Related papers: Twisted conjugacy classes in twisted Chevalley gro…

We study the notion of twisted conjugacy separability (essentially introduced in our previous paper for a proof of twisted version of Burnside-Frobenius theorem) and some related properties. We give examples of groups with and without this…

We give a geometric proof based on recent work of Eskin, Fisher and Whyte that the lamplighter group $L_n$ has infinitely many twisted conjugacy classes for any automorphism $\vp$ only when $n$ is divisible by 2 or 3, originally proved by…

Let G be a group and let W be an algebra over a field K. We will say that W is a G-graded twisted algebra if W can be written as a direct sum over the elements of G of one dimensional K-vector spaces. It is also assumed that W has no…

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…

Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois…

Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is…

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…

Let U be the group of units of an infinite twisted group algebra K_\lambda G over a field K. We describe the maximal FC-subgroup of U and give a characterization of U with finitely conjugacy classes. In the case of group algebras we obtain…

We solve the twisted conjugacy problem on Thompson's group F. We also exhibit orbit undecidable subgroups of Aut(F), and give a proof that Aut(F) and Aut_+(F) are orbit decidable provided a certain conjecture on Thompson's group T is true.…

Let G be a classical group over an algebraically closed field of characteristic 2 and let C be an elliptic conjugacy class in the Weyl group. In a previous paper the first named author associated to C a unipotent conjugacy class \Phi(C) in…

In this paper we introduce a new species of evolution algebras that we call Cayley evolution algebras. We show that if a field $k$ contains sufficiently many elements (for example if $k$ is infinite) then every finite group $G$ is…

Given a set $X$, the group $\mathrm{Sym}(X)$ consists of all bijections from $X$ to $X$, and $\mathrm{FSym}(X)$ is the subgroup of maps with finite support i.e. those that move only finitely many points in $X$. We describe the automorphism…

We introduce and study the class of groups graded by root systems. We prove that if {\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\Phi}, then under certain natural conditions on the grading,…

In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(\varphi, G)$ generalises the degree of…

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.

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…

A category is called {\em split} if for every morphism $s\colon X\to Y$ there exists a morphism $t\colon Y\to X$ such that $s\circ t\circ s = s$. Let $C$ be a finite split category, let $k$ be a field of characteristic 0 and let $\alpha$ be…

Let $G$ be a closed highly homogeneous subgroup of $S_{\infty}$ not involving circular orderings. We show that the closure of a conjugacy class from $G$ contains a conjugacy class which is comeagre in it. Furthermore, we show that the…

Let K be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety X is infinite, then X is uniruled.

Let G be an affine algebraic group over an algebraically closed field such that the identity component G^0 of G is reductive. Let W be the Weyl group of G and let D be a connected component of G whose image in G/G^0 is a unipotent element.…