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Valery Bardakov

Publications and source records attributed to Valery Bardakov.

5 recordsLinked to original sources
arXiv2013

On Cohen braids

For a general surface $M$ and an arbitrary braid $α$ from the surface braid group $B_{n-1}(M)$ we study the system of equations $d_1β=\cdots=d_{n}β=α, $ where operation $d_i$ is deleting of $i$-th strand. We obtain that if $M\not=S^2$ or $\mathbb RP^2$ this system of equations has a solution $β\in B_{n}(M)$ if and only if $d_1α=\ldots=d_nα. $ The set of braids satisfying the last system of equations we call Cohen braids. We also construct a set of generators for the groups of Cohen braids. In the cases of the sphere and the projective plane we give some examples for the small number of strands.

math.AT↗
arXiv2011

Generating groups by conjugation-invariant sets

Let S be a generating set of a group G. We say that G has FINITE WIDTH relative to S if G=(S\cup S^{-1})^k for a suitable natural number k. We say that a group G is a group of FINITE C-WIDTH if G has finite width with respect to all conjugation-invariant generating sets. We give a number of examples of groups of finite C-width, and, in particular, we prove that the commutator subgroup F' of Thompson's group F is a group of finite C-width. We also study the behaviour of the class of all groups of finite C-width under some group-theoretic constructions; it is established, for instance, that this class is closed under formation of group extensions.

math.GR↗
arXiv2003

The palindromic width of a free product of groups

Palindromes are those reduced words of free products of groups that coincide with their reverse words. We prove that a free product of groups $G$ has infinite palindromic width, provided that $G$ is not the free product of two cyclic groups of order two. This means that there is no a uniform bound $k$ such that every element of $G$ is a product of at most $k$ palindromes. Earlier the similar fact established for non-abelian free groups.

math.GR↗
arXiv2003

On the palindromic and primitive widths of a free group

Let G be a group and S a subset of G that generates G. For each x in G define the length l_S(x) of x relative to S to be the minimal k such that x is a product of k elements of S. The supremum of the values l_S(x), x \in G, is called the width of G with respect to S. Here we focus on a free group F. The width of F relative to the set of all primitive (respectively palindromic) elements is called the primitive (respectively palindromic) width of F. We prove that for a free group F_n of finite rank n, both widths are infinite. A result of independent interest is that every primitive element of F_2 is a product of at most two palindromes.

math.GR↗