Cohomology of braid groups with nontrivial coefficients

In the paper, the homology of the braid groups with twisted coefficients and the homology of commutator subgroups of the braid groups are calculated. The main tool is the multiplicative structure on the homology induced by the “addition” of braid groups. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 846–854, June, 1996. This research was partially supported by the International Science Foundation under grant MQO000.     

Three-page approach to knot theory. Universal semigroup

An explicit construction of a finitely presented semigroup whose central elements are in a one-to-one correspondence with the isotopy classes of unoriented links in the three-space is given, together with a finite presentation for the group of invertible elements of the semigroup. The group is presented by two generators and three relations. The commutator subgroup of the group is isomorphic to the braid group of infinite index. A similar construction is given for band-links. The kauffman theorems on the existence of polynomial band-link invariants satisfying some skein-relations are stated algebraically. This work is partially supported by Russian Foundation for Basic Research grant No. 99-01-00090. Moscow State University. Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 34, No. 1, pp. 29–40, January–March, 2000. Translated by I. a. Dynnikov     

On the group of conjugating automorphisms of a free group

The group of conjugating automorphisms of a free group and certain subgroups of this group, namely, the group of McCool basis-conjugating automorphisms and the Artin braid group are considered. The Birman theorem on the representation of a braid group by matrices is sharpened. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 92–108, July, 1996.     

On representations and <Emphasis Type="Italic">K</Emphasis>-theory of the braid groups

 Let Γ be the fundamental group of the complement of a K(Γ, 1) hyperplane arrangement (such as Artin's pure braid group) or more generally a homologically toroidal group as defined below. The triviality of bundles arising from orthogonal representations of Γ is characterized completely as follows. An orthogonal representation gives rise to a trivial bundle if and only if the representation factors through the spinor groups. Furthermore, the subgroup of elements in the complex K-theory of BΓ which arises from complex unitary representations of Γ is shown to be trivial. In the case of real K-theory, the subgroup of elements which arises from real orthogonal representations of Γ is an elementary abelian 2-group, which is characterized completely in terms of the first two Stiefel-Whitney classes of the representation. In addition, quadratic relations in the cohomology algebra of the pure braid groups which correspond precisely to the Jacobi identity for certain choices of Poisson algebras are shown to give the existence of certain homomorphisms from the pure braid group to generalized Heisenberg groups. These cohomology relations correspond to non-trivial Spin representations of the pure braid groups which give rise to trivial bundles. Received: 6 February 2002 / Revised version: 19 September 2002 / Published online: 8 April 2003 RID="⋆" ID="⋆" Partially supported by the NSF RID="⋆⋆" ID="⋆⋆" Partially supported by grant LEQSF(1999-02)-RD-A-01 from the Louisiana Board of Regents, and by grant MDA904-00-1-0038 from the National Security Agency RID="⋆" ID="⋆" Partially supported by the NSF Mathematics Subject Classification (2000): 20F36, 32S22, 55N15, 55R50     

Criteria of prime links in terms of pseudo-characters

Pseudo-characters of groups have recently found applications in the theory of classical knots and links in ℝ³. More precisely, there is a connection between pseudo-characters of Artin’s braid groups and properties of links represented by braids. In the present work, this connection is investigated and the notion of kernel pseudo-characters of braid groups is introduced. It is proved that a kernel pseudo-character ϕ and a braid β satisfy Ιϕ(β)І > C _ϕ, where C _ϕ is the defect of ϕ, then β represents a prime link (i.e., a link that is noncomposite, nonsplit, and nontrivial). Furthermore, the space of braid group pseudo-characters is studied and a way to obtain nontrivial kernel pseudo-characters from an arbitrary braid group pseudo-character that is not a homomorphisrn is described. This allows one to use an arbitrary nontrivial braid group pseudo-character for recognition of prime knots and links. Bibliography: 17 titles.     

On a width of verbal subgroups of certain free constructions

We study into widths of verbal subgroups of HNN-extensions, and of groups with one defining relation. It is proved that if a group G^* is an HNN-extension and the connected subgroups in G^* are distinct from a base of the extension, then every verbal subgroup V(G^*) has infinite width relative to a finite proper set V of words. A similar statement is proven to hold for groups presented by one defining relation and ≥3 generators. to Yurii I. Merzlyakov dedicated Supported by RFFR grant No. 93-01-01513. Translated fromAlgebra i Logika, Vol. 36, No. 5, pp. 494–517, September–October, 1997.     

On fundamental groups related to the Hirzebruch surface <Emphasis Type="Italic">F</Emphasis><Subscript>1</Subscript>

Given a projective surface and a generic projection to the plane, the braid monodromy factorization (and thus, the braid monodromy type) of the complement of its branch curve is one of the most important topological invariants, stable on deformations. From this factorization, one can compute the fundamental group of the complement of the branch curve, either in ℂ² or in ℂℙ². In this article, we show that these groups, for the Hirzebruch surface F _1,(a,b), are almost-solvable. That is, they are an extension of a solvable group, which strengthen the conjecture on degeneratable surfaces. This work was supported by the Emmy Noether Institute Fellowship (by the Minerva Foundation of Germany) and Israel Science Foundation (Grant No. 8008/02-3)     

Using Fox Derivatives in Treating Groups of the Form F/[R′, F]

For a factor group with respect to periodic part of a group of the form F/[R′, F], an embedding in the matrix group is defined. The criteria for a matrix to belong to an image of this group and for elements to be conjugate are specified. Some statements having a direct bearing on groups of the form in question are proved. Application of the results obtained allows us to refine the answer in [7] to a question by O. Chapuis concerning the universal classification of ∀-free soluble groups with two generators. Supported by RFBR grant No. 02-01-00293 and by FP “Universities of Russia” grant No. UR.04.01.227. __________ Translated from Algebra i Logika, Vol. 45, No. 1, pp. 114–125, January–February, 2006.     

Recognition of simple groups U 3(Q) by element orders

An exhaustive solution is given to the recognition-by-spectrum problem for finite, simple, three-dimensional unitary groups. For every such group, the number of non-isomorphic, finite, isospectral groups is determined. In particular, a new counterexample to Problem 13.63 in the Kourovka Notebook is furnished. Supported by RFBR grant No. 05-01-00797, and by SB RAS Young Researchers Support grant No. 29 and Integration Project No. 2006.1.2. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 185–202, March–April, 2006.     

Recognition of Virtual Braids

The virtual braid group VB_n containing the Artin braid group B_n as a subgroup is studied. A familiar complete invariant on B_n is extended to VB_n. It is conjectured that the new invariant is also complete. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 299, 2003, pp. 267–286.     

Associative nil-algebras and Golod groups

Non-nilpotent, finitely generated, associative nil-algebras are studied as well as their adjoint groups and Golod groups. Solutions are given to some problems in residually finite group theory, questions posed in the Kourovka Notebook included. Supported by RFBR grant No. 03-01-00356. __________ Translated from Algebra i Logika, Vol. 45, No. 2, pp. 231–238, March–April, 2006.     

Partially commutative metabelian groups: centralizers and elementary equivalence

For partially commutative metabelian groups, annihilators of elements of commutator subgroups are described; canonical representations of elements are defined; approximability by torsion-free nilpotent groups is proved; centralizers of elements are described. Also, it is proved that two partially commutative metabelian groups have equal elementary theories iff their defining graphs are isomorphic, and that every partially commutative metabelian group is embeddable in a metabelian group with decidable universal theory. Dedicated to V. N. Remeslennikov on the occasion of his 70th birthday Supported by RFBR (project No. 09-01-00099). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 309–341, May–June, 2009.     

Normal relatively convex subgroups of solvable orderable groups

Orderable solvable groups in which every relatively convex subgroup is normal are studied. If such a class is subgroup closed than it is precisely the class of solvable orderable groups which are locally of finite (Mal’tsev) rank. A criterion for an orderable metabelian group to have every relatively convex subgroup normal is given. Examples of an orderable solvable group G of length three with periodic G/G′ and of an orderable solvable group of length four with only one proper normal relatively convex subgroup are constructed. To the memory of N. Ya. Medvedev Supported by RFBR (project No. 03-01-00320). Translated from Algebra i Logika, Vol. 48, No. 3, pp. 291–308, May–June, 2009.     

<Emphasis Type="Italic">k</Emphasis>-Free-like groups

The following results are proved. In Theorem 1, it is stated that there exist both finitely presented and not finitely presented 2-generated nonfree groups which are k-free-like for any k ⩾ 2. In Theorem 2, it is claimed that every nonvirtually cyclic (resp., noncyclic and torsion-free) hyperbolic m-generated group is k-free-like for every k ⩾ m + 1 (resp., k ⩾ m). Finally, Theorem 3 asserts that there exists a 2-generated periodic group G which is k-free-like for every k ⩾ 3. Supported by NSF (grant Nos. DMS 0455881 and DMS-0700811). (A. Yu. Olshanskii, M. V. Sapir) Supported by RFBR project No. 08-01-00573. (A. Yu. Olshanskii) Supported by BSF grant (USA–Israel). (M. V. Sapir) Translated from Algebra i Logika, Vol. 48, No. 2, pp. 245–257, March–April, 2009.     

Extensions of lattice-ordered groups

A number of conditions are specified which are sufficient for totally ordered groups with polycyclic factor group to contain a finite normal series of convex subgroups whose factors possess good enough properties. In any case studying such totally ordered groups is reduced to treating extensions of these groups as well as their virtually o-equivalent extensions. The concept of a virtually o-equivalent extension is a particular case of the notion of an Archimedean extension. Supported by RFBR project No. 03-01-00320. Translated from Algebra i Logika, Vol. 47, No. 5, pp. 529–540, September–October, 2008.     

On presentations of generalizations of braids with few generators

In his initial paper on braids, E. Artin gave a presentation with two generators for an arbitrary braid group. We give analogs of Artin’s presentation for various generalizations of braids. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 23–32, 2005.     

Algebraic sets in metabelian groups

The research launched in [1] is brought to a close by examining algebraic sets in a metabelian group G in two important cases: (1) G = F_n is a free metabelian group of rank n; (2) G = W_n,k is a wreath product of free Abelian groups of ranks n and k. Supported by RFBR grant No. 05-01-00292. __________ Translated from Algebra i Logika, Vol. 46, No. 4, pp. 503–513, July–August, 2007.     

Locally nilpotent groups with the minimal condition on centralizers

It is proved that locally nilpotent groups with the minimal condition on centralizers are hypercentral, and that the Fitting subgroup of a group with the minimal condition on centralizers of normal subgroups is nilpotent. Supported by RFFR grant No. 96-01-00358. Translated fromAlgebra i Logika, Vol. 37, No. 3, pp. 270–278, May–June, 1998.     

Free products of groups have no outer normal automorphisms

An automorphism of an arbitrary group is called normal if all subgroups of this group are left invariant by it. Lubotski [1] and Lue [2] showed that every normal automorphism of a noncyclic free group is inner. Here we prove that every normal automorphism of a nontrivial free product of groups is inner as well. Supported by RFFR grant No. 13-011-1513. Translated fromAlgebra i Logika, Vol. 35, No. 5, pp. 562–566, September–October, 1996.