scholarly journals On the “largeness” of one-relator groups

1986 ◽  
Vol 29 (2) ◽  
pp. 263-269
Author(s):  
M. Edjvet
Keyword(s):  
Free Group ◽  
Finite Index ◽  
One Relator Groups ◽  
Rank 2

If G is a one-relator group on at least 3 generators, or is a one-relator group with torsion on at least 2 generators, then it follows from results in [1] and [6] that G has a subgroup of finite index which can be mapped homomorphically onto F2, the free group of rank 2. In the language of [2], G is equally as large as F2, written G⋍F2.

scholarly journals The SQ-universality of some finitely presented groups

1973 ◽  
Vol 16 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Peter M. Neumann
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Countable Group ◽  
Factor Group ◽  
Finitely Presented Groups ◽  
Rank 2 ◽  
Finitely Presented

Following a suggestion of G. Higman we say that the group G is SQ-universal if every countable group is embeddable in some factor group of G. It is a well-known theorem of G. Higman, B. H. Neumann and Hanna Neumann that the free group of rank 2 is sq-universal in this sense. Several different proofs are now available (see, for example, [1] or [9]). It is my intention to prove the LEmma. If H is a subgroup of finite index in a group G, then G is SQ-universal if and only if H is SQ-universal.

scholarly journals The isomorphism problem for a class of para-free groups

1997 ◽  
Vol 40 (3) ◽  
pp. 541-549 ◽  
Author(s):  
Benjamin Fine ◽  
Gerhard Rosenberger ◽  
Michael Stille
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Special Interest ◽  
Isomorphism Problem ◽  
Combinatorial Group Theory ◽  
History Of ◽  
One Relator Groups ◽  
Rank 2 ◽  
Combinatorial Group ◽  
Torsion Free

In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k ∈ ℕ, showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i ∈ ℕ is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 ≇ G1, 1 for all i > 1 and if i, k are primes then Gi, 1 ≅ Gk, 1 if and only if i = k.

Subgroups of Finite Index in Free Groups

1949 ◽  
Vol 1 (2) ◽  
pp. 187-190 ◽  
Author(s):  
Marshall Hall
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Recursion Formula ◽  
Independent Interest ◽  
Finiteness Conditions ◽  
Total Length ◽  
Free Generators

This paper has as its chief aim the establishment of two formulae associated with subgroups of finite index in free groups. The first of these (Theorem 3.1) gives an expression for the total length of the free generators of a subgroup U of the free group Fr with r generators. The second (Theorem 5.2) gives a recursion formula for calculating the number of distinct subgroups of index n in Fr.Of some independent interest are two theorems used which do not involve any finiteness conditions. These are concerned with ways of determining a subgroup U of F.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

scholarly journals Centralisers of Dehn twist automorphisms of free groups

2015 ◽  
Vol 159 (1) ◽  
pp. 89-114 ◽  
Author(s):  
MORITZ RODENHAUSEN ◽  
RICHARD D. WADE
Keyword(s):  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Dehn Twist ◽  
Classifying Space ◽  
Dehn Twists ◽  
Automorphisms Of Free Groups

AbstractWe refine Cohen and Lustig's description of centralisers of Dehn twists of free groups. We show that the centraliser of a Dehn twist of a free group has a subgroup of finite index that has a finite classifying space. We describe an algorithm to find a presentation of the centraliser. We use this algorithm to give an explicit presentation for the centraliser of a Nielsen automorphism in Aut(Fn). This gives restrictions to actions of Aut(Fn) on CAT(0) spaces.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

1979 ◽  
Vol 31 (6) ◽  
pp. 1329-1338 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Products

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

On the Invariance of a Quotient Group of the Center of F/[R, R]

1969 ◽  
Vol 12 (5) ◽  
pp. 653-660 ◽  
Author(s):  
Trueman MacHenry
Keyword(s):  
Free Group ◽  
Quotient Group ◽  
Rank 2

Let F be a free group of rank ⩾ 2, let F/R ≅ π, and let F0 = F/[R, R]. Auslander and Lyndon showed that the center of Fo is a subgroup of R/[R, R] = Ro, and that it is non-trivial if and only if π is finite [1, corollary 1.3 and theorem 2]. In this paper it will be shown that there is a canonically defined (and not always trivial) quotient group of the center of F which depends only on π.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

On 2-Symmetric Words and Verbal Subgroup of Metabelian Product of Abelian Groups

2006 ◽  
Vol 13 (03) ◽  
pp. 535-540
Author(s):  
Jiangmin Pan
Keyword(s):  
Free Group ◽  
Abelian Groups ◽  
Verbal Subgroup ◽  
Metabelian Product ◽  
Rank 2 ◽  
Torsion Groups

Let F be the free group of rank 2 with basis {x, y}, and G a metabelian product of some non-trivial abelian groups. If not all the factors of G are torsion groups, it is proved that the verbal subgroup of G in F equals F″. Moreover, all the 2-symmetric words of G are determined by using left Fox derivatives. In addition, we provide an example to illustrate that if all the factors of G are torsion groups, the above results need not be true.

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