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.

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.

Acylindrical hyperbolicity, non-simplicity and SQ\hbox{-}universality of groups splitting over ℤ

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Jack O. Button
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Group

AbstractWe show, using acylindrical hyperbolicity, that a finitely generated group splitting over

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

ALGEBRO-GEOMETRIC INVARIANTS OF GROUPS (THE DIMENSION SEQUENCE OF A REPRESENTATION VARIETY)

2011 ◽  
Vol 21 (04) ◽  
pp. 595-614 ◽  
Author(s):  
S. LIRIANO ◽  
S. MAJEWICZ
Keyword(s):  
Algebraic Group ◽  
Free Group ◽  
Algebraic Variety ◽  
Commutator Subgroup ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Geometric Invariants ◽  
Torus Knot ◽  
Irreducible Components ◽  
Irreducible Variety

If G is a finitely generated group and A is an algebraic group, then RA(G) = Hom (G, A) is an algebraic variety. Define the "dimension sequence" of G over A as Pd(RA(G)) = (Nd(RA(G)), …, N0(RA(G))), where Ni(RA(G)) is the number of irreducible components of RA(G) of dimension i (0 ≤ i ≤ d) and d = Dim (RA(G)). We use this invariant in the study of groups and deduce various results. For instance, we prove the following: Theorem A.Let w be a nontrivial word in the commutator subgroup ofFn = 〈x1, …, xn〉, and letG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety andV-1 = {ρ | ρ ∈ RSL(2, ℂ)(Fn), ρ(w) = -I} ≠ ∅, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). Theorem B.Let w be a nontrivial word in the free group on{x1, …, xn}with even exponent sum on each generator and exponent sum not equal to zero on at least one generator. SupposeG = 〈x1, …, xn; w = 1〉. IfRSL(2, ℂ)(G)is an irreducible variety, thenPd(RSL(2, ℂ)(G)) ≠ Pd(RPSL(2, ℂ)(G)). We also show that if G = 〈x1, . ., xn, y; W = yp〉, where p ≥ 1 and W is a word in Fn = 〈x1, …, xn〉, and A = PSL(2, ℂ), then Dim (RA(G)) = Max {3n, Dim (RA(G′)) +2 } ≤ 3n + 1 for G′ = 〈x1, …, xn; W = 1〉. Another one of our results is that if G is a torus knot group with presentation 〈x, y; xp = yt〉 then Pd(RSL(2, ℂ)(G))≠Pd(RPSL(2, ℂ)(G)).

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

Numerical upper bounds on growth of automaton groups

2021 ◽  
pp. 1-33
Author(s):  
Jérémie Brieussel ◽  
Thibault Godin ◽  
Bijan Mohammadi
Keyword(s):  
Group Theory ◽  
Upper Bounds ◽  
Geometric Invariant ◽  
Finitely Generated ◽  
Grigorchuk Group ◽  
Finitely Generated Group ◽  
Intermediate Growth ◽  
Mealy Automata ◽  
Optimal Bounds ◽  
Algorithmic Procedure

The growth of a finitely generated group is an important geometric invariant which has been studied for decades. It can be either polynomial, for a well-understood class of groups, or exponential, for most groups studied by geometers, or intermediate, that is between polynomial and exponential. Despite recent spectacular progresses, the class of groups with intermediate growth remains largely mysterious. Many examples of such groups are constructed using Mealy automata. The aim of this paper is to give an algorithmic procedure to study the growth of such automaton groups, and more precisely to provide numerical upper bounds on their exponents. Our functions retrieve known optimal bounds on the famous first Grigorchuk group. They also improve known upper bounds on other automaton groups and permitted us to discover several new examples of automaton groups of intermediate growth. All the algorithms described are implemented in GAP, a language dedicated to computational group theory.

On the existence of ergodic automorphisms in ergodic ℤd-actions on compact groups

2009 ◽  
Vol 30 (6) ◽  
pp. 1803-1816 ◽  
Author(s):  
C. R. E. RAJA
Keyword(s):  
London Math ◽  
Abelian Groups ◽  
Chain Condition ◽  
Compact Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Metrizable Group ◽  
Compact Abelian Groups ◽  
Descending Chain Condition

AbstractLet K be a compact metrizable group and Γ be a finitely generated group of commuting automorphisms of K. We show that ergodicity of Γ implies Γ contains ergodic automorphisms if center of the action, Z(Γ)={α∈Aut(K)∣α commutes with elements of Γ} has descending chain condition. To explain that the condition on the center of the action is not restrictive, we discuss certain abelian groups which, in particular, provide new proofs to the theorems of Berend [Ergodic semigroups of epimorphisms. Trans. Amer. Math. Soc.289(1) (1985), 393–407] and Schmidt [Automorphisms of compact abelian groups and affine varieties. Proc. London Math. Soc. (3) 61 (1990), 480–496].

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 Periodic groups with many permutable subgroups

1992 ◽  
Vol 53 (1) ◽  
pp. 116-119
Author(s):  
Patrizia Longobardi ◽  
Mercede Maj ◽  
Akbar Rhemtulla ◽  
Howard Smith
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Finitely Generated Group ◽  
Periodic Groups ◽  
Permutable Subgroups ◽  
Infinite Set

AbstractGroups in which every infinite set of subgroups contains a pair that permute were studied by M. Curzio, J. Lennox, A. Rhemtulla and J. Wiegold. The question whether periodic groups in this class were locally finite was left open. Here we show that if the generators of such a group G are periodic then G is locally finite. This enables us to get the following characterisation. A finitely generated group G is centre-by-finite if and only if every infinite set of subgroups of G contains a pair that permute.

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