


Groups, Equations, Noncommutative Algebraic Geometry / Groupes, équations, géométrie algébrique noncommutative Org: Olga Kharlampovich and/et Alexei G. Myasnikov (McGill)
 DMITRI BORMOTOV, The Graduate School of CUNY, Department of Computer Science,
365 Fifth Avenue, NY, NY 10016
Effective algorithm for solving onevariable equations in a
free group

We show that the task of finding all solutions of an onevariable
equation in a free group can be reduced to finding so called
pseudosolutions of cubic equations, and in turn, pseudosolutions of
quadratic equations. This results in a polynomial time algorithm with
complexity l^{4} * 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, torsionfree
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 treefree group is right orderable, but it has a locally invariant
order. The class of treefree groups includes all torsionfree
subgroups of SL_{2} (^{*}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, D70569
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 áXRñ, 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 onevariable 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 onevariable.
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
(KharlampovihMyasnikov 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 firstorder 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, Noncommutative 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={y_{1},z_{1},...,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 GrigorchukKurchanov, ComerfordEdmunds
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 nonstandard solutions of equations in
group

I will describe some approximation methods for solving equations in
groups and their relations with generic points, parametric and
nonstandard 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), 1679, may
succinctly be paraphrased by asserting that the class of Equationally
Noetherian groups is the direct union of a family of quasivarieties.

