ABSTRACT We prove that the natural HNN-extensions of the Fractional Fibonaccigroups are fundament... more ABSTRACT We prove that the natural HNN-extensions of the Fractional Fibonaccigroups are fundamental groups of high-dimensional knot complements.We give also some characterization and interpretation ofthese knots. In particular we shall show that some of them are 2knots.1 IntroductionThe fractional Fibonacci groups Fk=l(2; m) were introduced in [6] for integersk, l, m such that m 3 and (k; l) = 1 :Fk=l(2; m) = ha 1 ; a 2 ; : : : ; am j ali aki+1 = ali+2 ; i = 1; : : : ; mi...
We discuss the following question of G. Makanin from "Kourovka notebook": does there exist an alg... more We discuss the following question of G. Makanin from "Kourovka notebook": does there exist an algorithm to determine is for an arbitrary pair of words U and V of a free group Fn and an arbitrary automorphism ϕ ∈ Aut(Fn) the equation ϕ(X)U = V X solvable in Fn? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of Fn.
We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugac... more We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given.
The Fibonacci manifolds and the ten smallest known compact orientable three-dimensional hyperboli... more The Fibonacci manifolds and the ten smallest known compact orientable three-dimensional hyperbolic manifolds are obtained as two-fold coverings of the three-dimensional sphere. The corresponding branch sets are described. It is shown that every manifold under consideration admits a Heegaard splitting of genus two.
Rendiconti del Circolo Matematico di Palermo, 2001
Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, w... more Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, with few components but large finite symmetry groups. We consider the 3-manifolds obtained by equivariant surgeries on these links, most of them hyperbolic, and the quotient orbifolds obtained from these group actions, among them various of the smallest known hyperbolic 3-orbifolds. Also, various of the manifolds obtained by equivariant surgery on these links are maximally symmetric hyperbolic 3-manifolds.
In 1931 F. L~bell constructed the first example of a closed orientable three-dimensional hyperbol... more In 1931 F. L~bell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing LSbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.
This article studies the geometric structures of dihedral orbifolds with a knot and a bridge as i... more This article studies the geometric structures of dihedral orbifolds with a knot and a bridge as it's singular set and the three-dimensional sphere as the underlying space. These knotted graphs are connected with certain families of links factorized by their involution. In some of our cases the bridge is a tunnel of the knot. The greater part of the orbifolds described have a hyperbolic structure. Most of the fundamental groups appearing are two generator groups and moreover, they are generalized triangle groups, whose explicit representations are given.
ABSTRACT We prove that the natural HNN-extensions of the Fractional Fibonaccigroups are fundament... more ABSTRACT We prove that the natural HNN-extensions of the Fractional Fibonaccigroups are fundamental groups of high-dimensional knot complements.We give also some characterization and interpretation ofthese knots. In particular we shall show that some of them are 2knots.1 IntroductionThe fractional Fibonacci groups Fk=l(2; m) were introduced in [6] for integersk, l, m such that m 3 and (k; l) = 1 :Fk=l(2; m) = ha 1 ; a 2 ; : : : ; am j ali aki+1 = ali+2 ; i = 1; : : : ; mi...
We discuss the following question of G. Makanin from "Kourovka notebook": does there exist an alg... more We discuss the following question of G. Makanin from "Kourovka notebook": does there exist an algorithm to determine is for an arbitrary pair of words U and V of a free group Fn and an arbitrary automorphism ϕ ∈ Aut(Fn) the equation ϕ(X)U = V X solvable in Fn? We give the affirmative answer in the case when an automorphism is virtually inner, i.e. some its non-zero power is an inner automorphism of Fn.
We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugac... more We study the conjugacy problem in cyclic extensions of free groups. It is shown that the conjugacy problem is solvable in split extensions of finitely generated free groups by virtually inner automorphisms. An algorithm for construction of the unique representative (the conjugacy normal form) for each conjugacy class is given.
The Fibonacci manifolds and the ten smallest known compact orientable three-dimensional hyperboli... more The Fibonacci manifolds and the ten smallest known compact orientable three-dimensional hyperbolic manifolds are obtained as two-fold coverings of the three-dimensional sphere. The corresponding branch sets are described. It is shown that every manifold under consideration admits a Heegaard splitting of genus two.
Rendiconti del Circolo Matematico di Palermo, 2001
Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, w... more Starting from the regular Platonic solids we construct links, generalizing the Borromean rings, with few components but large finite symmetry groups. We consider the 3-manifolds obtained by equivariant surgeries on these links, most of them hyperbolic, and the quotient orbifolds obtained from these group actions, among them various of the smallest known hyperbolic 3-orbifolds. Also, various of the manifolds obtained by equivariant surgery on these links are maximally symmetric hyperbolic 3-manifolds.
In 1931 F. L~bell constructed the first example of a closed orientable three-dimensional hyperbol... more In 1931 F. L~bell constructed the first example of a closed orientable three-dimensional hyperbolic manifold. In the present paper we study properties of closed hyperbolic 3-manifolds generalizing LSbell's classical example. Explicit formulas for the volumes of these manifolds in terms of the Lobachevski function are obtained.
This article studies the geometric structures of dihedral orbifolds with a knot and a bridge as i... more This article studies the geometric structures of dihedral orbifolds with a knot and a bridge as it's singular set and the three-dimensional sphere as the underlying space. These knotted graphs are connected with certain families of links factorized by their involution. In some of our cases the bridge is a tunnel of the knot. The greater part of the orbifolds described have a hyperbolic structure. Most of the fundamental groups appearing are two generator groups and moreover, they are generalized triangle groups, whose explicit representations are given.
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Papers by Andrei Vesnin