scholarly journals Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups

2014 ◽  
Vol 57 (1) ◽  
pp. 132-140 ◽  
Author(s):  
T. Mubeena ◽  
P. Sankaran
Keyword(s):  
Hyperbolic Group ◽  
Complete Riemannian Manifold ◽  
Abelian Extension ◽  
Conjugacy Classes ◽  
Linear Groups ◽  
Congruence Subgroups ◽  
Group Automorphism ◽  
Abelian Extensions ◽  
Twisted Conjugacy ◽  
Torsion Free

AbstractGiven a group automorphismϕ: Γ → Γ, one has an action of Γ on itself byϕ-twisted conjugacy, namely,g.x = gxϕ(g-1). The orbits of this action are calledϕ-twisted conjugacy classes. One says that Γ has theR∞-property if there are infinitely manyϕ-twisted conjugacy classes for every automorphismϕof Γ. In this paper we show that SL(n; Z) and its congruence subgroups have the R8-property. Further we show that any (countable) abelian extension of Γ has the R8-property where Γ is a torsion free non-elementary hyperbolic group, or SL(n; Z); Sp(2n; Z) or a principal congruence subgroup of SL(n; Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

scholarly journals FREELY INDECOMPOSABLE GROUPS ACTING ON HYPERBOLIC SPACES

2004 ◽  
Vol 14 (02) ◽  
pp. 115-171 ◽  
Author(s):  
ILYA KAPOVICH ◽  
RICHARD WEIDMANN
Keyword(s):  
Hyperbolic Group ◽  
Conjugacy Classes ◽  
Hyperbolic Spaces ◽  
Gromov Hyperbolic ◽  
Torsion Free

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups.

scholarly journals Equations in the Q-completion of a torsion-free hyperbolic group

1999 ◽  
Vol 351 (7) ◽  
pp. 2961-2978
Author(s):  
O. Kharlampovich ◽  
E. Lioutikova ◽  
A. Myasnikov
Keyword(s):  
Hyperbolic Group ◽  
Torsion Free

scholarly journals Automorphisms of higher rank lamplighter groups

2015 ◽  
Vol 25 (08) ◽  
pp. 1275-1299 ◽  
Author(s):  
Melanie Stein ◽  
Jennifer Taback ◽  
Peter Wong
Keyword(s):  
Cayley Graph ◽  
Infinite Number ◽  
Conjugacy Classes ◽  
Generating Set ◽  
Twisted Conjugacy ◽  
Higher Rank ◽  
Lamplighter Groups

Let [Formula: see text] denote the group whose Cayley graph with respect to a particular generating set is the Diestel–Leader graph [Formula: see text], as described by Bartholdi, Neuhauser and Woess. We compute both [Formula: see text] and [Formula: see text] for [Formula: see text], and apply our results to count twisted conjugacy classes in these groups when [Formula: see text]. Specifically, we show that when [Formula: see text], the groups [Formula: see text] have property [Formula: see text], that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when [Formula: see text] the lamplighter groups [Formula: see text] have property [Formula: see text] if and only if [Formula: see text].

ON CONJUGACY CLASSES IN LINEAR GROUPS

1993 ◽  
Vol 26 (2) ◽  
Author(s):  
Jan Ambrosiewicz
Keyword(s):  
Conjugacy Classes ◽  
Linear Groups

Twisted conjugacy classes in saturated weakly branch groups

2008 ◽  
Vol 134 (1) ◽  
pp. 61-73 ◽  
Author(s):  
Alexander Fel’shtyn ◽  
Yuriy Leonov ◽  
Evgenij Troitsky
Keyword(s):  
Conjugacy Classes ◽  
Twisted Conjugacy

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 Chevalley Groups

2015 ◽  
Vol 53 (6) ◽  
pp. 481-501 ◽  
Author(s):  
T. R. Nasybullov
Keyword(s):  
Conjugacy Classes ◽  
Chevalley Groups ◽  
Twisted Conjugacy
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