RESEARCH INTERESTS AND ACCOMPLISHMENTS
I am going to divide the discussion of my work into three parts,
beginning with the most recent work.
During the last three years my research interests included
free groups, groups acting on $\Lambda$-trees,
hyperbolic and exponential groups, periodic
groups, groups with one defining relation
and geometric methods in group theory.
Together with A. Miasnikov I studied
first-order properties of free groups. This study culminated in a proof
that the elementary theories of all nonabelian free
groups coincide and are decidable, answering
two old questions
raised by Tarski around 1945.
There are three parts to this work each requiring very different techniques
from the others: algebraic geometry over free
groups (we described irreducible affine varieties over a free group),
the theory of free exponential groups, and the generalization of
Makanin-Razborov's
machinery which deals with equations over free groups and over affine groups
of irreducible varieties.
Currently
one of the mainstreams of combinatorial group theory
is the study of systems of equations over a group.
The problem of deciding if a finite system of equations in a group has a
solution generalizes the word problem and the conjugacy problem.
Makanin
and
Razborov
proved one of the most significant results in this area, namely
the algorithmic
solvability of systems of equations
in
free groups.
I obtained in collaboration with Miasnikov
an algebraic description
of existentially free groups ($\exists$-free groups are groups in which the class of $\exists$-formulas,
true in one of these groups, is the same as the class of $\exists$-formulas,
true in a nonabelian free group). To be $\exists$-free group is the same as to be
a fully residually free group. We proved that a finitely generated group is
$\exists$-free if and only if it is
embeddable in the ${\bf Z}[x]$-completion $F^{{\bf Z}[x]}$
of the free group, where
${\bf Z}[x]$ is the ring of polynomials of one variable
with integer coefficients. (Lyndon's attempts to solve Tarski's problem
led him to introduce $F^{{\bf Z}[x]}.$)
Our result implies that every finitely generated
fully residually free group is finitely presented.
An algorithm for solving equations
over hyperbolic groups was developed by E. Rips and Z. Sela.
G. Baumslag proved that the word problem is decidable in $\bf Q$-free
groups.
Together with Miasnikov I proved that there exists an algorithm
that decides if a given finite
system of
equations over a free $\bf Q$-group
has a solution, and if it does, the algorithm finds a solution.
We established the same result
for a tensor $\bf Q$-completion of a torsion-free
hyperbolic group (joint work with my student E. Lioutikova).
My own work in this direction continues and I have some other
interesting results.
Another direction that my collaboration with Miasnikov took
involves finding
necessary and
sufficient conditions for HNN-extensions and amalgamated free
products of hyperbolic groups over
abelian subgroups to preserve hyperbolicity.
We also proved some other results about HNN-extensions of hyperbolic
groups in much more general situations.
I will now turn to my early work and show how it leads to the middle period
of my development.
In the 1930's Turing, Church and others developed a precise concept
of algorithm and the algorithmic decidability of mathematical
problems. This also made it possible to prove the non-existence of
algorithms for solving specific mathematical problems. One of the
earliest results in this area was Markov and Post's proof of the
undecidability of the word problem for finitely presented
semigroups.
The analogous result for groups turned out to be much more
complicated. This problem had already been posed by Dehn in 1912,
but was only solved in 1952 by Novikov and then by Boone. I.e., he
proved that there exists a finitely presented group for which there
is no algorithm to determine whether or not a given element of the
group is the identity.
Novikov and Adyan then proposed as the next step that a finitely
presented group might be found, satisfying a nontrivial identity,
with undecidable word problem. An obvious identity is for all
commutators to be trivial; i.e., for the group to be abelian.
However, it is quite obvious that finitely generated abelian groups
have a decidable word problem. Since finitely generated metabelian
groups are residually finite (a result of Phillip Hall), the same
is true for them.
Novikov and Adyan's problem formally appeared in written form in
1973. In 1980, when I was an undergraduate student, I solved it by
constructing a 3-step solvable group, finitely presented in the
class of all groups, and with undecidable word problem.
A year earlier, I succeeded in giving a negative answer to a
question, posed in 1965 by Kargapolov (also known as Mal'tsev's
question), about the algorithmic decidability of the universal
theory of the class of all finite nilpotent groups.
For my undergraduate work I was awarded a gold medal from the
Soviet Academy of Sciences. One such medal is awarded every two
years to the best mathematics student in the Soviet Union.
In the middle period of my work
I investigated the connections between algebraic properties and
algorithmic properties of varieties of groups and Lie algebras. I looked, in particular, for some
boundary, in the class of all varieties of solvable groups (Lie algebras over a field of characteristic
0), between algorithmic decidability and undecidability. In the process, I obtained (modulo the difficult
problem about effective boundary for integer solutions of a certain exponential diophantine equation) an
almost complete description of the class of varieties of Lie algebras in which every (absolutely)
finitely presented algebra has decidable word problem. In the case of relatively finitely presented
algebras (or groups), I proved the surprising result that no boundary exists between varieties with
decidable and varieties with undecidable word problem. This I did, by demonstrating an infinite chain
in the lattice of varieties, in which varieties with decidable and undecidable word problem alternate.
I also proved the undecidability of the word problem in the Burnside variety of groups for a large
exponent.
In 1995 together with M. Sapir I published a survey, ``Algorithmic
Problems in Varieties'' in the International Journal of Algebra and Computation,
which summarized my and his research on algorithmic problems in algebra.
Apart from the algorithmic results alluded to above, I also proved various other results in
combinatorial group theory and the theory of Lie algebras, for example, the
Generalised Freiheitssatz for
solvable varieties of Lie algebras, and some results concerning residual finiteness in varieties of groups
and Lie algebras.