Pavel Shumyatsky
Let w be a multilinear commutator and n a positive integer. Suppose that G is a residually finite group in which every product of at most 896 w-values has order dividing n. Then the verbal subgroup w(G) is locally finite.
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In this paper we study the residual nilpotence of groups defined by basic commutators. We prove that the so-called Hydra groups as well as certain of their generalizations and quotients are, in the main, residually torsion-free nilpotent. By way of contrast we give an example of a group defined by two basic commutators which is not residually torsion-free nilpotent.
Let $m,n$ be positive integers. Suppose that $G$ is a residually finite group in which for every element $x \in G$ there exists a positive integer $q=q(x) \leqslant m$ such that $x^q$ is $n$-Engel. We show that $G$ is locally virtually nilpotent. Further, let $w$ be a multilinear commutator and $G$ a residually finite group in which for every product of at most $896$ $w$-values $x$ there exists a positive integer $q=q(x)$ dividing $m$ such that $x^q$ is $n$-Engel. Then $w(G)$ is locally virtually nilpotent.
Given a group-word w and a group G, the verbal subgroup w(G) is the one generated by all w-values in G. The word w is called concise if w(G) is finite whenever the set of w-values in G is finite. It is an open question whether every word is concise in residually finite groups. Let w=w(x_1,..,x_k) be a multilinear commutator word, n a positive integer and q a prime power. In the present article we show that the word [w^q,_n y] is concise in residually finite groups while the word [w,_n y] is boundedly concise in residually finite groups.
Let $q$ be a prime. Let $G$ be a residually finite group satisfying an identity. Suppose that for every $x \in G$ there exists a $q$-power $m=m(x)$ such that the element $x^m$ is a bounded Engel element. We prove that $G$ is locally virtually nilpotent. Further, let $d,n$ be positive integers and $w$ a non-commutator word. Assume that $G$ is a $d$-generator residually finite group in which all $w$-values are $n$-Engel. We show that the verbal subgroup $w(G)$ has $\{d,n,w\}$-bounded nilpotency class.
A group-word w is called concise if whenever the set of w-values in a group G is finite it always follows that the verbal subgroup w(G) is finite. More generally, a word w is said to be concise in a class of groups X if whenever the set of w-values is finite for a group $G\in X$, it always follows that w(G) is finite. P. Hall asked whether every word is concise. Due to Ivanov the answer to this problem is known to be negative. Dan Segal asked whether every word is concise in the class of residually finite groups. In this direction we prove that if w is a multilinear commutator and q is a prime…
Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a residually finite group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent. We also examine the question whether this is true in the case where G is locally graded rather than residually finite. We answer the question affirmatively in the case where m = 1. Moreover, we show that if u is a non-commutator word and G is a locally graded group in which all u-values are n-Engel, then the verbal subgroup u(G) is locally nilpotent.
A group G has restricted centralizers if for each g in G the centralizer C_G(g) either is finite or has finite index in G. A theorem of Shalev states that a profinite group with restricted centralizers is abelian-by-finite. In the present article we handle profinite groups with restricted centralizers of word-values. We show that if w is a multilinear commutator word and G a profinite group with restricted centralizers of w-values, then the verbal subgroup w(G) is abelian-by-finite.
Let $m,n$ be positive integers and $w$ a multilinear commutator word. Assume that $G$ is a finite group having subgroups $G_1,\ldots,G_m$ whose union contains all $w$-values in $G$. Assume further that all elements of the subgroups $G_1,\ldots,G_m$ are $n$-Engel in $G$. It is shown that the verbal subgroup $w(G)$ is $s$-Engel for some $\{m,n,w\}$-bounded number $s$.
Let $w$ be a multilinear commutator word, that is, a commutator of weight $n$ in $n$ different group variables. It is proved that if $G$ is a profinite group in which all pronilpotent subgroups generated by $w$-values are periodic, then the verbal subgroup $w(G)$ is locally finite.
Let $A$ be a residually finite dimensional algebra (not necessarily associative) over a field $k$. Suppose first that $k$ is algebraically closed. We show that if $A$ satisfies a homogeneous almost identity $Q$, then $A$ has an ideal of finite codimension satisfying the identity $Q$. Using well known results of Zelmanov, we conclude that, if a residually finite dimensional Lie algebra $L$ over $k$ is almost $d$-Engel, then $L$ has a nilpotent (resp. locally nilpotent) ideal of finite codimension if char $k=0$ (resp. char $k > 0$). Next, suppose that $k$ is finite (so $A$ is residually finite).…
Let w be a group-word. Suppose that the set of all w-values in a profinite group G is contained in a union of countably many cosets of subgroups. We are concerned with the question to what extent the structure of the verbal subgroup w(G) depends on the properties of the subgroups. We prove the following theorem. Let C be a class of groups closed under taking subgroups, quotients, and such that in any group the product of finitely many normal C-subgroups is again a C-subgroup. If w is a multilinear commutator and G is a profinite group such that the set of all w-values is contained in a union o…
A BFC-group is a group in which all conjugacy classes are finite with bounded size. In 1954 B. H. Neumann discovered that if G is a BFC-group then the derived group G' is finite. Let w=w(x_1,\dots,x_n) be a multilinear commutator. We study groups in which the conjugacy classes containing w-values are finite of bounded order. Let G be a group and let w(G) be the verbal subgroup of G generated by all w-values. We prove that if x^G has size at most m for every w-value x, then the derived subgroup of w(G) is finite of order bounded by a function of m and n. If x^{w(G)} has size at most m for every…
Cryptographic multilinear maps using pro-p groups
本文研究基于幂零群多线性换位子映射的非交互式密钥交换协议，分析其在有限 p-群上的安全性，并探索 pro-p 群作为替代平台。
We prove that if a group possesses a deficiency 1 presentation where one of the relators is a commutator then it is the integers times the integers, is large, or is as far as possible from being residually finite. Then we use this to show that a mapping torus of an endomorphism of a finitely generated free group is large if it contains the integers times the integers as a subgroup of infinite index, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We give applications to f…
Commutator relators of one-relator groups do not force Hopficity, residual finiteness, or automaticity
本文构造了一族单关系群，其关系为交换子，每个都有Baumslitar子群作为retract，从而证明了这些群不满足霍菲性、残余有限性和自动性。
Let m, n be positive integers, v a multilinear commutator word and w = v^m. We prove that if G is a locally graded group in which all w-values are n-Engel, then the verbal subgroup w(G) is locally nilpotent.
Let w be a group word. It is conjectured that if w has only countably many values in a profinite group G, then the verbal subgroup w(G) is finite. In the present paper we confirm the conjecture in the cases where w is a multilinear commutator word, or the word x^2, or the word [x^2,y].
Nonsoluble Length Of Finite Groups with Commutators of Small Order
本文证明，当 p 为奇素数且多重线性换位子 w 的取值阶一致有界时，Sylow p-子群满足该条件的有限群的非-p-可解长度存在统一上界。
Let m,n be positive integers, v a multilinear commutator word and w=v^m. We prove that if G is an orderable group in which all w-values are n-Engel, then the verbal subgroup v(G) is locally nilpotent. We also show that in the particular case where v=x the group G is nilpotent (rather than merely locally nilpotent).
Varieties of groups and the problem on conciseness of words
本文证明字 $w$ 在剩余有限群中的简洁性等价于在 virtually pro-$p$ 群中，并由此将 $w^q$ 和 $[w^q,_{n} y]$ 的简洁性推广至正整数 $q$。
Let m, n be positive integers, v a multilinear commutator word and w = v^m. Denote by v(G) and w(G) the verbal subgroups of a group G corresponding to v and w, respectively. We prove that the class of all groups G in which the w-values are n-Engel and w(G) is locally nilpotent is a variety (Theorem A). Further, we show that in the case where m is a prime-power the class of all groups G in which the w-values are n-Engel and v(G) has an ascending normal series whose quotients are either locally soluble or locally finite is a variety (Theorem B). We examine the question whether the latter result …
Groups with finitely many long commutators of maximal order
本文研究仅含有限多个极大阶长交换子的群，证明剩余有限群含有限指数子群使$\gamma_\ell(M)=1$，有限生成时$\gamma_\ell(G)$有限；还给出$r$-生成群中$\gamma_\ell(G)$阶数和子群指数的有界性条件。
Let w be a multilinear commutator word. We prove that if e is a positive integer and G is a finite group in which any nilpotent subgroup generated by w-values has exponent dividing e then the exponent of the corresponding verbal subgroup w(G) is bounded in terms of e and w only.
Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.
Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a prescribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.
If an outer (multilinear) commutator identity holds in a large subgroup of a group, then it holds also in a large characteristic subgroup. Similar assertions are valid for algebras and their ideals or subspaces. Varying the meaning of the word "large", we obtain many interesting facts. These results cannot be extended to arbitrary (non-multilinear) identities. As an application, we give a sharp estimate for the `virtual derived length' of (virtually solvable)-by-(virtually solvable) groups.
We show that if $w$ is a multilinear commutator word and $G$ a finite group in which every metanilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of the verbal subgroup $w(G)$ is bounded in terms of $r$ and $w$ only. In the case where $G$ is soluble we obtain a better result -- if $G$ is a finite soluble group in which every nilpotent subgroup generated by $w$-values is of rank at most $r$, then the rank of $w(G)$ is at most $r+1$.
We prove the following results. Let w be a multilinear commutator word. If G is a profinite group in which all w-values are contained in a union of countably many periodic subgroups, then the verbal subgroup w(G) is locally finite. If G is a profinite group in which all w-values are contained in a union of countably many subgroups of finite rank, then the verbal subgroup w(G) has finite rank as well. As a by-product of the techniques developed in the paper we also prove that if G is a virtually soluble profinite group in which all w-values have finite order, then w(G) is locally finite and has…
We present an algorithm which, given any finite presentation of a group as input, will terminate with answer yes if and only if the group is large. We use this to prove that a mapping torus of a finitely generated free group automorphism is large if it contains the integers times the integers as a subgroup of infinite index. We then extend this result to mapping tori of finitely generated free group endomorphisms, as well as showing that such a group is large if it contains a Baumslag-Solitar group of infinite index and has a finite index subgroup with first Betti number at least 2. We also sh…
A nilpotency criterion for some verbal subgroups
本文定义了单换位字，并证明了有限群中对应言子群的幂零性等价于互素阶w-值满足阶乘性条件，同时将该结论推广至w-值集合有限的剩余有限群。
Gate lattices and the stabilized automorphism group
该研究引入门格刻画剩余有限群上具最终填充性质的有限型子移位稳定自同构群的单整性，证明其整群为门格生成群的换位子群，且一维混合情形下稳定惰性自同构群是单群。
Residual Finiteness Growth in Two-Step Nilpotent Groups
本文研究两步幂零群的剩余有限性增长，改进其多对数上界并证明仅依赖复Mal'cev完备化，对低维换位子子群给出精确界并猜想一般成立。
On groups with BFC-covered word values
本文研究群的(n,s)-覆盖条件，证明满足该条件的群存在指数和导出长度(或幂零类)均有界的可解(或幂零)子群。
Probabilistic nilpotence in infinite groups
本文研究了无限群的 k 步 nilpotence 度，证明了 finitely 生成群的 nilpotence 度与 virtually k 步 nilpotent 性的关系，并推广了有限群和有限生成群的结果。
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