ON RESIDUAL FINITENESS OF GRAPHS OF NILPOTENT GROUPS

2004 ◽  
Vol 14 (04) ◽  
pp. 403-408
Author(s):  
E. RAPTIS ◽  
O. TALELLI ◽  
D. VARSOS
Keyword(s):  
Finite Groups ◽  
Finite Graph ◽  
Nilpotent Groups ◽  
Fundamental Groups ◽  
Residual Finiteness ◽  
Graph Of Groups ◽  
Residually Finite ◽  
Edge Group ◽  
Residually Finite Groups ◽  
Torsion Free

Here we characterize the residually finite groups G which are the fundamental groups of a finite graph of finitely generated torsion-free nilpotent groups. Namely we show that G is residually finite if and only if for each edge group of the graph of groups the two edge monomorphisms differ essentially by an isomorphism of certain subgroups of the Mal'cev completion of the corresponding vertex groups.

On the Residual Finiteness of Polygonal Products of Nilpotent Groups

1992 ◽  
Vol 35 (3) ◽  
pp. 390-399 ◽  
Author(s):  
Goansu Kim ◽  
C. Y. Tang
Keyword(s):  
Nilpotent Groups ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Isolated Subgroup ◽  
Torsion Free

AbstractIn general polygonal products of finitely generated torsion-free nilpotent groups amalgamating cyclic subgroups need not be residually finite. In this paper we prove that polygonal products of finitely generated torsion-free nilpotent groups amalgamating maximal cyclic subgroups such that the amalgamated cycles generate an isolated subgroup in the vertex group containing them, are residually finite. We also prove that, for finitely generated torsion-free nilpotent groups, if the subgroups generated by the amalgamated cycles have the same nilpotency classes as their respective vertex groups, then their polygonal product is residually finite.

Residually finite groups of finite rank

1989 ◽  
Vol 106 (3) ◽  
pp. 385-388 ◽  
Author(s):  
Alexander Lubotzky ◽  
Avinoam Mann
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Finite Rank ◽  
Prime Order ◽  
Residual Finiteness ◽  
Infinite Groups ◽  
Finiteness Conditions ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

The recent constructions, by Rips and Olshanskii, of infinite groups with all proper subgroups of prime order, and similar ‘monsters’, show that even under the imposition of apparently very strong finiteness conditions, the structure of infinite groups can be rather weird. Thus it seems reasonable to impose the type of condition that enables us to apply the theory of finite groups. Two such conditions are local finiteness and residual finiteness, and here we are interested in the latter. Specifically, we consider residually finite groups of finite rank, where a group is said to have rank r, if all finitely generated subgroups of it can be generated by r elements. Recall that a group is said to be virtually of some property, if it has a subgroup of finite index with this property. We prove the following result:Theorem 1. A residually finite group of finite rank is virtually locally soluble.

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

scholarly journals Residually finite tubular groups

2019 ◽  
Vol 150 (6) ◽  
pp. 2937-2951
Author(s):  
Nima Hoda ◽  
Daniel T. Wise ◽  
Daniel J. Woodhouse
Keyword(s):  
Finite Graph ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Graph Of Groups ◽  
Residually Finite ◽  
Single Vertex

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

The Residual Finiteness of Polygonal Products—Two Counterexamples

1994 ◽  
Vol 37 (4) ◽  
pp. 433-436 ◽  
Author(s):  
R. B. J. T. Allenby
Keyword(s):  
Abelian Groups ◽  
Nilpotent Groups ◽  
Residual Finiteness ◽  
Finitely Generated ◽  
Residually Finite ◽  
Cyclic Subgroups ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractWe show that, even under very favourable hypotheses, a polygonal product of finitely generated torsion free nilpotent groups amalgamating infinite cyclic subgroups is, in general, not residually finite, thus answering negatively a question of C. Y. Tang. A second example shows similar kinds of limitations apply even when the factors of the product are free abelian groups.

scholarly journals Residual finiteness of permutational products

1969 ◽  
Vol 10 (3-4) ◽  
pp. 423-428 ◽  
Author(s):  
R. J. Gregorac
Keyword(s):  
Free Product ◽  
Finite Groups ◽  
Residual Finiteness ◽  
Residually Finite ◽  
Generalized Free Product ◽  
Residually Finite Groups

Conditions sufficient to guarantee that a generalized free product of two residually finite groups A and B is again residually finite have been given by Baumslag [1]. We here show the same conditions guarantee that a certain permutational product of A and B is also residually finite.

scholarly journals Rokhlin dimension for actions of residually finite groups

2017 ◽  
Vol 39 (8) ◽  
pp. 2248-2304 ◽  
Author(s):  
GÁBOR SZABÓ ◽  
JIANCHAO WU ◽  
JOACHIM ZACHARIAS
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Metric Spaces ◽  
Nilpotent Groups ◽  
Asymptotic Dimension ◽  
Finitely Generated ◽  
Residually Finite ◽  
Compact Metric Spaces ◽  
Finite Dimensional ◽  
Residually Finite Groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

scholarly journals On Groups of Automorphisms of Residually Finite Groups

2000 ◽  
Vol 231 (2) ◽  
pp. 561-573
Author(s):  
Ulderico Dardano ◽  
Bettina Eick ◽  
Martin Menth
Keyword(s):  
Finite Groups ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Groups Of Automorphisms

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.

Sign in / Sign up

Export Citation Format

Share Document