scholarly journals TWISTED CONJUGACY CLASSES IN WREATH PRODUCTS

2006 ◽  
Vol 16 (05) ◽  
pp. 875-886 ◽  
Author(s):  
DACIBERG GONÇALVES ◽  
PETER WONG
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Conjugacy Classes ◽  
Finitely Generated ◽  
Twisted Conjugacy

Let G be a finitely generated abelian group and G ≀ ℤ be the wreath product. In this paper, we classify all such groups G for which every automorphism of G ≀ ℤ has infinitely many twisted conjugacy classes.

scholarly journals Minimal generating sets for some wreath products of groups

1973 ◽  
Vol 9 (1) ◽  
pp. 127-136
Author(s):  
Yeo Kok Chye
Keyword(s):  
Abelian Group ◽  
Nilpotent Group ◽  
Wreath Product ◽  
Wreath Products ◽  
Finitely Generated ◽  
Generating Set ◽  
Generating Sets ◽  
Products Of Groups ◽  
Minimal Generating Set ◽  
Standard Wreath Product

Let d(G) denote the minimum of the cardinalities of the generating sets of the group G. Call a generating set of cardinality d(G) a minimal generating set for G. If A is a finitely generated nilpotent group, B a non-trivial finitely generated abelian group and A wr B is their (restricted, standard) wreath product, then it is proved (by explicitly constructing a minimal generating set for A wr B ) that d(AwrB) = max{l+d(A), d(A×B)} where A × B is their direct product.

On Commutator Length and Square Length of the Wreath Product of a Group by a Finitely Generated Abelian Group

2010 ◽  
Vol 17 (spec01) ◽  
pp. 799-802 ◽  
Author(s):  
Mehri Akhavan-Malayeri
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Finitely Generated ◽  
Commutator Length

Let W = G ≀ H be the wreath product of G by an n-generator abelian group H. We prove that every element of W′ is a product of at most n+2 commutators, and every element of W2 is a product of at most 3n+4 squares in W. This generalizes our previous result.

Simple groups and irreducible lattices in wreath products

2020 ◽  
pp. 1-12 ◽  
Author(s):  
ADRIEN LE BOUDEC
Keyword(s):  
Finite Group ◽  
Wreath Product ◽  
Free Group ◽  
Wreath Products ◽  
Simple Groups ◽  
Finitely Generated ◽  
Locally Compact ◽  
Regular Tree ◽  
Finitely Generated Groups ◽  
A Finite Group

We consider the finitely generated groups acting on a regular tree with almost prescribed local action. We show that these groups embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C\wr F$ , where $C$ is a finite group and $F$ a non-abelian free group.

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 Geometry of Reidemeister classes and twisted Burnside theorem

2008 ◽  
Vol 2 (3) ◽  
pp. 463-506 ◽  
Author(s):  
Alexander Fel'shtyn ◽  
Evgenij Troitsky
Keyword(s):  
Conjugacy Classes ◽  
Type I ◽  
Finitely Generated ◽  
Dual Object ◽  
Finitely Generated Groups ◽  
Noncommutative Version ◽  
Twisted Conjugacy ◽  
Unitary Dual ◽  
Expository Paper ◽  
Definition Of

AbstractThe purpose of the present mostly expository paper (based mainly on [17, 18, 40, 16, 11]) is to present the current state of the following conjecture of A. Fel'shtyn and R. Hill [13], which is a generalization of the classical Burnside theorem.Let G be a countable discrete group, φ one of its automorphisms, R(φ) the number of φ-conjugacy (or twisted conjugacy) classes, and S(φ) = #Fix the number of φ-invariant equivalence classes of irreducible unitary representations. If one of R(φ) and S(φ) is finite, then it is equal to the other.This conjecture plays a important role in the theory of twisted conjugacy classes (see [26], [10]) and has very important consequences in Dynamics, while its proof needs rather sophisticated results from Functional and Noncommutative Harmonic Analysis.First we prove this conjecture for finitely generated groups of type I and discuss its applications.After that we discuss an important example of an automorphism of a type II1 group which disproves the original formulation of the conjecture.Then we prove a version of the conjecture for a wide class of groups, including almost polycyclic groups (in particular, finitely generated groups of polynomial growth). In this formulation the role of an appropriate dual object plays the finite-dimensional part of the unitary dual. Some counter-examples are discussed.Then we begin a discussion of the general case (which also needs new definition of the dual object) and prove the weak twisted Burnside theorem for general countable discrete groups. For this purpose we prove a noncommutative version of Riesz-Markov-Kakutani representation theorem.Finally we explain why the Reidemeister numbers are always infinite for Baumslag-Solitar groups.

scholarly journals Assouad–Nagata Dimension of Wreath Products of Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 245-253
Author(s):  
N. Brodskiy ◽  
J. Dydak ◽  
U. Lang
Keyword(s):  
Linear Function ◽  
Wreath Product ◽  
Wreath Products ◽  
Finitely Generated ◽  
Products Of Groups ◽  
Nagata Dimension

AbstractConsider the wreath product H ≀ G, where H ≠ 1 is finite and G is finitely generated. We show that the Assouad–Nagata dimension dimAN(H ≀ G) of H ≀ G depends on the growth of G as follows: if the growth of G is not bounded by a linear function, then dimAN(H ≀ G) = ∞; otherwise dimAN(H ≀ G) = dimAN(G) ≤ 1.

PROBABILISTIC GENERATION OF WREATH PRODUCTS OF NON-ABELIAN FINITE SIMPLE GROUPS, II

2006 ◽  
Vol 16 (03) ◽  
pp. 493-503 ◽  
Author(s):  
MARTYN QUICK
Keyword(s):  
Simple Group ◽  
Wreath Product ◽  
Inverse Limit ◽  
Wreath Products ◽  
Profinite Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finitely Generated

We show that the probability of generating an iterated wreath product of non-abelian finite simple groups converges to 1 as the order of the first simple group tends to infinity provided the wreath products are constructed with transitive and faithful actions. This has the consequence that the profinite group which is the inverse limit of these iterated wreath products is positively finitely generated.

CHARACTERIZATIONS OF SOME WREATH PRODUCTS OF GROUPS BY THEIR ENDOMORPHISM SEMIGROUPS

2008 ◽  
Vol 18 (02) ◽  
pp. 243-255 ◽  
Author(s):  
PEETER PUUSEMP
Keyword(s):  
Abelian Group ◽  
Wreath Product ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Wreath Products ◽  
Endomorphism Semigroup ◽  
Finite Abelian Groups ◽  
Products Of Groups

Let A be a cyclic group of order pn, where p is a prime, and B be a finite abelian group or a finite p-group which is determined by its endomorphism semigroup in the class of all groups. It is proved that under these assumptions the wreath product A Wr B is determined by its endomorphism semigroup in the class of all groups. It is deduced from this result that if A, B, A0,…, An are finite abelian groups and A0,…, An are p-groups, p prime, then the wreath products A Wr B and An Wr (…( Wr (A1 Wr A0))…) are determined by their endomorphism semigroups in the class of all groups.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

2014 ◽  
Vol 91 (2) ◽  
pp. 250-263 ◽  
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Hilbert Space ◽  
Wreath Product ◽  
Metric Spaces ◽  
Wreath Products ◽  
Finitely Generated ◽  
Product Construction ◽  
Finitely Generated Groups

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

scholarly journals Twisted conjugacy classes in unitriangular groups

2019 ◽  
Vol 22 (2) ◽  
pp. 253-266 ◽  
Author(s):  
Timur Nasybullov
Keyword(s):  
Characteristic Zero ◽  
Integral Domain ◽  
Nonempty Intersection ◽  
Conjugacy Classes ◽  
Finitely Generated ◽  
Twisted Conjugacy

Abstract Let R be an integral domain of characteristic zero. In this note we study the Reidemeister spectrum of the group {{\rm UT}_{n}(R)} of unitriangular matrices over R. We prove that if {R^{+}} is finitely generated and {n>2|R^{*}|} , then {{\rm UT}_{n}(R)} possesses the {R_{\infty}} -property, i.e. the Reidemeister spectrum of {{\rm UT}_{n}(R)} contains only {\infty} , however, if {n\leq|R^{*}|} , then the Reidemeister spectrum of {{\rm UT}_{n}(R)} has nonempty intersection with {\mathbb{N}} . If R is a field and {n\geq 3} , then we prove that the Reidemeister spectrum of {{\rm UT}_{n}(R)} coincides with {\{1,\infty\}} , i.e. in this case {{\rm UT}_{n}(R)} does not possess the {R_{\infty}} -property.

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