Orbit-blocking words in free groups

On the homology of free abelianized extensions of finite 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

We prove that there exists an algorithm which solves a conjugacy problem for finite subgroups in automorphism and outer automorphism groups of a free group of finite rank. Of independent interest is the construction of an algorithm of decomposing an arbitrary free-by-finite group into a fundamental group of a finite graph of finite groups, with the number of steps evaluated explicitly. In passing, we solve the conjugacy problem for finite subgroups in almost free groups. As a consequence, an algorithm is obtained computing generating sets for a group of fixed points in an arbitrary finite automorphism group of a free group of finite rank.

Automorphic orbits in free groups

Small conjugacy classes in the automorphism groups of relatively free groups

On the lower central quotients of the free groups of finite rank of a certain variety

We classify the groups that are the inverse limits of sequences of free groups of finite rank. Except free groups of finite rank, there exist four groups that are the inverse limits of sequences of free groups of finite rank.

The subject matter of the paper encompasses Abelian group theory, module theory, and topology. Hopfian groups appeared in combinatorial group theory, which appreciably determined characteristic features of the initial characterization of Hopfian groups. Methods and problems in combinatorial group theory served as tools for describing Hopfian groups and their connections with other classes of groups. In 1921 J. Nielsen published a paper [1] which improves on the research techniques for finitely generated subgroups of free groups developed in his previous works. Nielsen’s results imply an important consequence which says that a finitely generated free group cannot be isomorphic to its proper factor group; i.e., a homomorphic mapping of such a group onto itself always has a trivial kernel. Now this property, named after Heinz Hopf, is referred to as Hopficity. The reason for this is the following. In [2, 3], Hopf explored mappings of one two-dimensional variety into another. These mappings induce homomorphisms of the fundamental group of the first variety onto the fundamental group of its image. Using topological methods, Hopf showed that the groups mentioned cannot be isomorphic to any of their proper factor groups. The simplest case is one in which the fundamental group in question is free and has finite rank. Surprisingly, over a span of thirty years, nobody noticed that Nielsen had already given a purely algebraically proof of the fact that free groups of finite rank are Hopfian—in fact, ten years before Hopf posed his question. Hopfian groups (including Abelian ones) have been extensively studied by many algebraists (for a bibliography, see [4]). To some extent, the present paper can be regarded as a continuation of [4]. For the reader’s convenience, here we lay out some information concerning the notation and ∗Supported by the Russian Ministry of Education and Science, gov. contract No. 14.B37.21.0354.

We prove that non-abelian free groups of finite rank at least 3 or of countable rank are not [Formula: see text]-homogeneous. We answer three open questions from Kharlampovich, Myasnikov, and Sklinos regarding whether free groups, finitely generated elementary free groups, and non-abelian limit groups form special kinds of Fraïssé classes in which embeddings must preserve [Formula: see text]-formulas. We also provide interesting examples of countable non-finitely generated elementary free groups.

In this article we relate two different densities. Let $F_k$ be the free group of finite rank $k \ge 2$ and let $\alpha$ be the abelianization map from $F_k$ onto $ \mathbb{Z}^k$. We prove that if $S \subseteq \mathbb{Z}^k$ is invariant under the natural action of $SL(k, \mathbb{Z})$ then the asymptotic density of $S$ in $\mathbb Z^k$ and the annular density of its full preimage $\alpha^{-1}(S)$ in $F_k$ are equal. This implies, in particular, that for every integer $t\ge 1$, the annular density of the set of elements in $F_k$ that map to $t$-th powers of primitive elements in $\mathbb{Z}^k$ is equal to $\frac{1}{t^k\zeta(k)}$, where $\zeta$ is the Riemann zeta-function. An element $g$ of a group $G$ is called a \emph{test element} if every endomorphism of $G$ which fixes $g$ is an automorphism of $G$. As an application of the result above we prove that the annular density of the set of all test elements in the free group $F(a,b)$ of rank two is $1-\frac{6}{\pi^2}$. Equivalently, this shows that the union of all proper retracts in $F(a,b)$ has annular density $\frac{6}{\pi^2}$. Thus being a test element in $F(a,b)$ is an ``intermediate property'' in the sense that the probability of being a test element is strictly between $0$ and $1$.

The first section of this chapter contains algorithms about subgroups of finite index of an abstract free group of finite rank, e.g., how to decide whether an element of the free group belongs to a given subgroup of finite index or whether a subgroup of finite index is open in a certain topology. Given a prime number $p$ , a free abstract group $\varPhi$ (with an explicit basis $\mathbf{B}=\{b_{1}, \dots, b_{n}\}$ ) endowed with the pro- $p$ topology, and a finitely generated subgroup $H$ (whose generators are given in terms of $\mathbf{B}$ ), the second section of this chapter describes an algorithm to find a finite set of generators (written in terms of $\mathbf{B}$ ) of the closure of $H$ in that topology. The main result in that section presents a general approach to describing the closure of $H$ in a pro- $\mathcal{C}$ topology of $\varPhi$ (when $\mathcal{C}$ is an extension-closed pseudovariety of finite groups); one deduces from this theorem what are the main difficulties that arise when trying to make such a description algorithmic. The third part of the chapter contains several algorithms of interest in the theory of formal languages on a finite alphabet and in finite monoids; in particular, there is an algorithm that describes how to construct the kernel of a finite monoid, a problem posed by J. Rhodes.

Previous work showed that every pair of nontrivial Bernoulli shifts over a fixed free group are orbit equivalent. In this paper, we prove that if G_1 , G_2 are nonabelian free groups of finite rank then every nontrivial Bernoulli shift over G_1 is stably orbit equivalent to every nontrivial Bernoulli shift over G_2 . This answers a question of S. Popa.

Abstract.We will show that the inverse limit of finite rank free groups with surjective connecting homomorphism is isomorphic either to a finite rank free group or to a fixed universal group. In other words, any inverse system of finite rank free groups which is not equivalent to an eventually constant system has the universal group as its limit. This universal inverse limit is naturally isomorphic to the first shape group of the Hawaiian earring. We also give an example of a homomorphic image of the Hawaiian earring group which lies in the inverse limit of free groups but is neither a free group nor isomorphic to the Hawaiian earring group.

We answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed inFω. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ℵ1-homogeneous.

In this paper we investigate the special automata over finite rank free groups and estimate asymptotic characteristics of sets they accept‎. ‎We show how one can decompose an arbitrary regular subset of a finite rank free group into disjoint union of sets accepted by special automata or special monoids‎. ‎These automata allow us to compute explicitly generating functions‎, ‎$lambda-$measures and Cesaro measure of thick monoids‎. ‎Also we improve the asymptotic classification of regular subsets in free groups‎.