scholarly journals Finitely generated subgroups of branch groups and subdirect products of just infinite groups

2021 ◽  
Vol 85 ◽  
Author(s):  
Rostislav Ivanovich Grigorchuk ◽  
Paul-Henry Leemann ◽  
Tat'yana V Nagnibeda
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Subdirect Products

Reflections on some aspects of infinite groups

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
Dennis Spellman
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Limit Groups ◽  
Finitely Presented Groups ◽  
The Relationship ◽  
Finitely Presented

AbstractIn this paper we survey and reflect upon several aspects of the theory of infinite finitely generated and finitely presented groups that were originally motivated by work of Gilbert Baumslag. All but the last of the topics we have chosen are all related in one way or another to the theory of limit groups and the solution of the Tarski problems. These include the residually free and fully residually free properties and the big powers condition; Baumslag doubles and extensions of centralizers; residually-𝒳 groups and extensions of results of Benjamin Baumslag and finally the relationship between CT and CSA groups.

scholarly journals On non-periodic groups whose finitely generated subgroups are either permutable or weakly pronormal

2013 ◽  
Vol 21 ◽  
pp. 67
Author(s):  
T.V. Velychko
Keyword(s):  
Finitely Generated ◽  
Infinite Groups ◽  
Periodic Groups

We consider some infinite groups whose finitely generated subgroups are either permutable or weakly pronormal.

scholarly journals Presentability by products for some classes of groups

2017 ◽  
Vol 09 (01) ◽  
pp. 27-49
Author(s):  
P. de la Harpe ◽  
D. Kotschick
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Groups ◽  
Dense Subgroups ◽  
Finite Index Subgroups ◽  
Virtual Cohomological Dimension ◽  
By Products

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.

scholarly journals Groups isomorphic to their non-nilpotent subgroups

1998 ◽  
Vol 40 (2) ◽  
pp. 257-262 ◽  
Author(s):  
Howard Smith ◽  
James Wiegold
Keyword(s):  
Finite Groups ◽  
Proper Subgroup ◽  
Complete Classification ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finite Case ◽  
Locally Nilpotent ◽  
Infinite Case ◽  
Natural Generalisation ◽  
Abelian Subgroups

We were concerned in [12] with groups G that are isomorphic to all of their non-abelian subgroups. In order to exclude groups with all proper subgroups abelian, which are well understood in the finite case [7] and which include Tarski groups in the infinite case, we restricted attention to the class X of groups G that are isomorphic to their nonabelian subgroups and that contain proper subgroups of this type; such groups are easily seen to be 2-generator, and a complete classification was given in [12, Theorem 2] for the case G soluble. In the insoluble case, G/Z(G) is infinite simple [12; Theorem 1], though not much else was said in [12] about such groups. Here we examine a property which represents a natural generalisation of that discussed above. Let us say that a group G belongs to the class W if G is isomorphic to each of its non-nilpotent subgroups and not every proper subgroup of G is nilpotent. Firstly, note that finite groups in which all proper subgroups are nilpotent are (again) well understood [9]. In addition, much is known about infinite groups with all proper subgroups nilpotent (see, in particular, [8] and [13] for further discussion) although, even in the locally nilpotent case, there are still some gaps in our understanding of such groups. We content ourselvesin the present paper with discussing finitely generated W-groups— note that a W-group is certainly finitely generated or locally nilpotent. We shall have a little more to say about the locally nilpotent case below.

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.

Infinite groups whose group algebras satisfy the converse of Schur’s lemma

2019 ◽  
Vol 18 (10) ◽  
pp. 1950186
Author(s):  
Mohammed El Badry ◽  
Mostafa Alaoui Abdallaoui ◽  
Abdelfattah Haily
Keyword(s):  
Group Algebra ◽  
Finite Index ◽  
Abelian Subgroup ◽  
Sufficient Conditions ◽  
Group Algebras ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Soluble Groups ◽  
Schur’S Lemma ◽  
Schur's Lemma

In this work, we give some necessary and/or sufficient conditions for a group algebra of infinite group to satisfy the converse of Schur’s Lemma. Many classes of groups are investigated such as abelian groups, hypercentral groups, groups having abelian subgroup of finite index and finitely generated soluble groups.

scholarly journals On the residual and profinite closures of commensurated subgroups

2019 ◽  
Vol 169 (2) ◽  
pp. 411-432
Author(s):  
PIERRE–EMMANUEL CAPRACE ◽  
PETER H. KROPHOLLER ◽  
COLIN D. REID ◽  
PHILLIP WESOLEK
Keyword(s):  
Finite Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Groups Acting On Trees ◽  
Polycyclic Groups

AbstractThe residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

On Some Finiteness Properties in Infinite Groups

2008 ◽  
Vol 15 (01) ◽  
pp. 1-22 ◽  
Author(s):  
Gilbert Baumslag ◽  
Oleg Bogopolski ◽  
Benjamin Fine ◽  
Anthony Gaglione ◽  
Gerhard Rosenberger ◽  
...  
Keyword(s):  
Finite Groups ◽  
Elementary Proof ◽  
Free Groups ◽  
Basic Result ◽  
Surface Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Hall's Theorem ◽  
Finiteness Properties ◽  
Hall’S Theorem

We consider some questions concerning finiteness properties in infinite groups which are related to Marshall Hall's theorem. We call these properties Property S and Property R, and they are trivially true in finite groups. We give several elementary proofs using these properties for results on finitely generated subgroups of free groups as well as a new elementary proof of Hall's basic result. Finally, we consider these properties within surface groups and prove an analog of Hall's theorem in that context.

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