Sergei V. Ivanov

The free Burnside groups of sufficiently large exponents

The paper contains a self-contained construction of m-generated free Burnside groups B(m, n) of exponent n, where m>1, n≥248 and n is either odd or divisible by 29. As a corollary, one gets that the Burnside problem on the finiteness of finitely generated groups of exponent n is solved in the negative for almost all exponents.

Photoinduced Optical Anisotropy in Amorphous Films of Liquid Crystalline Polymers

Abstract Amorphous, optically isotropic films of photochromic liquid crystalline copolymers have been prepared and irradiated with linearly polarized light. In this way high values of dichroism and birefringence have been induced.

Photoinduced Optical Anisotropy in thin Films of Amorphous Photochromic Side Chain Polymers

Abstract High values of optical anisotropy have been induced in glassy films of amorphous copolymers containing azobenzene moieties and rod—like side groups by angular—dependent photoselection. The photoinduced reorientation of the photochromic groups causes a reorientation of the non—photochromic groups due to a co-operative effect.

Hyperbolic groups and their quotients of bounded exponents

In 1987, Gromov conjectured that for every non-elementary hyperbolic group G there is an n = n(G) such that the quotient group G/Gn is infinite. The article confirms this conjecture. In addition, a description of finite subgroups of G/Gn is given, it is proven that the word and conjugacy problem are solvable in G/Gn and that ⋂∞ k=1G k = {1}. The proofs heavily depend upon prior authors’ results on the Gromov conjecture for torsion free hyperbolic groups and on the Burnside problem for periodic groups of even exponents.

Boundary case of equality in optimal Loewner-type inequalities

We prove certain optimal systolic inequalities for a closed Riemannian manifold (X, G), depending on a pair of parameters, n and b. Here n is the dimension of X, while b is its first Betti number. The proof of the inequalities involves constructing Abel-Jacobi maps from X to its Jacobi torus T b , which are area-decreasing (on b-dimensional areas), with respect to suitable norms. These norms are the stable norm of G, the conformally invariant norm, as well as other L p -norms. Here we exploit L P -minimizing differential 1-forms in cohomology classes. We characterize the case of equality in our optimal inequalities, in terms of the criticality of the lattice of deck transformations of T b , while the Abel-Jacobi map is a harmonic Riemannian submersion. That the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of Federer and Fleming, under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the Abel-Jacobi map to the maximal free abelian cover.

INTERSECTING FREE SUBGROUPS IN FREE PRODUCTS OF GROUPS

A subgroup H of a free product of groups Gα, α∈ I, is termed factor free if for every and β∈I one has SHS-1∩Gβ= {1} (by Kurosh theorem on subgroups of free products, factor free subgroups are free). If K is a finitely generated free group, denote , where r(K) is the rank of K. It has earlier been proved by the author that if H, K are finitely generated factor free subgroups of then . It is proved in the article that this estimate is sharp and cannot be improved, that is, there are factor free subgroups H, K in so that and . It is also proved that if the factors Gα, α∈ I, are linearly ordered groups and H, K are finitely generated factor free subgroups of then .

On the intersection of free subgroups in free products of groups

Let (Gi j i 2 I) be a family of groups, let F be a free group, and let G = F ⁄ ⁄ i2I Gi, the free product of F and all the Gi.

On the Burnside problem on periodic groups

It is proved that the free

On one-relator inverse monoids and one-relator groups

m

ON THE INTERSECTION OF FINITELY GENERATED SUBGROUPS IN FREE PRODUCTS OF GROUPS

-generated Burnside groups