associative ring.

These results show that a theorem φ of the theory of a topological algebraic system A may not hold in approximations of A. Nevertheless we introduce the bounded formulas of the signature of A and their approximations. We prove that a positive bounded formula φ holds in A if all precise enough approximations of φ hold in all precise enough approximations of A.

This is a joint work with L.Yu. Glebsky (UASLP, Mexico) and C.W. Henson (UIUC, USA).

An anti-Ramsey theorem Greg Hjorth

                     greg@math.ucla.edu

We give a kind of general negative Ramsey result for ultrahomogeneous structures and go on to describe a generalization for isometry groups. This second result may have certain connections with distortion problems.

Local modularity in homogeneous structures Tapani Hyttinen

                tapani.hyttinen@helsinki.fi

We study ω-stable large homogeneous structures M of bounded U-rank. Our main interest will be in consequences of the property that every strongly minimal (also called quasi-minimal) set in M is locally modular.

Fully saturated extensions of standard universe Vladimir Kanovei ^{*}, Michael Reeken, Saharon Shelah^{*}

      kanovei@mccme.ru, reeken@math.uni-wuppertal.de,
                   shelah@math.huji.ac.il

It seems that it has been taken for granted that there is no distinguished, definable nonstandard model of the reals. (This means a countably saturated elementary extension of the reals.) Of course the axiom of constructibility implies the existence of such an extension (take the first one in the sense of the canonical well-ordering of L), but the existence provably in ZFC was established quite recently in [3]. Note that without Choice the existence of any elementary extension of the reals, containing an infinitely large integer, is not provable.

The problem of the existence of a definable fully saturated (that is κ-saturated for any cardinal κ) elementary extension of the whole set universe of ZFC is even more challenging. In this case even the permission to use arbitrary sets as parameters of definition does not make things easier because any intended construction of an Ord-long chain of consecutive ultrapowers with increasing amount of saturation needs an Ord-long sequence of choices of suitable ultrafilters, which is hardly possible provably in ZFC. Nevertheless we prove Theorem 1 There exists, provably in ZFC, a definable fully saturated elementary extension of the whole set universe of ZFC.

*Supported by RFFI and DFG

*Supported by The Israel Science Foundation.

Such an extension can be viewed as an interpretation of bounded set theory BST in ZFC. The theory BST is an improved, foundations-friendly version of Nelson’s internal set theory IST, in which every set belongs to a standard set.

Corollary 2 There exists, provably in ZFC, an interpretation of BST in ZFC such that the standard core of the interpretation coincides with the ZFC universe.

The existence of such an interpretation of BST was earlier known only on the base of Global Choice extensions of ZFC [2]. Note that IST itself does not admit such an interpretation, and essentially by the same reason discovered in [2] Theorem 1 cannot be strengthened to provide the saturation with respect to all V-size families (V being the ZFC universe) and accordingly an interpretation of IST in ZFC satisfying the condition mentioned in Corollary 2 since any minimal transitive model M of ZFC in the role of V provides a counterexample [2], 2.8.

The proof of Theorem 1 is a combination of several arguments, most notably the iterated ultrapower construction (with finite support) in the sense of [1], 6.5. and the construction of Ord-long chains of consecutive ultrapower-like extensions.

References

[1] C. C. Chang and H. J. Keisler, Model Theory, 3rd ed., North Holland, 1992.

[2] V. Kanovei and M. Reeken, Internal approach to external sets and universes, part 1, Studia Logica, 1995, 55, 2, pp. 229–257.

[3] V. Kanovei and S. Shelah, A definable nonstandard model of the reals, JSL 2004, 69, no. 1, pp. 159–164.

Ramsey classes and homogeneous structures Jaroslav Nesetril

                  nesetril@kam.mff.cuni.cz

We shall survey recent result related to the classification program for Ramsey classes of structures.

The random geometry of the 6-vertex model Nicolai Reshetikhin

                 reshetik@math.berkeley.edu

No abstract available.

Geometric Cardinal Invariants, Maximal Functions and a

Measure Theoretic Pigeonhole Principle Juris Steprans

                 steprans@mathstat.yorku.ca

It is shown to be consistent with set theory that every set of reals of size ℵ_{1} is null yet there are ℵ_{1} planes in Euclidean 3-space whose union is not null. Similar results will be obtained for circles and other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain maximal operators and a measure theoretic pigeonhole principle. This talk will emphasize the pigeonhole principle used.

Minimalities and Modules over Group Rings Carlo Toffalori

                 carlo.toffalori@unicam.it

(Joint work with Sonia L’Innocente and Vera Puninskaya)

We are interested in several notions of model theoretic minimalities (strong, pseudo-strong, weak, quasi minimality) within modules over certain classes of rings, including group rings D[G] where D is a Dedekind domain and G is a finite group [LPT] as well as some related classes of rings (such as pullbacks of Dedekind domains and Dedekind-like rings) [LPT1]. In these frameworks we compare the various minimality notions and we classify the modules satisfying them.

References

[LPT] S. L’Innocente, V. Puninskaya, C. Toffalori, Strongly minimal modules over group rings, Preprint 2003

[LPT1] S. L’Innocente, V. Puninskaya, C. Toffalori, Minimalities and modules over Dedekind-like rings, Preprint 2003

Shelah’s Categoricity Conjecture for Tame Abstract Elementary Classes Monica VanDieren

                       mvd@umich.edu

We prove the generalization of Morley’s Theorem for a broad range of non-elementary classes. This captures some of the most interesting cases of Shelah’s 30 year old categoricity conjecture for L_{ω_{1},ω} which may have applications in number theory similar to Zilber’s work with pseudo-exponentiation. A new component of our work is an upward categoricity transfer theorem. This is joint work with Rami Grossberg.

Set theoretic aspects of the Control Measure Problem Boban Velickovic

                  boban@logique.jussieu.fr

We survey the history and present some recent results on the Control Measure Problem and its connections with Set Theory. The origin of the problem is due to Von Neumann in the 1930s who asked if certain algebraic properties of measure algebras are sufficient to characterize them. In its final form the problem was asked by Maharam in the 1940s. She defined the concept of a continuous exhaustive submeasure and asked if any such measure is equivalent to a measure. This problem is significant to Set Theory since it is related to a fundamental problem of Prikry who asked if Cohen and random forcing are the only nontrivial basic forcing notions with the countable chain condition. While these problems are still open in full generality we show that, assuming the well known P-ideal dichotomy, the Von Neumann problem is equivalent to Maharam’s problem. Moreover, it is a consequence of our results and some previous work of Shelah that in the context of Souslin forcing notions the full Prikry conjecture reduces to the Control Measure Problem, as well.

Universality, Randomness,Complexity Anatoly Vershik

                    vershik@pdmi.ras.ru

Several examples of the universal objects (metric spaces, graphs etc) and its group of automotrphisms will be considered. The notion of randomness based on the theory of random matrices (tensors), the relevant notion of complexity of such objects must be based on Kolmogorov’s complexity. The main theorem claimed that in some categories randomness and universality coincide. Some of resent results concerned to group of automorphisms were obtained together with P.Cameron. There are a lot of open questions and one of the main problem to get a new link between logic, combinatorics algebra and analysis.

A genaralization of o-minimality for expansions of Boolean

algebras Roman Wencel

                  wencel@maths.leeds.ac.uk

Assume that (M, ≤, …) is a partially ordered L-structure. We say that it is o-minimal iff every L-definable subset of M is a finite Boolean combination of sets defined by formulas of the form x ≤ a and x ≥ b, where a, b ∈ M. This definition was introduced by Toffalori in 1998 and investigated in case of structures with partial orderings derived from Boolean algebras (Toffalori, Leonesi, Newelski and myself). The assumption that an expansion of a Boolean algebra is o-minimal implies that the underlying Boolean algebra has finitely many atoms. Here we propose the following definition, which coincides with the Toffalori’s notion of o-minimality in case of Boolean algebras with finitely many atoms.

Definition (B, …), an expansion of a Boolean algebra B to the language L is o-minimal iff every L-definable subset of B is definable in the language of pure Boolean algebras. We say that the theory T = T h(B, …) is o-minimal iff every model of T is o-minimal.

We investigate this notion in case of expansions of Boolean algebras admitting weak elimination of imaginaries. In this context we discuss the following problems

• weak orthogonality of certain types,

• existence and uniqueness of prime models over sets,

• Vaught conjecture,

• o-minimality for a structure and for its theory.

Conjugately dense subgroups of free products of groups Sergey Zyubin^{*}

                      szubin@yandex.ru

A subgroup of any group is called conjugately dense if it has nonempty intersection with each class of conjugate elements of the group. In the group GL_{n}(K) over algebraically closed field, examples of proper conjugately dense subgroups are triangular subgroup, all it’s conjugate, and overgroups of these subgroups. In general, Borel subgroups and also parabolic subgroups are conjugately dense in any connected affine algebraic group. P. Neumann set the following problem in “Kourovka Notebook”. Describe all irreducible conjugately dense subgroups H of the group GL_{n}(K) over arbitrary field K. In the same place P. Neumann conjectured that H = GL_{n}(K) with the exception of a case when n = charK = 2, K is quadratically closed field and H is conjugated to monomial subgroup [1, Problem 6.38a]. Finally he asked the following question. As far as it true that every conjugately dense subgroup of group of Lie type is parabolic? [1, Problem 6.38b] The author and V.M. Levchuk confirmed Neumann’s conjecture for the group GL_{2}(K) over a locally finite field K [2] and for the locally finite Chevalley groups of Lie rank 1 [3]. Further, the author proved that the conjecture is true for the group GL_{3}(K) over a locally finite field K [4], see also [5]. In this paper we prove the next theorem.

Theorem 1. Suppose the group G is nontrivial free product G = G_{1} ∗ G_{2}_{A} with amalgamated subgroup A, where index of a sub-

group A in group G_{1} or G_{2} greater than 2. Let B be largest normal subgroup of G that lying in A. If quotient group G/B contain α conjugacy classes of elements of G and α ⩽ max{|G_{1} : A|, |G_{2} : A|, ℵ_{0}}; then G have max{|G_{1} : A|, |G_{2} : A|, 2^{α}} distinct conjugately dense subgroups.

Then we use Ihara’s decomposition [6] of the group SL_{2}(K) over field K with discrete valuation into free product with amalgamation and prove the following.

*This research was supported by the Russian Foundation for Basic Research (Grant 03–01–00905).

Theorem 2. Let K be a field with nontrivial, discrete valuation and k be it’s residue class field. Suppose |K| = max{|k|, ℵ_{0}} then the group SL_{2}(K) has 2^{|K|} mutually nonconjugated conjugately dense subgroups.

The group SL_{2}(K) has only two conjugacy classes of parabolic subgroups: the group itself and Borel subgroups. It follows that Neumann’s conjecture is false for the group SL_{2}(K) over a field K that satisfies conditions of theorem 2. An example of such field is field of rational numbers with any p-adic valuation. Also we have another consequences of Theorem 1.

Corollary 1. Modular group P SL_{2}(ℤ) has continuum mutually nonconjugated conjugately dense subgroups.

Corollary 2. Free group F (X) with rank > 1 has 2^{max{|X|, ℵ_{0}}} mutually nonconjugated conjugately dense subgroups.

References

[1] “The Kourovka Notebook”, Unsolved Problems in Group Theory, 15th ed. (2002), Inst. Mat. (Novosibirsk).

[2] Zyubin S.A. and Levchuk V.M., “Conjugately dense subgroups of group GL_{2}(K) over locally finite field K” (in Russian), Sbornik trudov konf. ”Symm. and Diff. Equations”, Krasnoyarsk (2000), 110-112.

[3] Zyubin S. A. and Levchuk V. M., “Conjugately dense subgroups of locally finite Chevalley groups of Lie rank 1”, Siberian Math. J., 44 (2003), N4, 742–748.

[4] Zyubin S.A., Conjugately dense subgroups of 3-dimensional linear groups over locally finite field, Int. J. of Algebra and Computation: Proc. of Conf. on Group Theory, Gaeta, Italy 2003, (to be published).

[5] Zyubin S.A., Conjugatively dense subgroups of linear groups, Int. Conf. on Group Theory, Gaeta, Italy, 2003, p. 112.

[6] Ihara Y., On diskrete subgroups of the two by two projective linear groupy over p-adic fields, J. Math. Soc. Japan, 18 (1966), 219-235.