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Groups, Equations, Non-commutative Algebraic Geometry / Groupes, équations, géométrie algébrique non-commutative
Org: Olga Kharlampovich and/et Alexei G. Myasnikov (McGill)

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DMITRI BORMOTOV, The Graduate School of CUNY, Department of Computer Science, 365 Fifth Avenue, NY, NY 10016
Effective algorithm for solving one-variable equations in a free group

We show that the task of finding all solutions of an one-variable equation in a free group can be reduced to finding so called pseudo-solutions of cubic equations, and in turn, pseudo-solutions of quadratic equations. This results in a polynomial time algorithm with complexity l⁴ * d, where l is the length of the equation and d is its degree. The algorithm is efficiently implemented in C++.

This talk is based on a joint paper with Alexei G. Myasnikov.

IAN CHISWELL, School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, England
Locally invariant orders on groups

The idea of a locally invariant order on a group was introduced by D. Promislow. The class of groups having a locally invariant order is intermediate between the class of right orderable groups and the class of unique product groups. It is unknown if any of these classes coincide. Recent work of T. Delzant and S. Hair shows that certain groups are unique product groups by, in effect, showing they have a locally invariant order. This includes, for example, torsion-free Fuchsian groups. The class of groups having a locally invariant order has some interesting properties, for example, it is closed under free products and restricted direct products. It is unknown whether or not a tree-free group is right orderable, but it has a locally invariant order. The class of tree-free groups includes all torsion-free subgroups of SL₂ (^*Z), where ^*Z is an ultrapower of Z.

EVELINA DANIYAROVA, Omsk Branch of Institute of Mathematics, 13 Pevtsova St., 644099, Omsk, Russian Federation
Algebraic Geometry over Metabelian Lie Algebras

For the universal closure of free metabelian Lie algebra F_r of finite rank r ³ 2 over a finite field k we find a convenient set of axioms in two distinct languages: with constants and without them. We give a description of:

(1) the structure of finitely generated algebras from universal closure of F_r in both mentioned languages;
(2) the structure of irreducible algebraic sets over F_r and the respective coordinate algebras.

We also prove that the universal theory of free metabelian Lie algebra over a finite field is decidable in both languages.

VOLKER DIEKERT, Universität Stuttgart, FMI, Universitätsstr. 38, D-70569 Stuttgart, Germany
Word equations over graph products

At the FSTTCS conference 2003 in Mumbay we showed that for a restricted class of monoids the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of some of the factor monoids and the existential theories of the remaining factors. Both results also include suitable recognizable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

Our results are a continuation of works by Makanin, Schulz, Matiyasevich, Plandowski, Gutierrez and work done in our reseach group at Stuttgart.

This is a joint work with Markus Lohrey.

ANDREW DUNCAN, Newcastle University
Centraliser dimension in partially commutative groups

The motivation for the work reported in this talk, is the study of the algebraic geometry of partially commutative groups. A partially commutative (pc) group is one with a finite presentation áX|Rñ, where each element of R has the form [x,y] for x and y in R.

Chiswell and Remeslennikov have proved that the irreducible algebraic sets of one-variable systems of equations over a finitely generated free group F are either equal to F; or contain a unique element; or have the form uCv where u and v are elements of F and C is the centraliser of an element of F. This indicates that the structure of the centraliser lattice is crucial to the understanding of irreducible sets, at least in one-variable.

As a first step towards the classification of such sets in pc groups the centraliser lattice of these groups has been studied (in collaboration with I. Kazatchkov and V. Remeslennikov). We show that there is a global bound on the length of chains of centralisers (so pc groups have finite centraliser dimension). Moreover we show precisely how this bound depends on the commutation graph of the pc group in question.

BEN FINE, Fairfield University, Fairfield, CT 06430
Elementary Free Groups and Tame Automorphisms

As an outgrowth of the solution of the Tarski problem (Kharlampovih-Myasnikov and independently Sela) it is possible to characterize groups which share the same elementary theory as free groups. Such groups are called elementary free groups. The class of elementary free groups is wider than solely the class of free groups and includes in particular the the orientable surface groups. For a group G a tame automorphsim is an automorphism induced by a free group automorphism. A result of Zieschang and extended by Rosenberger and others shows that every automorphism of a surface group is tame. In this talk we examine the relationship between elementary free groups and the property that every automorphism is tame.

This is joint work with O. Kharlampovich, A. Myasnikov and V. Remesslennikov.

ANTONY GAGLIONE, U.S. Naval Academy
On FP infinite Torsion Groups and a Question of V. H. Dyson

This talk does not settle the issue of the existence of such groups. Assuming a first-order language, L, for group theory, the universal theory of a class of groups is just the set of all universal sentences of L true for every group in the class. Sometime ago Verena Huber Dyson asked whether or not the universal theory of torsion groups coincides with the universal theory of finite groups. Here we show that if these theories do coincide, then there cannot exist a finitely presented (f.p.) infinite group of any finite exponent. We cannot, however, say anything about the converse.

BOB GILMAN, Stevens Institute of Technology
One variable equations over hyperbolic groups

The subject of this talk is recent progress on solving one variable equations over torsion free word hyperbolic groups. In particular we will focus on progress toward an algorithm for writing down complete solutions in closed form. These results are joint work with Alexei Myasnikov and Dima Bormotov.

JOINT PANEL DISCUSSION
Participating Sessions

This is a joint Panel Discussion of the following sessions:

• Groups, Equations, Non-commutative Algebraic Geometry
  
  
• Combinatorial and Geometric Group Theory
  
  
• Interactions between Algebra and Computer Science
  
  

ILYA KAZATCHKOV, Omsk Branch of Institute of Mathematics, 13 Pevtsova St., 644099, Omsk, Russian Federation
A Gathering Process in Artin Braid Groups

In the current talk I shall construct a gathering process by the means of which I obtain new normal forms in braid groups. The new normal forms generalise Artin normal forms and possess an extremely natural geometric description. Then I plan to discuss the implementation of the introduced gathering process and derive some interesting corollaries and, in particular, offer a method of generating a random braid.

OLGA KHARLAMPOVICH, McGill University
Finiteness results in algebraic geometry for a free group

We introduce some basic objects of algebraic geometry for groups similar to the ones from commutative algebra: a system of equations, an algebraic variety, a radical, a coordinate group, a Zariski topology and so on. Suppose we have two irreducible systems of equations T(X)=1 and S(X,Y)=1 over a free group.

Let H be the coordinate group of the first system, K be the coordinate group of the second system. Then it is known that K and H are fully residually free groups. Suppose that H is embedded into K. We say that K does not have a sufficient splitting modulo H if K either does not split as an amalgamated free product (HNN extension) with abelian edge group and H being elliptic or it splits but does not split as a free product and "minimal" solutions of S(X,Y)=1 with respect to the group of automorphisms of K corresponding to these splittings define the algebraic variety with the same coordinate group K. (Example: X={x}, Y={y₁,z₁,...,y_m,z_m}, T(x) is trivial, S(X,Y) = xP_i=1^m[y_i,z_i].) We introduce the notion of solutions of finite type of the system S(X,Y)=1, and will show that one can effectively find a constant N such that for every solution of T(X)=1 there are at most N finite type solutions of S(X,Y)=1.

As one of the applications we will show that the number of finite type solutions of quadratic equations depends only on the number of variables and does not depend on the coefficients. This is interesting to compare with Grigorchuk-Kurchanov, Comerford-Edmunds description of algebraic varieties for quadratic equations in free groups.

These are joint results with A. Myasnikov.

ALEXEI G. MYASNIKOV, McGill University, Dept. of Math. and Stat., 805 Sherbrooke St. W., Montreal, QC, Canada
Approximations and non-standard solutions of equations in group

I will describe some approximation methods for solving equations in groups and their relations with generic points, parametric and non-standard solutions.

This talk is based on joint work with O. Kharlampovich, R. Gilman, and V. Remeslennikov.

VLADIMIR REMESLENNIKOV, Omsk Branch of Institute of Mathematics, 13 Pevtsova St., 644099, Omsk, Russian Federation
Algebraic Geometry over Free Lie Algebras

In this talk I shall classify all bounded algebraic sets over free Lie algebras in three different languages:

(1) in the language of algebraic sets,
(2) using the language of radical ideals, and
(3) the language of coordinate algebras.

DENNIS SERBIN, McGill University, Montreal
Diophantine Problem over fully residually free groups

In this talk we give a solution of the Diophantine Problem over fully residually free groups. In other words we introduce the decision algorithm for verifying consistency of a system of equations over a finitely generated fully residually free group G. This is done by reducing the original problem to the one for free groups using the representation of elements of G by infinite words as well as the notion of cancellation tables on infinite words.

This is a joint result with Olga Kharlampovich and Alexei G. Myasnikov.

DENNIS SPELLMAN, Temple University, Philadelphia, PA
Unions of Varieties and Quasivarieties

We characterize the following four kinds of classes of groups by means of closure properties:

1. Unions of Varieties
2. Direct Unions of Varieties
3. Unions of Quasivarieties
4. Direct Unions of Quasivarieties.

As an application/example we make the observation that closure of the class of Equationally Noetherian groups under all of the particular properties explicitly mentioned in G. Baumslag, A. G. Myasnikov and V. N. Remeslennikov, Algebraic geometry over groups I. Algebraic sets and ideal theory, J. Algebra 219(1999), 16-79, may succinctly be paraphrased by asserting that the class of Equationally Noetherian groups is the direct union of a family of quasivarieties.