arXiv subjects

Valeriy G. Bardakov

Publications and source records attributed to Valeriy G. Bardakov.

51 records · Page 3Linked to original sources

arXiv2017

Automorphisms of pure braid Groups

In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to an automorphism of $B_n$.

math.GR↗

arXiv2016

Extensions and automorphisms of Lie algebras

Let $0 \to A \to L \to B \to 0$ be a short exact sequence of Lie algebras over a field $F$, where $A$ is abelian. We show that the obstruction for a pair of automorphisms in $\Aut(A) \times \Aut(B)$ to be induced by an automorphism in $\Aut(L)$ lies in the Lie algebra cohomology $\Ha^2(B;A)$. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in $\Aut\big(L_{n,2}^{(1)}\big) \times \Aut\big(L_{n,2}^{ab}\big)$ to be induced by an automorphism in $\Aut\big(L_{n,2}\big)$, where $L_{n,2}$ is a free nilpotent Lie algebra of rank $n$ and step $2$.

math.RA↗

arXiv2015

Monoid and group of pseudo braids

In this paper we define a monoid of pseudo braids and prove that this monoid is isomorphic to a singular braid monoid. We also prove an analogue of Markov's theorem for pseudo braids.

math.GT↗

arXiv2015

Palindromic Automorphisms of Free Nilpotent Groups

In this paper, we initiate the study of palindromic automorphisms of groups that are free in some variety. More specifically, we define palindromic automorphisms of free nilpotent groups and show that the set of such automorphisms is a group. We find a generating set for the group of palindromic automorphisms of free nilpotent groups of step 2 and 3. In particular, we obtain a generating set for the group of central palindromic automorphisms of these groups. In the end, we determine central palindromic automorphisms of free nilpotent groups of step 3 which satisfy the necessary condition of Bryant-Gupta-Levin-Mochizuki for a central automorphism to be tame.

math.GR↗

arXiv2015

Palindromic Automorphisms of Free Groups

Let $F_n$ be the free group of rank $n$ with free basis $X=\{x_1,\dots,x_n \}$. A palindrome is a word in $X^{\pm 1}$ that reads the same backwards as forwards. The palindromic automorphism group $ΠA_n$ of $F_n$ consists of those automorphisms that map each $x_i$ to a palindrome. In this paper, we investigate linear representations of $ΠA_n$, and prove that $ΠA_2$ is linear. We obtain conjugacy classes of involutions in $ΠA_2$, and investigate residual nilpotency of $ΠA_n$ and some of its subgroups. Let $IA_n$ be the group of those automorphisms of $F_n$ that act trivially on the abelianisation, $P I_n$ be the palindromic Torelli group of $F_n$, and $E ΠA_n$ be the elementary palindromic automorphism group of $F_n$. We prove that $PI_n=IA_n \cap E ΠA_n'$. This result strengthens a recent result of Fullarton.

math.GR↗

arXiv2015

On Palindromic Widths of Nilpotent and Wreathe Products

We prove that the nilpotent product of a set of groups $A_{1},\dots, A_{s}$ has finite palindromic width if and only if the palindromic widths of $A_{i}, i=1,\dots, s,$ are finite. We give a new proof that the commutator width of $F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n \geq 2$ and $K$ a finite group. This result, combining with a result of Fink \cite{f1} gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.

math.GR↗

arXiv2015

Some Problems on Knots, Braids, and Automorphism Groups

We present and discuss some open problems formulated by participants of the International Workshop "Knots, Braids, and Auto\-mor\-phism Groups" held in Novosibirsk, 2014. Problems are related to palindromic and commutator widths of groups; properties of Brunnian braids and two-colored braids, corresponding to an amalgamation of groups; extreme properties of hyperbolic 3-orbifold groups, relations between inner and quasi-inner automorphisms of groups; and Staic's construction of symmetric cohomology of groups.

math.GT↗

arXiv2015

Example of non-linearizable quasi-cyclic subgroup of automorphism group of polynomial algebra

It is well known that every finite subgroup of automorphism group of polynomial algebra of rank 2 over the field of zero characteristic is conjugated with a subgroup of linear automorphisms. We prove that it is not true for an arbitrary torsion subgroup. We construct an example of abelian $p$-group of automorphism of polynomial algebra of rank 2 over the field of complex numbers, which is not conjugated with a subgroup of linear automorphisms.

math.GR↗

arXiv2014

Upper central series for the group of unitriangular automorphisms of a free associative algebra

We study some subgroups of the group of unitriangular automorphisms $U_n$ of a free associative algebra over a field of characteristic zero. We find the center of $U_n$ and describe the hypercenters of $U_2$ and $U_3$. In particular, we prove that the upper central series for $U_2$ has infinite length. As consequence, we prove that the groups $U_n$ are non-linear for all $n \geq 2$.

math.GR↗

arXiv2014

On palindromic width of certain extensions and quotients of free nilpotent groups

In arXiv:1303.1129, the authors provided a bound for the palindromic width of free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group ${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$. We denote by $\widetilde{G}_n = G_n / \langle \langle x_1^2, \ldots, x_n^2 \rangle \rangle$ the quotient of group $G_n = \langle x_1, \ldots, x_n \rangle$, which is free in some variety by the normal subgroup generated by $x_1^2, \ldots, x_n^2$. We prove that the palindromic width of the quotient $\widetilde{AN}_n$ is finite and bounded by $3n$. We also prove that the palindromic width of the quotient $\widetilde{\rm N}_{n, 2}$ is precisely $2(n-1)$. We improve the lower bound of the palindromic width for ${\rm N}_{n, r}$. We prove that the palindromic width of ${\rm N}_{n, r}$, $r\geq 2$ is at least $2(n-1)$. We also improve the bound for palindromic widths of free metabelian groups. We prove that the palindromic width of free metabelian group of rank $n$ is at most $4n-1$.

math.GR↗

arXiv2014

Palindromic Width of Finitely Generated Solvable Groups

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated $3$-step solvable group has finite palindromic width. More generally, we show the finiteness of palindromic width for finitely generated abelian-by-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step $\geq 3$, we prove that if $G$ is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of $G$ is finite. We also prove that the palindromic width of $\mathbb Z \wr \mathbb Z$ with respect to the set of standard generators is $3$.

math.GR↗

arXiv2014

Palindromic width of free nilpotent groups

In this paper we consider the palindromic width of free nilpotent groups. In particular, we prove that the palindromic width of a finitely generated free nilpotent group is finite. We also prove that the palindromic width of a free abelian-by-nilpotent group is finite.

math.GR↗

arXiv2012

Groups of virtual and welded links

We define new notions of groups of virtual and welded knots (or links) and we study their relations with other invariants, in particular the Kauffman group of a virtual knot.

math.GT↗

arXiv2010

Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from these group. Also we give a more simple system of generators for the group of tame automorphisms than the system from Umirbaev's paper.

math.GR↗

arXiv2007

On certain questions of the free group automorphisms theory

Certain subgroups of the groups $Aut(F_n)$ of automorphisms of a free group $F_n$ are considered. Comparing Alexander polynomials of two poly-free groups $Cb_4^+$ and $P_4$ we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from \cite{CVW}. The questions of linearity of subgroups of $Aut(F_n)$ are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type $Aut(G*\mathbb Z)$ for certain groups $G$ and the subgroup of $IA$-automorphisms $IA(F_n)\subset Aut(F_n)$ are not linear for $n\geq 3$. This generalizes the recent result of Pettet that $IA(F_n)$ are not linear for $n\geq 5$.

math.GR↗