Related papers: Reidemeister classes in lamplighter type groups

Given an endomorphism $\varphi: G \to G$ on a group $G$, one can define the Reidemeister number $R(\varphi) \in \mathbb{N} \cup \{\infty\}$ as the number of twisted conjugacy classes. The corresponding Reidemeister zeta function…

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 $F$,…

A group $G$ is said to have the property $R_\infty$ if every automorphism $\phi \in {\rm Aut}(G)$ has an infinite number of $\phi$-twisted conjugacy classes. Recent work of Gon\c{c}alves and Kochloukova uses the $\Sigma^n$…

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…

We show that the number of twisted conjugacy classes is infinite for any automorphism of non-elementary, Gromov hyperbolic group . An analog of Selberg theory for twisted conjugacy classes is proposed.

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…

Reidemeister numbers of group automorphisms encode the number of twisted conjugacy classes of groups and might yield information about self-maps of spaces related to the given objects. Here we address a question posed by Gon\c{c}alves and…

The RP-property of Fel'shtyn and Troitsky is proved for wreath products of finitely generated Abelian groups with the group of integers. Such wreath products become the first known example of finitely generated RP-groups being not almost…

We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$,…

We show that an accessible group with infinitely many ends has property $R_{\infty}$. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property $R_{\infty}$ is undecidable…

The Reidemeister number $R(\varphi)$ of a group automorphism $\varphi \in \mathrm{Aut}(G)$ encodes the number of orbits of the $\varphi$-twisted conjugation action of $G$ on itself, and the Reidemeister spectrum of $G$ is defined as the set…

We investigate fixed-point properties of automorphisms of groups similar to R. Thompson's group $F$. Revisiting work of Gon\c{c}alves-Kochloukova, we deduce a cohomological criterion to detect infinite fixed-point sets in the…

Given a morphism $\varphi : G \to A \wr B$ from a finitely presented group $G$ to a wreath product $A \wr B$, we show that, if the image of $\varphi$ is a sufficiently large subgroup, then $\mathrm{ker}(\varphi)$ contains a non-abelian free…

Given an automorphism $\phi:\Gamma\to \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$…

A group $G$ has the $R_{\infty}$--property if the number $R(\varphi)$ of twisted conjugacy classes is infinite for any automorphism $\varphi$ of $G$. For such a group $G$, the $R_{\infty}$--nilpotency index is the least integer $c$ such…

Let $f$ be an automorphism of a group $G$. Two elements $x, y$ in $G$ are said to be in the same $f$-twisted conjugacy class if there exists an element $z$ in $G$ such that $y=z x f(z^{-1})$. This is an equivalence relation known as…

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.…

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

It is well known there is no finitely generated abelian group which has the $R_\infty$ property. We will show that also many non-finitely generated abelian groups do not have the $R_\infty$ property, but this does not hold for all of them.…

We say that $x,y\in \Gamma$ are in the same $\phi$-twisted conjugacy class and write $x\sim_\phi y$ if there exists an element $\gamma\in \Gamma$ such that $y=\gamma x\phi(\gamma^{-1})$. This is an equivalence relation on $\Gamma$ called…