Magnus Embedding Research Articles

On embedding of partially commutative metabelian groups to matrix groups

‎The Magnus embedding of a free metabelian group induces the embedding of partially commutative metabelian group $S_Gamma$ in a group of matrices $M_Gamma$. Properties and the universal theory of the group $M_Gamma$ are studied.

Parallel identity testing for skew circuits with big powers and applications

Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labeled with powers [Formula: see text] for binary encoded numbers [Formula: see text]. It is shown that polynomial identity testing for powerful skew arithmetic circuits belongs to [Formula: see text], which generalizes a corresponding result for (standard) skew circuits. Two applications of this result are presented: (i) Equivalence of higher-dimensional straight-line programs can be tested in [Formula: see text]; this result is even new in the one-dimensional case, where the straight-line programs produce words. (ii) The compressed word problem (or circuit evaluation problem) for certain wreath products of finitely generated abelian groups belongs to [Formula: see text]. Using the Magnus embedding, it follows that the compressed word problem for a free metabelian group belongs to [Formula: see text].

The Conjugacy Problem in Free Solvable Groups and Wreath Products of Abelian Groups is in TC0

We show that the conjugacy problem in a wreath product A ≀ B is uniform-TC0-Turing-reducible to the conjugacy problem in the factors A and B and the power problem in B. If B is torsion free, the power problem in B can be replaced by the slightly weaker cyclic submonoid membership problem in B. Moreover, if A is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-AC0-many-one-reducible to the conjugacy problem in A ≀ B. Furthermore, under certain natural conditions, we give a uniform TC0 Turing reduction from the power problem in A ≀ B to the power problems of A and B. Together with our first result, this yields a uniform TC0 solution to the conjugacy problem in iterated wreath products of abelian groups – and, by the Magnus embedding, also in free solvable groups.

A remark on spherical equations in free metabelian groups

AbstractI. Lysenok and A. Ushakov proved that the Diophantine problem for spherical quadric equations in free metabelian groups is solvable. The present paper proves this result by using the Magnus embedding.

Magnus embedding and algorithmic properties of groups 𝐹/𝑁^{(𝑑)}

In this paper we further study properties of Magnus embedding, give a precise reducibility diagram for Dehn problems in groups of the form F / N ( d ) F/N^{(d)} , and provide a detailed answer to Problem 12.98 in The Kourovka notebook. We also show that most of the reductions are polynomial time reductions and can be used in practical computation.

Metric behaviour of the Magnus embedding

The classic Magnus embedding is a very effective tool in the study of abelian extensions of a finitely generated group \(G\), allowing us to see the extension as a subgroup of a wreath product of a free abelian group with \(G\). In particular, the embedding has proved to be useful when studying free solvable groups. An equivalent geometric definition of the Magnus embedding is constructed and it is used to show that it is \(2\)-bi-Lipschitz, with respect to an obvious choice of generating sets. This is then applied to obtain a non-zero lower bound on \(L_p\) compression exponents in free solvable groups.

THE MAGNUS EMBEDDING IS A QUASI-ISOMETRY

We show that the Magnus embedding, which embeds the free solvable group Sd, r of rank r and degree d into the wreath product ℤr ≀ Sd-1, r, is a quasi-isometry.

Polynomial time conjugacy in wreath products and free solvable groups

We prove that the complexity of the Conjugacy Problems for wreath products and for free solvable groups is decidable in polynomial time. For the wreath product AwrB, we must assume the decidability in polynomial time of the Conjugacy Problems for A and B and of the power problem in B. We obtain the result by making the algorithm for the Conjugacy Problem described in a paper of Matthews run in polynomial time. Using this result and properties of the Magnus embedding, we show that the Conjugacy and Conjugacy Search Problems in free solvable groups are computable in polynomial time.

The word and geodesic problems in free solvable groups

We study the computational complexity of the Word Problem (WP) in free solvable groups S r , d S_{r,d} , where r ≥ 2 r \geq 2 is the rank and d ≥ 2 d \geq 2 is the solvability class of the group. It is known that the Magnus embedding of S r , d S_{r,d} into matrices provides a polynomial time decision algorithm for WP in a fixed group S r , d S_{r,d} . Unfortunately, the degree of the polynomial grows together with d d , so the uniform algorithm is not polynomial in d d . In this paper we show that WP has time complexity O ( r n log 2 ⁡ n ) O(r n \log _2 n) in S r , 2 S_{r,2} , and O ( n 3 r d ) O(n^3 r d) in S r , d S_{r,d} for d ≥ 3 d \geq 3 . However, it turns out, that a seemingly close problem of computing the geodesic length of elements in S r , 2 S_{r,2} is N P NP -complete. We prove also that one can compute Fox derivatives of elements from S r , d S_{r,d} in time O ( n 3 r d ) O(n^3 r d) ; in particular, one can use efficiently the Magnus embedding in computations with free solvable groups. Our approach is based on such classical tools as the Magnus embedding and Fox calculus, as well as on relatively new geometric ideas; in particular, we establish a direct link between Fox derivatives and geometric flows on Cayley graphs.

The Magnus embedding as a special case of a homological construction

Let E be a group extension with Abelian kernel. Then it can be assigned an extension E′ of modules over the group ring of the quotient group. As a consequence, an embedding of the initial extension in some splitting extension arises. We prove that the celebrated Magnus embedding is a special case of this general construction.

The Magnus Embedding for Right-Symmetric Algebras

We construct a basis for the universal multiplicative enveloping algebra U(A) of a right-symmetric algebra A. We prove an analog of the Magnus embedding for right-symmetric algebras; i.e., we prove that a right-symmetric algebra A/R2, where A is a free right-symmetric algebra, is embedded into the algebra of triangular matrices of the second order.

FREE SOLVABLE P-ALGEBRAS

The construction of a free solvable P-algebra of finite degree k in the variety of all solvable P-algebras of degree at most k (k ≥ 1) has been given. Some properties of the same have been studied. The structure of the free solvable P-algebra has been viewed as a module over a ring with several objects. The Magnus embedding theorem associated with the Fox-derivative in a free group ring has been considered to prove properties associated with the partial (Fox) derivative in a free associative ring. Residual nilpotency and triviality of the center of a free metabelian P-algebra has been proved. Various properties of a homomorphism associated with a free metabelian P-algebra of finite rank have been studied. The non-embedding property of a free solvable P-algebra of degree k of higher rank in a lower rank has also been presented here.

Shmel’kin embeddings for abstract and profinite groups

The Magnus embedding is well known: given a group A=F/R, where F is a free group, the group F/[R, R] can be represented as a subgroup of a semidirect product AT, where T is an additive group of a free Z A-module. Shmel’kin genralized this construction and found an embedding for F/V(R), where V(R) is the verbal subgroup of R corresponding to a variety V. Later, he treated F as a free product of arbitrary groups, and on condition that R is contained in a Cartesian subgroup of the product, pointed out an embedding for F/V(R). Here, we combine both these Shmel’kin embeddings and weaken the condition on R, by assuming that F is a free product of groups Ai (ieI) and a free group X, and that its normal subgroup R has trivial intersection with each factor Ai. Subject to these conditions, an embedding for F/V(R) is found; we cell it the generalized Shmel’kin embedding. For the case where V is an Abelian variety of groups, a criterion is specified determining whether elements of AT belong to an embedded group F/V(R). Similar results are proved also for profinite groups.

Metabelian groups with a single defining relation and the Magnus embedding

Metabelian groups with a single defining relation and the Magnus embedding

Generators for the automorphism groups of free metabelian pro-p-groups

The automorphism groups of pro-finite groups attract attention of researchers. A number of inter- esting results and conjectures are contained in the articles by A. Lubotzky, D. Gildenhuys,. W. Herfort, and L. Ribes [1- 3]. At the conference on pro-finite groups in Oberwolfach (1990), A. Lubotzky for- mulated the problem: is the automorphism group of a pro-p-group of finite rank finitely generated? The generation is understood in the topological sense. Several well-known results on generators of the automorphism group of a finite rank free group in a nilpotent variety could be of value in describing generators of the automorphism group of a free pro-p-group. Different finite generating sets for the above-mentioned groups were given in [4-7]. Also, therein the principal obstacle to transferring finite generation to the case of pro-p-groups was delineated. The fact is that, for a free group of rank n _> 2 in a nilpotent Variety of class k > 3, the number of the automorphism group generators essentially depends on k in the descriptions available. In all probability this dependence is unavoidable. The present article is devoted to studying generators of the automorphism and IA-automorphism groups for free groups of finite rank in varieties of metabelian pro-p-groups. Preliminary information is collected in w Some profinite analogues of the Fox derivations are also defined there. They originated in the generalization, of the well-known Magnus embedding of a free metabelian group into a matrix group to the case of free metabelian pro-p-groups, that was obtained by V. N. Remeslennikov in [8]. In order to analyze the automorphism group of'a free metabelian pro-p-group, in w we present a profinite analogue of the well-known Bachmuth embedding [9] of the IA-automorphism group of a free metabelian group into the group of matrices over the ring of Laurent polynomials with integer coefficients. In our case, the group is represented by matrices over the ring of power series over p-adic integers. We also consider the case in which coefficients are chosen in a field with p elements. It is a free metabelian pro-p-group with the commutator subgroup of prime exponent that arises naturally in the situation in question. It turned out that the questions dealt with are related to the following: are the p-adic modules (or the modules over a field with p elements) that arise in studying the above series finitely generated (in the usual or generalized sense)? The key part of this paper, w is entirely devoted to the power series. Main results are duly stated and proved in w in the statements, p is prime. It is established that the IA-automorphism group of a free metabelian pro-p-group of rank n _> 9. with the commutator subgroup of prime exponent is infinitely generated (Theorem 1). This implies that the IA-automorphism groups of free metabelian groups and free pro-p-groups of rank n >_ 2 are infinitely generated (Corollary 1). Moreover, the number of generators of the IA-automorphism group of a free metabelian nilpotent group of class p and rank n _> 2 is proved to be infinitely increasing in k (Theorem 4). An analogous assertion holds without the assumption that the group is metabelian (Corollary 3). The automorphism group of a free rnetabelian pro-p-group of rank 2 with the commutator sub- group of prime exponent is finitely generated (Theorem 2). Its subgroup of automorphisms with determinant 1 (which is calculated according to the Bachmuth embedding) coincides with the inner Omsk. Translated from

On the quotient of a free group by the commutator of two normal subgroups

Let F be a noncyclic free group and let K and L be normal subgroups of F. In trying to describe the group F [K, L] in terms of F K and F L , it is important to understand the group A=K∩ F [K, L] . The main theorem gives an exact sequence into which the group A fits. The sequence reduces in special cases to Hopf's Formula for H 2( F K, Z ) and to the Magnus embedding or relation sequence embedding the relation module in a free module. The sequence is 0→H 2(S)⊕H 2(T)→A→P→Δ(C)⊗ zcZB→0 . where B= F KL , C= F K ∩L, Δ(C) denotes the augmentation ideal of C, P is a free ZB-module on a basis in one to one correspondence with a basis of F, and the sequence is of ZB-modules. As one application of this, we show that if each of K and L is the normal closure of a single element, then A is free abelian. A second application gives that if l>2 and F n , F ( n) are the nth terms of the lower central and derived series respectively of F, then F [F … F (f)] contains an infinite elementary abelian 2-group.

On the Magnus–Smelkin embedding

The generalization of the Magnus embedding [7] proved by Smelkin [9] may bestated as follows. Let L be a free group freely generated by the set xi(i∈I), and let R be a normal subgroup of L with G = L/R. If V is any variety of groups and ∏ is the V-freegroup with free generating set the symbols [g, xi] (g∈G, i∈I), then L/V(R) is embeddedin the semidirect product ∏ ⋊ G (where the action of G on ∏ is given by h · [g, xi] = [hg, xi], for h, g ∈ G).

On the Conjugacy Problem for F/R �

Let FIR be a finitely generated recursively presented group with the solvable conjugacy problem and the solvable power problem. The conjugacy problem for F/R' when F/R is torsion free has been solved by Remeslennikov and Sokolov (Algebra and Logic 9 (1970), 342-349). In this note we prove that F/R' has the solvable conjugacy problem even when F/R is not torsion free. This allows us, in particular, to conclude the solvability of the conjugacy problem for F/[Fl, FI] in terms of the conjugacy problem for F/Fn. Summary. If the conjugacy problem and the power problem for F/R is solvable, then the conjugacy problem for F/R' is solvable. Introduction. Let F/R be a finitely generated recursively presented group with the solvable conjugacy problem and the solvable power problem. The conjugacy problem for F/R' when F/R is torsion free has been solved by Remeslennikov and Sokolov [6]. In this note we are able to prove that F/R' has the solvable conjugacy problem even when F/R is not torsion free. This allows us, in particular, to conclude the solvability of the conjugacy problem for F/[Fn, Fn] in terms of the conjugacy problem for F/FI. As in [6] our tool is the Magnus embedding of F/R' into a group of 2 X 2 matrices and a result of Matthews [5] about the conjugacy problem for the restricted wreath product of two groups. Pre iminary results. Let F be a noncyclic free group on a finite set X and R a normal subgroup of F. Let Ql be the free left Z(F/R)-module with basis {X$jx E X}. Then every element of Q2 is of the form fZE fxX fx E Z(F/R). xX For w E F, we set Rw = w and consider the group M F/R Q2 O RIR) of 2 X 2 matrices of the form (w f 0O 1) w E F, f E ?2. It is easily verified that M is isomorphic to the restricted wreath product of a free abelian group of rank IXI by F/R. Thus, by a result of Matthews [5], we have Received by the editors October 6, 1980. 1980 Mathematics Subject Classification. Primary 20F10. lResearch supported by the Natural Sciences and Engineering Research Council Canada. i) 1982 American Mathematical Society 0002-9939/81/0000-1066/$02.25 149 This content downloaded from 157.55.39.111 on Wed, 03 Aug 2016 05:17:24 UTC All use subject to http://about.jstor.org/terms

EMBEDDING THEOREMS FOR PROFINITE GROUPS

Suppose that the profinite group G is an extension of A by H. In this paper the profinite subgroups of the topological group of continuous maps from H to A are investigated. The results obtained are used to prove topological analogues for profinite groups of the Frobenius and Magnus embedding theorems. Moreover, a sufficient condition is formulated for a pro-p-group that is an extension of an abelian group by a finitely presented group to be finitely presented, in the language of complete tensor products of abelian pro-p-groups; and this condition is used to prove that a finitely generated metabelian pro-p-group is a subgroup of a finitely presented metabelian pro-p-group.Bibliography: 15 titles.