scholarly journals The free group of rank 2 is a limit of Thompson’s group F

2010 ◽  
pp. 433-454 ◽  
Author(s):  
Matthew Brin
Keyword(s):  
Free Group ◽  
Thompson’S Group ◽  
Rank 2 ◽  
Thompson's Group F

Thompson’s Group F

2018 ◽  
pp. 37-66
Author(s):  
Marianna C. Bonanome ◽  
Margaret H. Dean ◽  
Judith Putnam Dean
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group 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.

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 ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

scholarly journals Random subgroups of Thompson’s group F

2010 ◽  
pp. 91-126 ◽  
Author(s):  
Sean Cleary ◽  
Murray Elder ◽  
Andrew Rechnitzer ◽  
Jennifer Taback
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group F

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.

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