scholarly journals Some quadratic equations in the free group of rank 2

2008 ◽  
Author(s):  
Daciberg Gonçalves ◽  
Elena Kudryavtseva ◽  
Heiner Zieschang
Keyword(s):  
Free Group ◽  
Quadratic Equations ◽  
Rank 2

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.

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 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.

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.

Certain maximal characteristic subgroups of the free group of rank 2

1997 ◽  
Vol 25 (4) ◽  
pp. 1047-1077 ◽  
Author(s):  
Derek Holt ◽  
Murray Macbeath
Keyword(s):  
Free Group ◽  
Rank 2

scholarly journals The automorphism group of the free group of rank 2 is a CAT(0) group

2010 ◽  
Vol 59 (2) ◽  
pp. 297-302 ◽  
Author(s):  
Adam Piggott ◽  
Kim Ruane ◽  
Genevieve Walsh
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Rank 2

scholarly journals A classification of the commutator subgroup of the group of a boundary link

1974 ◽  
Vol 18 (2) ◽  
pp. 216-221
Author(s):  
Bai Ching Chang
Keyword(s):  
Free Product ◽  
Free Group ◽  
Commutator Subgroup ◽  
Free Product With Amalgamation ◽  
Rank 2

In Neuwirth's book “Knot Groups” ([2]), the structure of the commutator subgroup of a knot is studied and characterized. Later Brown and Crowell refined Neuwith's result ([1], and we thus know that ifGis the groups of a knotK, then [G, G] is either free of rank 2g, wheregis the genus ofK, or a nontrivial free product with amalgamation on a free group of rank 2g, and may be written in the form, whereFis free of rank 2g, and the amalgamations are all proper and identical.

scholarly journals AUTOMORPHIC ORBITS IN FREE GROUPS: WORDS VERSUS SUBGROUPS

2010 ◽  
Vol 20 (04) ◽  
pp. 561-590 ◽  
Author(s):  
PEDRO V. SILVA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Finitely Generated ◽  
Primitive Elements ◽  
Group Of Automorphisms ◽  
Rational Subset ◽  
Rank 2 ◽  
Higher Rank

We show that the following problems are decidable in a rank 2 free group F2: Does a given finitely generated subgroup H contain primitive elements? And does H meet the orbit of a given word u under the action of G, the group of automorphisms of F2? Moreover, decidability subsists if we allow H to be a rational subset of F2, or alternatively if we restrict G to be a rational subset of the set of invertible substitutions (a.k.a. positive automorphisms). In higher rank, the following weaker problem is decidable: given a finitely generated subgroup H, a word u and an integer k, does H contain the image of u by some k-almost bounded automorphism? An automorphism is k-almost bounded if at most one of the letters has an image of length greater than k.

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