Valeriy G. Bardakov

A Generalization of Fibonacci Groups

We study the class of cyclically presented groups which contain Fibonacci groups and Sieradski groups. Conditions are specified for these groups to be finite, pairwise isomorphic, or aspherical. As a partial answer to the question of Cavicchioli, Hegenbarth, and Repovš, it is stated that there exists a wide subclass of groups with an odd number of generators cannot appear as fundamental groups of hyperbolic three-dimensional manifolds of finite volume.

Palindromic automorphisms of free groups

Abstract Let F n be the free group of rank n with free basis X = { x 1 , … , x n } . A palindrome is a word in X ± 1 that reads the same backwards as forwards. The palindromic automorphism group Π A n of F n consists of those automorphisms that map each x i to a palindrome. In this paper, we investigate linear representations of Π A n , and prove that Π A 2 is linear. We obtain conjugacy classes of involutions in Π A 2 , and investigate residual nilpotency of Π A n and some of its subgroups. Let IA n be the group of those automorphisms of F n that act trivially on the abelianisation, PI n be the palindromic Torelli group of F n , and let E Π A n be the elementary palindromic automorphism group of F n . We prove that PI n = IA n ∩ E Π A n ′ . This result strengthens a recent result of Fullarton [2] .

On Certain Questions of the Free Group Automorphisms Theory

Certain subgroups of the groups Aut(Fn) of automorphisms of a free group Fn are considered. Comparing Alexander polynomials of two poly-free groups and P4 we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohenet al. (preprint). The questions of linearity of subgroups of Aut(Fn) 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 * ℤ) for certain groups G and the subgroup of IA-automorphisms IA(Fn) ⊂ Aut(Fn) are not linear for n ≥ 3. This generalizes the recent result of Pettet that IA(Fn) are not linear for n ≥ 5 (Pettet, 2006).

On the Pure Virtual Braid Group PV3

We investigate various properties of the pure virtual braid group PV3. Out of its presentation, we get a free product decomposition of PV3. As a consequence, we show that PV3 is residually torsion free nilpotent, what implies that the set of the finite type invariants in the sense of Goussarov–Polyak–Viro is complete for virtual pure braids with three strands. Moreover, we prove that the presentation of PV3 is aspherical. We determine also the cohomology ring and the associated graded Lie algebra of PV3.

Representations of virtual braids by automorphisms and virtual knot groups

In the present paper the representation of the virtual braid group

Twisted conjugacy classes of the unit element

VB_n

Automorphism groups of quandles arising from groups

into the automorphism group of free product of the free group and free abelian group is constructed. This representation generalizes the previously constructed ones. The fact that these already known representations are not faithful for

Palindromic Width of Finitely Generated Solvable Groups

n \geq 4

Automorphism groups of quandles and related groups

is verified. Using representations of

Virtual link groups

VB_n