scholarly journals Non-∀-homogeneity in free groups

2020 ◽  
pp. 2050019
Author(s):  
Olga Kharlampovich ◽  
Christopher Natoli
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Finitely Generated ◽  
Limit Groups ◽  
Open Questions ◽  
Countable Rank

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.

On Torsion-Free Groups of Finite Rank

1984 ◽  
Vol 36 (6) ◽  
pp. 1067-1080 ◽  
Author(s):  
David Meier ◽  
Akbar Rhemtulla
Keyword(s):  
Equivalence Class ◽  
Equivalence Relation ◽  
Finite Rank ◽  
Maximal Element ◽  
Free Groups ◽  
Solvable Groups ◽  
Finitely Generated ◽  
Isolated Subgroup ◽  
Image Position ◽  
Torsion Free

This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the setWe say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

scholarly journals The Herzog–Schönheim conjecture for finitely generated groups

2019 ◽  
Vol 29 (06) ◽  
pp. 1083-1112 ◽  
Author(s):  
Fabienne Chouraqui
Keyword(s):  
Metric Space ◽  
Finite Rank ◽  
Free Groups ◽  
Sufficient Conditions ◽  
Combinatorial Approach ◽  
Finitely Generated ◽  
Covering Spaces ◽  
Finitely Generated Groups

Let [Formula: see text] be a group and [Formula: see text] be subgroups of [Formula: see text] of indices [Formula: see text], respectively. In 1974, Herzog and Schönheim conjectured that if [Formula: see text], [Formula: see text], is a coset partition of [Formula: see text], then [Formula: see text] cannot be distinct. We consider the Herzog–Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define [Formula: see text] the space of coset partitions of [Formula: see text] and show [Formula: see text] is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood [Formula: see text] in [Formula: see text] such that all the partitions in [Formula: see text] satisfy also the conjecture.

scholarly journals Detecting conjugacy stability of subgroups in certain classes of groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Isabel Fernández Martínez ◽  
Denis Serbin
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Hyperbolic Groups ◽  
Effective Procedure ◽  
Finitely Generated ◽  
Limit Groups ◽  
Torsion Free ◽  
Effective Procedures

Abstract In this paper, we consider the conjugacy stability property of subgroups and provide effective procedures to solve the problem in several classes of groups. In particular, we start with free groups, that is, we give an effective procedure to find out if a finitely generated subgroup of a free group is conjugacy stable. Then we further generalize this result to quasi-convex subgroups of torsion-free hyperbolic groups and finitely generated subgroups of limit groups.

Finitely generated subgroups of free groups as formal languages and their cogrowth

2021 ◽  
Vol volume 13, issue 2 ◽  
Author(s):  
Arman Darbinyan ◽  
Rostislav Grigorchuk ◽  
Asif Shaikh
Keyword(s):  
Free Group ◽  
Finite Automaton ◽  
Finite Rank ◽  
Regular Language ◽  
Free Groups ◽  
Formal Languages ◽  
Finitely Generated ◽  
Deterministic Finite Automaton ◽  
Method Of Calculation ◽  
Reduced Words

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

scholarly journals Folding-like techniques for CAT(0) cube complexes

2021 ◽  
pp. 1-12
Author(s):  
MICHAEL BEN–ZVI ◽  
ROBERT KROPHOLLER ◽  
RYLEE ALANZA LYMAN
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Coxeter Groups ◽  
Free Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Seminal Paper ◽  
Finite Graphs ◽  
Cube Complexes ◽  
Positively Curved

Abstract In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings’s methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani–Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

scholarly journals Soluble groups isomorphic to their non-nilpotent subgroups

1999 ◽  
Vol 67 (3) ◽  
pp. 399-411 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Rank ◽  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Torsion Free

AbstractA group G belongs to the class W if G has non-nilpotent proper subgroups and is isomorphic to all of them. The main objects of study are the soluble groups in W that are not finitely generated. It is proved that there are no torsion-free groups of this sort, and a reasonable classification is given in the finite rank case.

scholarly journals Finitely generated subgroups of free groups as formal languages and their cogrowth

2021 ◽  
Vol volume 13, issue 2 ◽  
Author(s):  
Arman Darbinyan ◽  
Rostislav Grigorchuk ◽  
Asif Shaikh
Keyword(s):  
Free Group ◽  
Finite Automaton ◽  
Finite Rank ◽  
Regular Language ◽  
Free Groups ◽  
Formal Languages ◽  
Finitely Generated ◽  
Deterministic Finite Automaton ◽  
Method Of Calculation ◽  
Reduced Words

For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the minimal deterministic finite automaton that recognizes $L_H$. Then we characterize the f.g. subgroups $H$ for which $L_H$ is irreducible and for such groups explicitly construct ergodic automaton that recognizes $L_H$. This construction gives us an efficient way to compute the cogrowth series $L_H(z)$ of $H$ and entropy of $L_H$. Several examples illustrate the method and a comparison is made with the method of calculation of $L_H(z)$ based on the use of Nielsen system of generators of $H$.

scholarly journals FULLY RESIDUALLY FREE GROUPS AND GRAPHS LABELED BY INFINITE WORDS

2006 ◽  
Vol 16 (04) ◽  
pp. 689-737 ◽  
Author(s):  
ALEXEI G. MYASNIKOV ◽  
VLADIMIR N. REMESLENNIKOV ◽  
DENIS E. SERBIN
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Limit Groups ◽  
Infinite Words ◽  
Integer Coefficients ◽  
Ring Of Polynomials

Let F = F(X) be a free group with basis X and ℤ[t] be a ring of polynomials with integer coefficients in t. In this paper we develop a theory of (ℤ[t],X)-graphs — a powerful tool in studying finitely generated fully residually free (limit) groups. This theory is based on the Kharlampovich–Myasnikov characterization of finitely generated fully residually free groups as subgroups of the Lyndon's group Fℤ[t], the author's representation of elements of Fℤ[t] by infinite (ℤ[t],X)-words, and Stallings folding method for subgroups of free groups. As an application, we solve the membership problem for finitely generated subgroups of Fℤ[t], as well as for finitely generated fully residually free groups.

scholarly journals Quantitative residual properties of Γ-limit groups

2014 ◽  
Vol 24 (02) ◽  
pp. 207-231
Author(s):  
Brent B. Solie
Keyword(s):  
Hyperbolic Group ◽  
Finitely Generated ◽  
Limit Group ◽  
Limit Groups ◽  
Residual Properties

Let Γ be a fixed hyperbolic group. The Γ-limit groups of Sela are exactly the finitely generated, fully residually Γ groups. We introduce a new invariant of Γ-limit groups called Γ-discriminating complexity. We further show that the Γ-discriminating complexity of any Γ-limit group is asymptotically dominated by a polynomial.

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