Deficiency and commensurators

2018 ◽  
Vol 21 (3) ◽  
pp. 511-530
Author(s):  
Jonathan A. Hillman
Keyword(s):  
Fundamental Group ◽  
Natural Homomorphism ◽  
Finitely Generated ◽  
Locally Finite ◽  
Finitely Generated Group ◽  
Locally Nilpotent

Abstract We show that if π is the fundamental group of a 4-dimensional infrasolvmanifold then {-2\leq\mathrm{def}(\pi)\leq 0} , and give examples realizing each value allowed by our constraints, for each possible value of the rank of {\pi/\pi^{\prime}} . We also consider the abstract commensurators of such groups. Finally, we show that if G is a finitely generated group, the kernel of the natural homomorphism from G to its abstract commensurator {\mathrm{Comm}(G)} is locally nilpotent by locally finite, and is finite if {\mathrm{def}(G)>1} .

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.

On Certain Classes of Locally Soluble Groups

1962 ◽  
Vol 58 (2) ◽  
pp. 185-195
Author(s):  
J. E. Roseblade
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Soluble Groups ◽  
Locally Nilpotent

A group G is called locally soluble if every finitely generated subgroup of G is soluble. Terms like ‘locally nilpotent’ and ‘locally finite’ are defined similarly.

On locally nilpotent groups

1956 ◽  
Vol 52 (1) ◽  
pp. 5-11 ◽  
Author(s):  
D. H. McLain ◽  
P. Hall
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Direct Product ◽  
Finite Subset ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
A Finite Group

1. If P is any property of groups, then we say that a group G is ‘locally P’ if every finitely generated subgroup of G satisfies P. In this paper we shall be chiefly concerned with the case when P is the property of being nilpotent, and will examine some properties of nilpotent groups which also hold for locally nilpotent groups. Examples of locally nilpotent groups are the locally finite p-groups (groups such that every finite subset is contained in a finite group of order a power of the prime p); indeed, every periodic locally nilpotent group is the direct product of locally finite p-groups.

scholarly journals The Klein bottle group is not strongly verbally closed, though awfully close to being so

2020 ◽  
pp. 1-7
Author(s):  
Anton A. Klyachko
Keyword(s):  
Fundamental Group ◽  
Closed Subgroup ◽  
Klein Bottle ◽  
Finitely Generated ◽  
Finitely Generated Group

Abstract According to Mazhuga’s theorem, the fundamental group H of anyconnected surface, possibly except for the Klein bottle, is a retract of each finitely generated group containing H as a verbally closed subgroup. We prove that the Klein bottle group is indeed an exception but has a very close property.

scholarly journals BNS INVARIANTS AND ALGEBRAIC FIBRATIONS OF GROUP EXTENSIONS

2021 ◽  
pp. 1-15
Author(s):  
Stefan Friedl ◽  
Stefano Vidussi
Keyword(s):  
Fundamental Group ◽  
Broad Class ◽  
Interesting Case ◽  
Finitely Generated ◽  
Group Extensions ◽  
Finitely Generated Group ◽  
Surface Bundles ◽  
Surface Bundle

Abstract Let G be a finitely generated group that can be written as an extension $$ \begin{align*} 1 \longrightarrow K \stackrel{i}{\longrightarrow} G \stackrel{f}{\longrightarrow} \Gamma \longrightarrow 1 \end{align*} $$ where K is a finitely generated group. By a study of the Bieri–Neumann–Strebel (BNS) invariants we prove that if $b_1(G)> b_1(\Gamma ) > 0$ , then G algebraically fibres; that is, admits an epimorphism to $\Bbb {Z}$ with finitely generated kernel. An interesting case of this occurrence is when G is the fundamental group of a surface bundle over a surface $F \hookrightarrow X \rightarrow B$ with Albanese dimension $a(X) = 2$ . As an application, we show that if X has virtual Albanese dimension $va(X) = 2$ and base and fibre have genus greater that $1$ , G is noncoherent. This answers for a broad class of bundles a question of J. Hillman ([9, Question 11(4)]). Finally, we show that there exist surface bundles over a surface whose BNS invariants have a structure that differs from that of Kodaira fibrations, determined by T. Delzant.

scholarly journals Hopfian and co-Hopfian groups

1997 ◽  
Vol 56 (1) ◽  
pp. 17-24 ◽  
Author(s):  
Satya Deo ◽  
K. Varadarajan
Keyword(s):  
Fundamental Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Finitely Generated Groups ◽  
Aspherical Manifold ◽  
Finitely Presented

The main results proved in this note are the following:(i) Any finitely generated group can be expressed as a quotient of a finitely presented, centreless group which is simultaneously Hopfian and co-Hopfian.(ii) There is no functorial imbedding of groups (respectively finitely generated groups) into Hopfian groups.(iii) We prove a result which implies in particular that if the double orientable cover N of a closed non-orientable aspherical manifold M has a co-Hopfian fundamental group then π1(M) itself is co-Hopfian.

scholarly journals Planar Transitive Graphs

10.37236/7888 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Matthias Hamann
Keyword(s):  
Fundamental Group ◽  
Homology Group ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Transitive Graph ◽  
Locally Finite ◽  
Transitive Graphs ◽  
Finitely Presented

We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible. 

scholarly journals On profinite groups in which commutators are Engel

2001 ◽  
Vol 70 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Pavel Shumyatsky
Keyword(s):  
Finite Order ◽  
Profinite Group ◽  
Profinite Groups ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Nilpotent ◽  
Additional Assumptions

AbstractWe show that if G is a finitely generated profinite group such that [x1, x2, …, xk] is Engel for any x1, x2, …, xk ∈ G, then γ(G) is locally nilpotent, and if [x1, x2, …, xk] has finite order for any x1, x2, …, xk ∈ G then, under some additional assumptions, γk(G) is locally finite.

Groups with every subgroup ascendant-by-finite

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Sergio Camp-Mora
Keyword(s):  
Finite Group ◽  
Finite Index ◽  
Locally Finite Group ◽  
Locally Finite ◽  
Locally Nilpotent

AbstractA subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

scholarly journals Some properties of the growth and of the algebraic entropy of group endomorphisms

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Anna Giordano Bruno ◽  
Pablo Spiga
Keyword(s):  
Finite Groups ◽  
Addition Theorem ◽  
Finitely Generated ◽  
Growth Type ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Algebraic Entropy ◽  
Finitely Generated Groups ◽  
Identity Automorphism ◽  
Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

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