Writing certain commutators as products of cubes in free groups

In the work of the positive solution of the conjugation of words in HNN-extension with the system of entrance letters. The base HNN-extensions is a wood product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in HNN-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letter. The conjugacy problem for words is of interest in free designs groups. The problem was solved in free groups with cyclic subgroups by S.Lipshutz, in the HNNextension of a free group by an associate of cyclic subgroups by A. Friedman, in HNN-extension of a tree product with the association cyclic groups associated with cyclic subgroups by author with V.N. Bezverkhny. In this paper a positive solution of the conjugation problem for words in HNNextension with the system of entrance letters. The base HNN-extensions is a tree product of the infinite cyclic groups with cyclic subgroups. The result is a generalization of the conjugacy problem in HNN-extension of a wood product of cyclic groups associated cyclic subgroups with one entrance letters. Assertion is proved for any number of entrance letters using the method of mathematical induction. In the proof of the main theorem the author proved self result assertion : - algorithmic solvability of intersection of finitely generated subgroup of the core group with an associated sub-group; - algorithmic solvability of intersection of the related class of finitely generated subgroup of the core group with an associated sub-group.

Let D=F1 x F2 x... x Fn be a direct product of n free groups F1, F2, * , F* * , ox an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed, T an infinite cyclic group and F another free group. Let D x a T be the semidirect product of D and T with respect to a and (D x a T) x aXIdT F the semidirect product of D xa Tand F with respect to the automorphism x id T of D Xa T induced by a. We prove that the Whitehead group of (D xa, T) X 2xidT F and the projective class group of the integral group ring Z((D x a T) X aXidT F) are trivial. These results extend that of [3]. Let G be a group. We denote the Whitehead group of G by Wh G and the projective class group of the integral group ring Z(G) of G by kOZ(G). We recall the definition of semidirect product of groups and the definition of twisted group ring. For undefined terminologies used in the paper, we refer to [3] and [4]. Let oc be an automorphism of G and F a free group generated by {tA}. If w is a word in tA defining an element in F, we denote by Iwl the total exponent sum of the tA appearing in w. The semidirect product G xa F of G and F with respect to a is defined as follows: G x . F=GxF as sets and multiplication in G x . Fis given by (g, w)(g', w') = (go-lwl(g'), ww'), for any (g, w), (g', w') in G x F. In particular, if F is an infinite cyclic group T= (t) generated by t, we have the semidirect product G x a T of G and T with respect to oc. Let R be an associative ring with identity and oc an automorphism of R. Let F be a free group (or free semigroup) generated by {tA}. The otwisted group ring R,[F] of F over R is defined as follows: additively R,[F]=R[F], the group ring of F over R, so that its elements are finite linear combinations of elements in F with coefficients in R. Multiplication in R,[F] is given by (rw)(rIw')=roc-1I1(r')ww', for any rw, r'w' in R,[F]. In particular, if F is a free group (resp. free semigroup) generated by t, we Received by the editors May 25, 1973. AMS (MOS) subject classfiJcations (1970). Primary 13D15, 16A26, 18F25; Secondary 16A06, 16A54.

We exploit the combinatorial properties of surface maps into free groups to prove several new results in the field of stable commutator length and bounded cohomology. We show that random homomorphisms between free groups are isometries of scl; we prove interesting properties of the scl unit ball; we describe a transfer construction for quasimorphisms and give an infinite family of chains whose scl it certifies; we linearize the dynamics of endomorphisms on free groups and use this to prove that random endomorphisms can be realized by surface immersions, which provides many examples of surface subgroups of HNN extensions of free groups; and finally, we give an algorithm to compute scl in free products of finite or infinite cyclic groups that generalizes and improves previous work.

A ring R is an IPQ (isomorphic proper quotient)-ring if R ⋍ R/A for every proper ideal A ⋪ R. If every ideal A ⋬ R satisfies: either R ⋍ A or R ⋍ R/A, then R is called an SE (self extending)-ring. It is shown that with one exception, an abelian group G is the additive group of an IPQ-ring if and only if G is the additive group of an SE-ring. The one exception is the infinite cyclic group Z. The zeroring with additive group Z is an SE-ring, but a ring with infinite cyclic additive group is not an IPQ-ring. Since the structure of the additive groups of IPQ-rings is known, the structure of the additive groups of SE-rings is completely determined.

Explicit Helfgott type growth in free products and in limit groups

This paper deals with a problem raised in a paper by J. de Groot (1): Do there exist fields Ω whose full automorphism group is isomorphic to the additive group of integers Z?The answer to this question is yes. In this paper we construct, given any subfield k of the complex numbers, extension fields Ω of k such that the automorphism group G(Ω/k) of Ω with respect to k is infinite cyclic. Fields having the infinite cyclic group as a full group of automorphisms are obtained by choosing the base field k in such a way that it does not contain any subfield k0 so that k possesses non-trivial automorphisms leaving k0 pointwise fixed.

Let $D = {F_1} \times {F_2} \times \cdots \times {F_n}$ be a direct product of $n$ free groups ${F_1},{F_2}, \cdots ,{F_n},\alpha$ an automorphism of $D$ which leaves all but one of the noncyclic factors in $D$ pointwise fixed, $T$ an infinite cyclic group and $F$ another free group. Let $D{ \times _\alpha }T$ be the semidirect product of $D$ and $T$ with respect to $\alpha$ and $(D{ \times _\alpha }T){ \times _{\alpha \times \text {id}T}}F$ the semidirect product of $D{ \times _\alpha }T$ and $F$ with respect to the automorphism $\alpha \times idT$ of $D{ \times _\alpha }T$ induced by $\alpha$. We prove that the Whitehead group of $(D{ \times _\alpha }T){ \times _{\alpha \times idT}}F$ and the projective class group of the integral group ring $Z((D{ \times _\alpha }T){ \times _{\alpha \times idT}}F)$ are trivial. These results extend that of [3].

It is proved that any countable abelian group can be embedded as a centre into a -generated group such that the quotient group is isomorphic to the free Burnside group of rank and of odd period . The proof is based on a certain modification of the method that was used by Adian in his monograph in 1975 for a positive solution of Kontorovich’s famous problem from the Kourovka Notebook on the existence of a finitely generated noncommutative analogue of the additive group of rational numbers with any number of generators (in contrast to the abelian case). More precisely, he proved that the desired analogues in which the intersection of any two non-trivial subgroups is infinite, can be constructed as a central extension of the free Burnside group , where , and is an odd number, using the infinite cyclic group as its centre. The paper also discusses other applications of the proposed generalization of Adian’s technique. In particular, the free groups of the variety defined by the identity and the Schur multipliers of the free Burnside groups for any odd are described. Bibliography: 14 titles.

A sufficient condition that a closed connected definite 4-manifold with infinite cyclic fundamental group is TOP-split is given. By this condition, it is shown that every closed connected definite smooth 4-manifold with infinite cyclic fundamental group is TOP-split. By combining with an earlier result, it is confirmed that every closed connected oriented smooth 4-manifold with infinite cyclic fundamental group is TOP-split. This also implies that every smooth sphere-knot in a closed simply connected smooth 4-manifold is topologically unknotted if the fundamental group of the complement is infinite cyclic.

This article is a revised version of the author's earlier paper on a TOP-splitting of a closed connected oriented 4-manifold with infinite cyclic fundamental group. We show that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if it is virtually TOP-split. As a consequence, we see that a closed connected oriented 4-manifold with infinite cyclic fundamental group is TOP-split if the intersection form is indefinite. This also implies that every closed connected oriented smooth spin 4-manifold with infinite cyclic fundamental group is TOP-split.

A scheme of construction of infinite groups, other than simple groups, free groups of infinite rank and the infinite cyclic group, which are isomorphic to all their non-trivial normal subgroups is presented. Some results about the automorphism groups of simple infinite groups are also obtained. In particular, it is proved that there is an infinite group G of any sufficiently large prime exponent p (or which is torsion-free) all of whose proper subgroups are cyclic, and such that the groups Aut G and Out G are isomorphic. The proofs use the technique of graded diagrams developed by A. Yu. Ol'shanskii. 1991 Mathematics Subject Classification: 20F05, 20F06.

Lattice-ordered loops and quasigroups

In the theory of one-relator groups, Magnus subgroups, which are free subgroups obtained by omitting a generator that occurs in the given relator, play an essential structural role. In a previous article, the author proved that if two distinct Magnus subgroups M and N of a one-relator group, with free bases S and T are given, then the intersection of M and N is either the free subgroup P generated by the intersection of S and T or the free product of P with an infinite cyclic group. The main result of this article is that if M and N are Magnus subgroups (not necessarily distinct) of a one-relator group G and g and h are elements of G, then either the intersection of gMg^{-1} and hNh^{-1} is cyclic (and possibly trivial), or gh^{-1} is an element of NM in which case the intersection is a conjugate of the intersection of M and N.

Let D = F 1 × F 2 × ⋯ × F n D = {F_1} \times {F_2} \times \cdots \times {F_n} be a direct product of n n free groups F 1 , F 2 , ⋯ , F n , α {F_1},{F_2}, \cdots ,{F_n},\alpha an automorphism of D D which leaves all but one of the noncyclic factors in D D pointwise fixed and T T an infinite cyclic group. Let D × α T D{ \times _\alpha }T be the semidirect product of D D and T T with respect to α \alpha . We prove that the Whitehead group of D × α T D{ \times _\alpha }T and the projective class group of the integral group ring Z ( D × α T ) Z(D{ \times _\alpha }T) are trivial. The second result implies that the projective class group of the integral group ring over the fundamental group of a surface is trivial.

Intersections of Magnus subgroups and embedding theorems for cyclically presented groups