We study twisted conjugacy classes of the unit element in different groups. Fel'shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).
This paper initiates the study of effective twisted conjugacy separability for finitely generated groups, which measures the complexity of separating distinct twisted conjugacy classes via finite quotients. The focus is on nilpotent groups, and our main result shows that there is a polynomial upper bound for twisted conjugacy separability. That allows us to study regular conjugacy separability in the case of virtually nilpotent groups, where we compute a polynomial upper bound as well. As another application, we improve the work of the second author by giving a precise calculation of conjugacy…
Let $ϕ:G\to G$ 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 infinitely many $ϕ$-twisted conjugacy classes. As an application, we show that if $G$ is the fundamental group of a three-manifold that is not irreducible, then $G$ has property $R_\infty$, that is, there are infinitely many $ϕ$-twisted conjugacy classes in $G$ for every automorphism $ϕ$ of $G$.
A group $G$ has property $R_\infty$ if for every $ϕ\in Aut(G)$, there are an infinite number of $ϕ$-twisted conjugacy classes of elements in $G$. In this note, we determine the $R_\infty$-property for $G=π_1(M)$ for all geometric $3$-manifolds $M$.
Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $Γ$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely, that there are infinitely $ϕ$-twisted conjugacy classes for every automorphism $ϕ$ of $Γ$. Also, we show that any lattice in $G$ has the $R_\infty$-property, extending our earlier result for irreducible lattices.
Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $φ\in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $φ$-twisted conjugacy classes is infinite for every $φ\in Aut(G)$.
A group $G$ is said to have property $R_{\infty}$ if for every automorphism $φ\in {\rm Aut}(G)$, the cardinality of the set of $φ$-twisted conjugacy classes is infinite. Many classes of groups are known to have such property. However, very few examples are known for which $R_{\infty}$ is {\it geometric}, i.e., if $G$ has property $R_{\infty}$ then any group quasi-isometric to $G$ also has property $R_{\infty}$. In this paper, we give examples of groups and conditions under which $R_{\infty}$ is preserved under commensurability. The main tool is to employ the Bieri-Neumann-Strebel invariant.
Let $G$ be a group and $φ$ be an automorphism of $G$. Two elements $x, y$ of $G$ are said to be $φ$-twisted conjugate if $y=gxφ(g)^{-1}$ for some $g\in G$. A group $G$ has the $R_{\infty}$-property if the number of $φ$-twisted conjugacy classes is infinite for every automorphism $φ$ of $G$. In this paper we prove that the big mapping class group $MCG(S)$ possesses the $R_{\infty}$-property under some suitable conditions on the infinite-type surface $S$. As an application we also prove that the big mapping class group possesses the $R_\infty$-property if and only if it satisfies the $S_{\infty}…
Let $F$ be a subfield of the algebraic closure of a finite field $\mathbb{F}_p$, $p \ne 2$, and let $R$ denote any ring such that $F[t] \subset R \subsetneq F(t)$. Let $G$ be a classical Chevalley group of adjoint type defined over $R$. We prove that the group $G(R)$ has the $R_{\infty}$-property.
Twisted conjugacy in free groups and Makanin's question
我们讨论了G. Makanin从"Kourovka笔记本“：是否存在一个算法来确定是一个自由群$F_n$和任意自同构$ϕ\in Aut(F_n)$的任意一对词$U$和$V$的方程$ϕ(X) U = V X$可解在$F_n$？当一个自同构是虚内自同构时，即当它的某个非零幂是$F_n$的内自同构时，我们给出了肯定的回答.
We construct two practical algorithms for twisted conjugacy classes of polycyclic-by-finite groups. The first algorithm determines whether two elements of a group are twisted conjugate for two given endomorphisms, under the condition that the Reidemeister coincidence number of these endomorphisms is finite. The second algorithm determines representatives of the Reidemeister coincidence classes of two endomorphisms if their Reidemeister coincidence number is finite, or returns "fail" if the Reidemeister coincidence number is infinite.
Let $R$ be an integral domain of zero characteristic. 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, then we prove that the Reidemeister spectrum of ${\rm UT}_n(R)$ coincides with $\{1,\infty\}$, i. e…
Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(φ)$ denote the set of all orbits of the $φ$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gxφ(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(φ)$ is infinite for every $φ\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solva…
Let $ϕ:G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation $\sim_ϕ$ on $G$ defined as $x\sim_ϕy$ if there exists a $z\in G$ such that $y=zxϕ(z^{-1})$. The equivalence classes are called $ϕ$-twisted conjugacy classes and the set $G/\!\!\sim_ϕ$ of equivalence classes is denoted $\mathcal R(ϕ)$. The cardinality $R(ϕ)$ of $\mathcal R(ϕ)$ is called the Reidemeister number of $ϕ$. We write $R(ϕ)=\infty$ when $\mathcal R(ϕ)$ is infinite. We say that $G$ has the $R_\infty$-{\it property} if $R(ϕ)=\infty$ for every automorphism $ϕ$ of $G$. We show that the groups $G=GL_…
We establish a connection between the generalized conjugacy problem for a $G$-by-$\mathbb{Z}$ group, $GCP(G \rtimes \mathbb{Z})$, and two algorithmic problems for $G$: the generalized Brinkmann's conjugacy problem, $GBrCP(G)$, and the generalized twisted conjugacy problem, $GTCP(G)$. We explore this connection for generalizations of different kinds: relative to finitely generated subgroups, to theirs cosets, or to recognizable, rational, context-free or algebraic subsets of the group. Using this result, we are able to prove that $GBrCP(G)$ is decidable (with respect to cosets) when $G$ is a vi…
We calculate asymptotic estimates for the conjugacy growth function of finitely generated class 2 nilpotent groups whose derived subgroup is infinite cyclic, including the so-called higher Heisenberg groups. We prove that these asymptotics are stable when passing to commensurable groups, by understanding their twisted conjugacy growth. We also use these estimates to prove that, in certain cases, the conjugacy growth series cannot be a holonomic function.
