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34

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## Publications

Publications (34)

We prove that the Diophantine problem for orientable quadratic equations in free metabelian groups is decidable and furthermore, NP-complete. In the case when the number of variables in the equation is bounded, the problem is decidable in polynomial time.

Let $G$ be the first Grigorchuk group. We show that the commutator width of $G$ is $2$: every element $g\in [G,G]$ is a product of two commutators, and also of six conjugates of $a$. Furthermore, we show that every finitely generated subgroup $H\leq G$ has finite commutator width, which however can be arbitrarily large, and that $G$ contains a subg...

We provide an algorithm which, for a given quadratic equation in the
Grigorchuk group determines if it has a solution. As a corollary to our
approach, we prove that the group has a finite commutator width.

We prove that the Diophantine problem for spherical quadratic equations in
free metabelian groups is solvable and, moreover, NP-complete

We provide polynomial upper bounds on the size of a shortest solution for
quadratic equations in a free group. A similar bound is given for parametric
solutions in the description of solutions sets of quadratic equations in a free
group.

We prove that the problems of deciding whether a quadratic equation over a free group has a solution is NP-complete.
KeywordsEquations over free groups-NP-completeness

In this paper we prove that the Conjugacy Problem in the Grigorchuk group $\Gamma$ has polynomial time complexity.

We prove that the problems of deciding whether a quadratic equation over
a free group has a solution is NP-complete.

We establish conditions guaranteeing that a group possesses the following property: there is a number such that if elements , of generate a finite subgroup then lies in the normalizer of . These conditions are of a quite special form. They hold for groups with relations of the form which appear as approximating groups for the free Burnside groups o...

For any odd number n≥1003, the authors construct an infinite 2-generator group each of whose proper subgroups is contained in a cyclic subgroup of order n. This result strengthens analogous results of Ol'shanskiĭ for prime n>1075 and Atabekyan and Ivanov for odd n>1080. The proof is carried out in the original language of Novikov-Adyan theory.
Bibl...

For hyperbolic groups the author establishes the solvability of the algorithmic
problems of extracting a root of an element, determining the order of an element,
membership of a cyclic subgroup, and existence of a solution of an arbitrary quadratic
equation. It is proved that every hyperbolic group has a finite presentation for which
the word probl...

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We give a linear lower bound on the exponential growth rate of a non-elementary subgroup of a word hyperbolic group, with
respect to the number of generators for the subgroup.

We show that non-elementary word hyperbolic groups are growth tight. This means that, given such a group G and a finite set A of its generators, for any infinite normal subgroup N of G, the exponential growth rate of G/N with respect to the natural image of A is strictly less than the exponential growth rate of G with respect to A.

We say that a group G is abelian if the group operation (usually written as addition) is commutative. The theory of abelian groups can be considered on the one hand as a part of the general theory of groups and also on the other hand as a part of module theory since every abelian group is a module over the ring ℤ of integers. But at the same time t...

We present a modified version of the Novikov-Adian theory for free Burnside groups of exponent n=16k≥8000. On the basis of this theory, we obtain a negative solution to the Burnside problem for even values of n≥8000 and prove a number of assertions on Burnside groups of exponent n=16k≥8000.

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