scholarly journals Subgroups of finitely presented metabelian groups

1973 ◽  
Vol 16 (1) ◽  
pp. 98-110 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented

In 1961 Graham Higman [1] proved that a finitely generated group is a subgroup of a finitely presented group if, and only if, it is recursively presented. Therefore a finitely generated metabelian group can be embedded in a finitely presented group.

Dehn functions of finitely presented metabelian groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wenhao Wang
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Dehn Function ◽  
Metabelian Groups ◽  
Dehn Functions ◽  
Finitely Presented

Abstract In this paper, we compute an upper bound for the Dehn function of a finitely presented metabelian group. In addition, we prove that the same upper bound works for the relative Dehn function of a finitely generated metabelian group. We also show that every wreath product of a free abelian group of finite rank with a finitely generated abelian group can be embedded into a metabelian group with exponential Dehn function.

Whitehead Graphs and the Σ1-Invariants of Infinite Groups

1998 ◽  
Vol 08 (01) ◽  
pp. 23-34 ◽  
Author(s):  
Susan Garner Garille ◽  
John Meier
Keyword(s):  
Local Structure ◽  
Universal Cover ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Group ◽  
Finite Presentation ◽  
Finitely Presented Group ◽  
Finitely Presented

Let G be a finitely generated group. The Bieri–Neumann–Strebel invariant Σ1(G) of G determines, among other things, the distribution of finitely generated subgroups N◃G with G/N abelian. This invariant can be quite difficult to compute. Given a finite presentation 〈S:R〉 for G, there is an algorithm, introduced by Brown and extended by Bieri and Strebel, which determines a space Σ(R) that is always contained in, and is sometimes equal to, Σ1(G). We refine this algorithm to one which involves the local structure of the universal cover of the standard 2-complex of a given presentation. Let Ψ(R) denote the space determined by this algorithm. We show that Σ(R) ⊆ Ψ ⊆ Σ1(G) for any finitely presented group G, and if G admits a staggered presentation, then Ψ = Σ1(G). By casting this algorithm in terms of connectivity properties of graphs, it is shown to be computationally feasible.

scholarly journals Groups with a quotient that contains the original group as a direct factor

1992 ◽  
Vol 45 (3) ◽  
pp. 513-520 ◽  
Author(s):  
Ron Hirshon ◽  
David Meier
Keyword(s):  
Free Group ◽  
Normal Subgroups ◽  
Direct Factor ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Generated Groups ◽  
Original Group ◽  
Hopfian Group ◽  
Finitely Presented

We prove that given a finitely generated group G with a homomorphism of G onto G × H, H non-trivial, or a finitely generated group G with a homomorphism of G onto G × G, we can always find normal subgroups N ≠ G such that G/N ≅ G/N × H or G/N ≅ G/N × G/N respectively. We also show that given a finitely presented non-Hopfian group U and a homomorphism φ of U onto U, which is not an isomorphism, we can always find a finitely presented group H ⊇ U and a finitely generated free group F such that φ induces a homomorphism of U * F onto (U * F) × H. Together with the results above this allows the construction of many examples of finitely generated groups G with G ≅ G × H where H is finitely presented. A finitely presented group G with a homomorphism of G onto G × G was first constructed by Baumslag and Miller. We use a slight generalisation of their method to obtain more examples of such groups.

scholarly journals Subgroups of finitely presented metabelian groups of finite rank

1976 ◽  
Vol 22 (4) ◽  
pp. 501-508 ◽  
Author(s):  
James Boler
Keyword(s):  
Finite Rank ◽  
Metabelian Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Presented

AbstractLet G be a finitely generated metabelian group whose derived group G′ has finite rank. It is shown that G can be embedded in a finitely presented metabelian group H with H′ of finite rank.

scholarly journals An algebraic characterization of groups with soluble word problem

1974 ◽  
Vol 18 (1) ◽  
pp. 41-53 ◽  
Author(s):  
William W. Boone ◽  
Graham Higman
Keyword(s):  
Word Problem ◽  
Focal Point ◽  
Algebraic Characterization ◽  
Necessary And Sufficient Condition ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Necessary And Sufficient ◽  
Finitely Presented

The following theorem is the focal point of the present paper. It stipulates an algebraic condition equivalent, in any finitely generated group, to the solubility of the word problem.THEOREM I. A necessary and sufficient condition that a finitely generated group G have a soluble word problem is that there exist a simple group H, and a finitely presented group K, such that G is a subgroup of H, and H is a subgroup of K.

scholarly journals Calibrating word problems of groups via the complexity of equivalence relations

2016 ◽  
Vol 28 (3) ◽  
pp. 457-471 ◽  
Author(s):  
ANDRÉ NIES ◽  
ANDREA SORBI
Keyword(s):  
Word Problem ◽  
Equivalence Relation ◽  
Word Problems ◽  
Truth Table ◽  
Equivalence Relations ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Presented Group ◽  
Finitely Presented ◽  
Computably Enumerable

(1) There is a finitely presented group with a word problem which is a uniformly effectively inseparable equivalence relation. (2) There is a finitely generated group of computable permutations with a word problem which is a universal co-computably enumerable equivalence relation. (3) Each c.e. truth-table degree contains the word problem of a finitely generated group of computable permutations.

A finitely generated, infinitely related group with trivial multiplicator

1971 ◽  
Vol 5 (1) ◽  
pp. 131-136 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Related Group ◽  
Finitely Related

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

On the profinite topology on solvable groups

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Khadijeh Alibabaei
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Solvable Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Finitely Generated Group ◽  
Hnn Extension ◽  
Profinite Topology

AbstractWe show that the wreath product of a finitely generated abelian group with a polycyclic group is a LERF group. This theorem yields as a corollary that finitely generated free metabelian groups are LERF, a result due to Coulbois. We also show that a free solvable group of class 3 and rank at least 2 does not contain a strictly ascending HNN-extension of a finitely generated group. Since such groups are known not to be LERF, this settles, in the negative, a question of J. O. Button.

scholarly journals ISOPERIMETRIC FUNCTIONS OF GROUPS ACTING ON Lδ-SPACES

2007 ◽  
Vol 49 (1) ◽  
pp. 23-28
Author(s):  
JON CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Metric Space ◽  
Finitely Generated ◽  
Isoperimetric Function ◽  
Finitely Generated Group ◽  
Isoperimetric Functions ◽  
Finitely Presented

Abstract.A finitely generated group acting properly, cocompactly, and by isometries on an Lδ-metric space is finitely presented and has a sub-cubic isoperimetric function.

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