相关论文: On the Reidemeister spectrum and the $R_{\infty}$ …

Let $F$ be either a free nilpotent group of a given class and of finite rank or a free solvable group of a certain derived length and of finite rank. We show precisely which ones have the $R_{\infty}$ property. Finally, we also show that…

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 $r = 3$…

We establish a strong link between two open problems: determining the Reidemeister spectrum of free nilpotent groups and determining the coefficients in the Schur expansion of plethysms of Schur functions. Specifically, we show that the…

A ring $R$ satisfies the {\it strong rank condition} (SRC) if, for every natural number $n$, the free $R$-submodules of $R^n$ all have rank $\leq n$. Let $G$ be a group and $R$ a ring strongly graded by $G$ such that the base ring $R_1$ is…

We say a group $G$ has property $R_\infty$ if the number $R(\varphi)$ of twisted conjugacy classes is infinite for every automorphism $\varphi$ of $G$. For such groups, the $R_\infty$-nilpotency degree is the least integer $c$ such that…

In this paper we study the Reidemeister spectrum of 2-step nilpotent groups associated to graphs. We develop three methods, based on the structure of the graph, that can be used to determine the Reidemeister spectrum of the associated group…

We investigate the Reidemeister spectrum of direct products of nilpotent groups. More specifically, we prove that the Reidemeister spectra of the individual factors yield complete information for the Reidemeister spectrum of the direct…

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…

We extend the notions of "$R_\infty$-property" and "full (extended) Reidemeister spectrum" to finite groups in a meaningful way. We provide examples of finite groups admitting these properties, if they exist, by looking at groups of small…

It is well known that when $G$ is the fundamental group of a closed surface of negative Euler characteristic, it has the $R_{\infty}$ property. In this work we compute the least integer $c$, {\it called the $R_{\infty}$-nilpotency degree of…

Let $M$ be a flat manifold. We say that $M$ has $R_\infty$ property if the Reidemeister number $R(f) = \infty$ for every homeomorphism $f \colon M \to M.$ In this paper, we investigate a relation between the holonomy representation $\rho$…

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…

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 of equivalence classes for this relation is the Reidemeister…

In this note we study the Reidemeister spectrum for metabelian groups of the form ${\mathbb Q}^n\rtimes \mathbb Z$ and ${\mathbb Z[1/p]}^n\rtimes \mathbb Z$. Particular attention is given to the $R_{\infty}$ property of a subfamily of these…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

We prove that a saturated weakly branch group $G$ has the property $R_\infty$ (any automorphism $\phi:G\to G$ has infinite Reidemeister number) in each of the following cases: 1) any element of $Out(G)$ has finite order; 2) for any $\phi$…

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…

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…

We introduce a model of random f.g., torsion-free, $2$-step nilpotent groups (in short, $\tau_2$-groups). To do so, we show that these are precisely the groups that admit a presentation of the form $ \label{tau2pres_0}\langle A, C \mid…

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…