arXiv subjects
Publications and source records attributed to V. Roman'kov.
We show that the attacks based on the linear decomposition method introduced by the author and the span-method introduced by Tsaban allow one to find the transmitted message in the cryptosystem and the exchanged key in the protocol which are proposed by Baba et al.
The paper is intended to be a survey on some topics within the framework of automorphisms of a relatively free groups of infinite rank. We discuss such properties as tameness, primitivity, small index, Bergman property, and so on.
We prove that the unitriangular automorphism group of a free group of rank $n$ has a faithful representation by matrices over a field, or in other words, it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a description of relatively free groups with linear the unitriangular automorphism groups. This description was initiated by Erofeev and the author in \cite{Erofeev}, where proper varieties of groups have been considered.
We prove that every verbally closed subgroup of a free group $F$ of a finite rank is a retract of $F.$
Let $F$ be a free non-abelian group. We show that for any group word $w$ the set $w[F]$ of all values of $w$ in $F$ is rational in $F$ if and only if $w[F] = 1$ or $w[F] = F.$ We generalize this to a wide class of free products of groups.
An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $φ, ψ: G \to G$ and any pair of elements $u, v \in G,$ whether or not the equation $(xφ)u = v (xψ)$ has a solution $x \in G.$ Thus it is shown that the problem of the title is decidable. Also we present an algorithm that produces a finite set of generators of the subgroup (equalizer) $Eq_{φ, ψ}(G) \leq G$ of all elements $u \in G$ such that $u φ= u ψ.$
We describe the Reidemeister spectrum $Spec_RG$ for $G = N_{rc},$ the free nilpotent group of rank $r$ and class $c,$ in the cases: $r \in {\mathbb N}$ and $c = 1;$ $r = 2, 3$ and $c = 2;$ $ r = 2$ and $c = 3,$ and prove that any group $N_{2c}$ for $c \geq 4$ satisfies to the $R_{\infty}$ property. As a consequence we obtain that every free solvable group $S_{2t}$ of rank 2 and class $t \geq 2$ (in particular the free metabelian group $M_2 = S_{22}$ of rank 2) satisfies to the $R_{\infty}$ property. Moreover, we prove that any free solvable group $S_{rt}$ of rank $r \geq 2$ and class $t$ big enough also satisfies to the $R_{\infty}$ property.
Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $ϕ\in Aut(N)$ defines the Reidemeister number $R(ϕ).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$ and class $c \geq 4r,$ or rank $r \geq 4$ and class $c \geq 2r,$ belongs to the class $R_{\infty}.$
We study the computational complexity of the Word Problem (WP) in free solvable groups $S_{r,d}$, where $r \geq 2$ is the rank and $d \geq 2$ is the solvability class of the group. It is known that the Magnus embedding of $S_{r,d}$ into matrices provides a polynomial time decision algorithm for WP in a fixed group $S_{r,d}$. Unfortunately, the degree of the polynomial grows together with $d$, so the uniform algorithm is not polynomial in $d$. In this paper we show that WP has time complexity $O(r n \log_2 n)$ in $S_{r,2}$, and $O(n^3 r d)$ in $S_{r,d}$ for $d \geq 3$. However, it turns out, that a seemingly close problem of computing the geodesic length of elements in $S_{r,2}$ is $NP$-complete. We prove also that one can compute Fox derivatives of elements from $S_{r,d}$ in time $O(n^3 r d)$, in particular one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as, on a relatively new geometric ideas, in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.