Related papers: Reidemeister classes, wreath products and solvabil…

Let $G$ be a group and $\varphi$ an automorphism of $G$. Two elements $x,y \in G$ are said to be $\varphi$-conjugate if there exists a third element $z \in G$ such that $z x \varphi(z)^{-1} = y$. Being $\varphi$-conjugate defines an…

We study the product formula for Reidemeister numbers on finitely generated torsion-free nilpotent groups in two ways. On the one hand, we generalise the product formula to central extensions. On the other hand, we derive general results…

The Farrell-Jones Fibered Isomorphism Conjecture for the stable topological pseudoisotopy theory has been proved for several classes of groups. For example for discrete subgroups of Lie groups, virtually poly-infinite cyclic groups, Artin…

We study groups $G$ where the $\varphi$-conjugacy class $[e]_{\varphi}=\{g^{-1}\varphi(g)~|~g\in G\}$ of the unit element is a subgroup of $G$ for every automorphism $\varphi$ of $G$. If $G$ has $n$ generators, then we prove that the $k$-th…

Given a group $G$ and an automorphism $\varphi$ 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 $R(\varphi)$ of equivalence classes with respect to this relation is…

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^…

We introduce a new class of groups called wreath-like products. These groups are close relatives of the classical wreath products and arise naturally in the context of group theoretic Dehn filling. Unlike ordinary wreath products, many…

Let G be a group and {\phi} be an automorphism of G. Two elements x, y of G are said to be {\phi}-twisted if y = gx{\phi}(g)^{-1} for some g in G. We say that a group G has the R_{\infty}-property if the number of {\phi}-twisted conjugacy…

A group is said to have the $R_\infty$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_\infty$ property when $G$ is a finitely generated torsion-free nilpotent…

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…

Given a sequence of $(G_i)_{i \in \N}$ of finite transitive groups of degree $n_i$, let $W_\infty$ be the inverse limit of the iterated permutational wreath products of the first m groups. We prove that $W_\infty$ is (topologically)…

Wreath products such as Z wr Z are not finitely-presentable yet can occur as subgroups of finitely presented groups. Here we compute the distortion of Z wr Z as a subgroup of Thompson's group F and as a subgroup of Baumslag's metabelian…

Let $G$ be a group and $\varphi$ be an automorphism of $G$. Two elements $x, y$ of $G$ are said to be $\varphi$-twisted conjugate if $y=gx\varphi(g)^{-1}$ for some $g\in G$. A group $G$ has the $R_{\infty}$-property if the number of…

In this paper we study the number of twisted conjugacy classes (the Reidemeister number) for automorphisms of crystallographic groups. We present two main algorithms for crystallographic groups whose holonomy group has finite normaliser in…

Let $G$ be a non-compact semisimple Lie group with finite centre and finitely many components. We show that any finitely generated group $\Gamma$ which is quasi-isometric to an irreducible lattice in $G$ has the $R_\infty$-property, namely,…

An algorithm is constructed that, when given an explicit presentation of a finitely generated nilpotent group $G,$ decides for any pair of endomorphisms $\varphi, \psi : G \to G$ and any pair of elements $u, v \in G,$ whether or not the…

In [F. Caselli, Involutory reflection groups and their models, J. Algebra 24 (2010), 370--393] it is constructed a uniform Gelfand model for all non-exceptional irreducible complex reflection groups which are involutory. This model can be…

The Reidemeister number of an endomorphism of a group is the number of twisted conjugacy classes determined by that endomorphism. The collection of all Reidemeister numbers of all automorphisms of a group $G$ is called the Reidemeister…

Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman's theorem states that they precisely specify pseudovarieties, i.e. classes of finite algebras closed under finite products,…

We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the power problem for $B$…