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 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 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.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

scholarly journals A 5′ splice site mutation affecting the pre-mRNA splicing of two upstream exons in the collagen COL1A1 gene. Exon 8 skipping and altered definition of exon 7 generates truncated proα1(I) chains with a non-collagenous insertion destabilizing the triple helix

1994 ◽  
Vol 302 (3) ◽  
pp. 729-735 ◽  
Author(s):  
J F Bateman ◽  
D Chan ◽  
I Moeller ◽  
M Hannagan ◽  
W G Cole
Keyword(s):  
Splice Site ◽  
De Novo ◽  
Donor Site ◽  
Mrna Splicing ◽  
Splice Donor Site ◽  
Sequence Motif ◽  
Type I ◽  
Splice Donor ◽  
Site Mutation ◽  
Definition Of

A heterozygous de novo G to A point mutation in intron 8 at the +5 position of the splice donor site of the gene for the pro alpha 1(I) chain of type I procollagen, COL1A1, was defined in a patient with type IV osteogenesis imperfecta. The splice donor site mutation resulted not only in the skipping of the upstream exon 8 but also unexpectedly had the secondary effect of activating a cryptic splice site in the next upstream intron, intron 7, leading to re-definition of the 3′ limit of exon 7. These pre-mRNA splicing aberrations cause the deletion of exon 8 sequences from the mature mRNA and the inclusion of 96 bp of intron 7 sequence. Since the mis-splicing of the mutant allele product resulted in the maintenance of the correct codon reading frame, the resultant pro alpha 1(I) chain contained a short non-collagenous 32-amino-acid sequence insertion within the repetitive Gly-Xaa-Yaa collagen sequence motif. At the protein level, the mutant alpha 1(I) chain was revealed by digestion with pepsin, which cleaved the mutant procollagen within the protease-sensitive non-collagenous insertion, producing a truncated alpha 1(I). This protease sensitivity demonstrated the structural distortion to the helical structure caused by the insertion. In long-term culture with ascorbic acid, which stimulates the formation of a mature crosslinked collagen matrix, and in tissues, there was no evidence of the mutant chain, suggesting that during matrix formation the mutant chain was unable to stably incorporated into the matrix and was degraded proteolytically.

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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