Related papers: Twisted Burnside-Frobenius theory for endomorphism…

For a wide class of groups including polycyclic and finitely generated polynomial growth groups it is proved that the Reidemeister number of an automorphism f is equal to the number of finite-dimensional fixed points of the induced map f^…

It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms $\phi$ and $\psi$ is equal to the number of coincidence points of $\wh\phi$ and $\wh\psi$ on the unitary dual, if the Reidemeister number is…

Let $\phi:G \to G$ be a group endomorphism where $G$ is a finitely generated group of exponential growth, and denote by $R(\phi)$ the number of twisted $\phi$-conjugacy classes. Fel'shtyn and Hill \cite{fel-hill} conjectured that if $\phi$…

We prove that for any automorphism $\phi$ of the restricted wreath product $\mathbb{Z}_2 \mathrm{wr} \mathbb{Z}^k$ and $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d}$ the Reidemeister number $R(\phi)$ is infinite, i.e. these groups have the…

The purpose of the present paper is to discuss the following conjecture of Fel'shtyn and Hill, which is a generalization of the classical Burnside theorem: Let G be a countable discrete group, f its automorphism, R(f) the number of…

Let $\phi:G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation $\sim_\phi$ on $G$ defined as $x\sim_\phi y$ if there exists a $z\in G$ such that $y=zx\phi(z^{-1})$. The equivalence classes are called…

Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (the number of $\varphi$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite…

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one…

Let $N$ be a finitely generated nilpotent group. Algorithm is constructed such, that for every automorphism $\phi \in Aut(N)$ defines the Reidemeister number $R(\phi).$ It is proved that any free nilpotent group of rank $r = 2$ or $r = 3$…

Among restricted wreath products $G\wr \mathbb Z^k $, where $G$ is a finite Abelian group, we find three large classes of groups admitting an automorphism $\varphi$ with finite Reidemeister number $R(\varphi)$ (number of $\varphi$-twisted…

Given a group $G$ and an endomorphism $\varphi$ of $G$, two elements $x, y \in G$ are said to be $\varphi$-conjugate if $x = gy \varphi(g)^{-1}$ for some $g \in G$. The number of equivalence classes for this relation is the Reidemeister…

We provide two alternative ways to determine the number of (bi-)twisted conjugacy classes in a finite group: one by counting certain irreducible characters and one by counting certain twisted conjugacy classes of other endomorphisms. In…

We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $\varphi$ has infinite Reidemeister number $R(\varphi)$, in exactly two cases: (1) for…

Reidemeister (or twisted conjugacy) classes are considered in restricted wreath products of the form $G\wr \mathbb{Z}^k$, where $G$ is a finite group. For an automorphism $\varphi$ of finite order (supposed to be the same for the torsion…

We consider twisted conjugacy classes of continuous automorphisms $\varphi$ of a Lie group $G$. We obtain a necessary and sufficient condition on $\varphi$ for its Reidemeister number, the number of twisted conjugacy classes, to be infinite…

We prove for residually finite groups the following long standing conjecture: the number of twisted conjugacy classes of an automorphism of a finitely generated group is equal (if it is finite) to the number of finite dimensional…

Let $\phi:G\to G$ be an automorphism of a group which is a free-product of finitely many groups each of which is freely indecomposable and two of the factors contain proper finite index characteristic subgroups. We show that $G$ has…

In this short article, we prove that any automorphism of the R. Thompson's group $F$ has infinitely many twisted conjugacy classes. The result follows from the work of Matthew Brin, together with a standard facts on R. Thompson's group $F$,…

Let $G$ be a finitely generated polyfree group. If $G$ has nonzero Euler characteristic then we show that $Aut(G)$ has a finite index subgroup in which every automorphism has infinite Reidemeister number. For certain $G$ of length 2, we…

A group $G$ is said to have property $R_{\infty}$ if for every automorphism $\varphi \in {\rm Aut}(G)$, the cardinality of the set of $\varphi$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property.…