Whitehead groups of certain semidirect products of 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 = FlxF2x- xF" be a direct product of n free groups Ft, F2, , Fn, a an automorphism of D which leaves all but one of the noncyclic factors in D pointwise fixed and T an infinite cyclic group.Let D x T be the semidirect product of D and T with respect to a.We prove that the Whitehead group of D x T and the projective class group of the integral group ring Z(D x 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.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 K0Z(G).Let a be an automorphism of G and let F be an infinite cyclic group.Then we denote by G x x F the semidirect product of G and F with respect to a.Let M be a connected 2-dimensional manifold and trx(M) the fundamental group of M. If M is open, then ttx(M) is a free group so that K0Z(ttx(M)) is trivial by a theorem of Bass (cf.[I]).Next, if Misasphereor a projective plane, then trx(M)=0 or T2 (cyclic group of order 2) and so K0Z(ttx(Mj) =0 (cf.[7, p. 419]).Now, if Mis closed and is not a sphere or projective plane, then Farrell-Hsiang[4] have shown that trx(M) is just the semidirect product F xxT, where F is a free group.The purpose of this paper is to show that K0Z(F xx T)=0 and so the projective class group of the integral group ring over the fundamental group of a surface is always trivial.In fact, we prove:

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.

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].

The unsolvability of some algorithmic problems is proved for equations in free groups and semigroups, namely, some simple properties of the solutions of the equations are determined and the absence of an algorithm permitting the determination of whether an arbitrary equation in a free group or semigroup has a solution with the properties introduced is proved.

This paper attempts to study the irreducibility on complete prefix code (CPC-irreducibility) of a Markov shift over a free group, a topological mixing property first considered for shift spaces over free semigroups that induces chaotic behavior such as the existence of a dense set of periodic points. An example shows that the ({textsf{d}},{textsf{c}})-reduction, an effective algorithm of determination of CPC-irreducibility of Markov shifts over free semigroups (Ban et al. in J Stat Phys 177:1043–1062, 2019), fails for general Markov shifts over free groups. This paper reveals an algorithm for determining the CPC-irreducibility of Markov shifts over both free semigroups and groups. Furthermore, such an examination is finitely checkable, and an upper bound for the complexity of the algorithm is provided.

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.

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 . Let α be an automorphism of G and T an infinite cyclic group. Then we denote by G ×αT the semidirect product of G and T with respect to α. For undefined terminologies used in the paper, we refer to [3] and [7].

We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by terms of the lower central series of knot groups. Also, we study decomposition of virtual knot groups as semi direct product and free product with amalgamation. In particular, we prove that the groups of some virtual knots are extensions of finitely generated free groups by infinite cyclic groups.

Explicit Helfgott type growth in free products and in limit groups

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.

Writing certain commutators as products of cubes in free groups

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.

Let 〈x〉 be an infinite cyclic group and Ri〈x〉 its group ring over a ring (with identity) Ri, for i = l and 2. Let J(Ri) be the Jacobson radical of Ri. In this note we study the question of whether or not R1〈x〉≃R2〈x〉 implies R1≃R2. We prove that this is so if Zi the centre of Ri is semi-perfect and J(Zi〈x〉) = J(Zi〈)x〉 for i = l and 2. In particular, when Zi is perfect the second condition is satisfied and the isomorphism of group rings Ri〈x〉 implies the isomorphism of Ri.

Let $X$ be an infinite cyclic group. An example of two noncommutative nonisomorphic rings $R$, $S$ such that their group rings $RX$, $SX$ are isomorphic has been given in [1]. In the present note, we show that there also exist commutative nonisomorphic noetherian domains $A$, $B$ of Krull dimension 2 such that the group rings $AX$, $BX$ are isomorphic. That solves Problem 27 of [4] in the negative.