Speciality:
01.01.06 (Mathematical logic, algebra, and number theory)
E-mail: Keywords: group theory; knot theory; braid groups; automorphism groups; permutation groups; width of verbal subgroups; equations in groups; partial differential equation; multidimensional inverse problems; evolution equations; integral geometry; systems of kinetic equations.
Subject:
It is proved that any verbal subgroup $V(B_n)$ of braid group $B_n$, $n>2$, defined by a finite set of words $V$ has infinite width. Similar result is proved for some Artin groups. We investigate decompositions of automorphisms of various free modules into products of transvections and dilations. In particular, as a corollary, we obtain that for $n>2$ the width of the spetial linear group $SL_n(Z)$ over ring of integer $Z$, with respect to the set of commutators, does not exceed 10. It is shown that for any integers $k>3$ and $m>0$ any permutation in the alternating group $A_{km}$ of degree $km$ can be expressed as the product of two permutations each of them has $m$ cycles of length $k$ in its cycle decomposition. This proves a conjecture by J. L. Brenner and R. J. Evans. It is proved that if in the HNN-extension $G^*$, the subgroups $A$ and $B$ are distinct from the base group $G$, then every subgroup $V(G^*)$ defined by a finite proper set $V$ of words has infinite width relative to $V$. Similar results are proved for groups with one relation and number generations greater 2 and for some free produts with amalgamations. It is described a new algorithm for calculating the commutator length in free groups and some equations in free groups are investigated. It is proved that there exist continuum many non-isomorphic two-generated groups which each possess a regularly exhausting sequence with polynomial growth and which are but not groups with polynomial growth.
Main publications:
Bardakov V. G. Two inverse problems for a system of quantum kinetic equations // J. Inv.Ill-Posed Problems, 7, 2(1998), 105–119.
