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.

scholarly journals Recognizing powers in nilpotent groups and nilpotent images of free groups

2007 ◽  
Vol 83 (2) ◽  
pp. 149-156
Author(s):  
Gilbert Baumslag
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Factor Group ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Proper Power ◽  
Torsion Free

AbstractAn element in a free group is a proper power if and only if it is a proper power in every nilpotent factor group. Moreover there is an algorithm to decide if an element in a finitely generated torsion-free nilpotent group is a proper power.

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 On a Nonsingular Equation of Length 9 Over Torsion Free Groups

2019 ◽  
Vol 12 (2) ◽  
pp. 590-604
Author(s):  
M. Fazeel Anwar ◽  
Mairaj Bibi ◽  
Muhammad Saeed Akram
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Torsion Free Group ◽  
Torsion Free

In \cite{levin}, Levin conjectured that every equation is solvable over a torsion free group. In this paper we consider a nonsingular equation $g_{1}tg_{2}t g_{3}t g_{4} t g_{5} t g_{6} t^{-1} g_{7} t g_{8}t \\ g_{9}t^{-1} = 1$ of length $9$ and show that it is solvable over torsion free groups modulo some exceptional cases.

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

scholarly journals A FAST ALGORITHM FOR STALLINGS' FOLDING PROCESS

2006 ◽  
Vol 16 (06) ◽  
pp. 1031-1045 ◽  
Author(s):  
NICHOLAS W. M. TOUIKAN
Keyword(s):  
Free Group ◽  
Fast Algorithm ◽  
Free Groups ◽  
Membership Problem ◽  
Finitely Generated ◽  
Folding Process ◽  
Algorithmic Problems ◽  
The Given

Stalling's folding process is a key algorithm for solving algorithmic problems for finitely generated subgroups of free groups. Given a subgroup H = 〈J1,…,Jm〉 of a finitely generated nonabelian free group F = F(x1,…,xn) the folding porcess enables one, for example, to solve the membership problem or compute the index [F : H]. We show that for a fixed free group F and an arbitrary finitely generated subgroup H (as given above) we can perform the Stallings' folding process in time O(N log *(N)), where N is the sum of the word lengths of the given generators of H.

scholarly journals Finitely generated cyclic extensions of free groups are residually finite

1971 ◽  
Vol 5 (1) ◽  
pp. 87-94 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Free Group ◽  
Principal Ideal ◽  
Free Groups ◽  
Cyclic Extension ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Residually Finite ◽  
Ideal Domain ◽  
Direct Sum

We establish the result that a finitely generated cyclic extension of a free group is residually finite. This is done, in part, by making use of the fact that a finitely generated module over a principal ideal domain is a direct sum of cyclic modules.

scholarly journals ON RESTRICTING SUBSETS OF BASES IN RELATIVELY FREE GROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250030
Author(s):  
LUCAS SABALKA ◽  
DMYTRO SAVCHUK
Keyword(s):  
Solvable Group ◽  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Free Solvable Group ◽  
Relatively Free Group

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.

MALNORMALITY IS DECIDABLE IN FREE GROUPS

1999 ◽  
Vol 09 (06) ◽  
pp. 687-692 ◽  
Author(s):  
GILBERT BAUMSLAG ◽  
ALEXEI MYASNIKOV ◽  
VLADIMIR REMESLENNIKOV
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated

We prove here that there is an algorithm whereby one can decide whether or not any finitely generated subgroup of a finitely generated free group is malnormal.

scholarly journals Groups with many permutable subgroups

1990 ◽  
Vol 48 (3) ◽  
pp. 397-401 ◽  
Author(s):  
Mario Curzio ◽  
John Lennox ◽  
Akbar Rhemtulla ◽  
James Wiegold
Keyword(s):  
Free Groups ◽  
Finitely Generated ◽  
Soluble Groups ◽  
Cyclic Subgroups ◽  
Prove Theorem ◽  
Permutable Subgroups ◽  
Infinite Set ◽  
Torsion Free

AbstractWe consider the influence on a group G of the condition that every infinite set of cyclic subgroups of G contains a pair that permute and prove (Theorem 1) that finitely generated soluble groups with this condition are centre-by-finite, and (Theorem 2) that torsion free groups satisfying the condition are abelian.

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.

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