对于包括多环和多项式增长群在内的广泛的一类群，证明了自同构f的Reidemeister数等于酉对偶上诱导映射f^的有限维不动点的个数，如果这些个数之一是有限的.这个定理是经典的Burnside-Frobenius定理到无限群的自然推广。这个定理在拓扑动力学中也有重要的结果，在某种意义上是对J. P. Serre。本文所取得的主要技术成果为今后的进一步研究提供了工具。
英文摘要：
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^ on the unitary dual, if one of these numbers is finite. This theorem is a natural generalization of the classical Burnside-Frobenius theorem to infinite groups. This theorem also has important consequences in topological dynamics and in some sense is a reply to a remark of J.-P. Serre. The main technical results proved in the paper yield a tool for a further progress.
We prove that the restricted wreath product ${\mathbb{Z}_n \mathbin{\mathrm{wr}} \mathbb{Z}^k}$ has the $R_\infty$-property, i. e. every its automorphism $φ$ has infinite Reidemeister number $R(φ)$, in exactly two cases: (1) for any $k$ and even $n$; (2) for odd $k$ and $n$ divisible by 3. In the remaining cases there are automorphisms with finite Reidemeister number, for which we prove the finite-dimensional twisted Burnside--Frobenius theorem (TBFT): $R(φ)$ is equal to the number of equivalence classes of finite-dimensional irreducible unitary representations fixed by the action ${[ρ]\mapsto…
Let $R(ϕ)$ be the number of $ϕ$-conjugacy (or Reidemeister) classes of an endomorphism $ϕ$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(ϕ)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(ϕ)<\infty$. Applying the result to iterations of $ϕ$ we obtain Gauss congruences for Reidemeister numbers. In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_\infty$ pr…
We describe the generalized Matsuda's theorem, and some results of a Burnside ring extend a partial Burnside ring. In particular, we give isomorphism between partial Burnside rings of different groups. Moreover, we consider the relationship between image of Frobenius-Wielandt homomorphism, a partial Burnside ring, and structure of a group.
We introduce new zeta functions related to an endomorphism $ϕ$ of a discrete group $Γ$. They are of two types: counting numbers of fixed ($ρ\sim ρ\circϕ^n$) irreducible representations for iterations of $ϕ$ from an appropriate dual space of $Γ$ and counting Reidemeister numbers $R(ϕ^n)$ of different compactifications. Many properties of these functions and their coefficients are obtained. In many cases it is proved that these zeta functions coincide. The Gauss congruences are proved. Useful asymptotic formulas for the zeta functions are found. Rationality is proved for some examples, which giv…
Suppose, $G$ is a residually finite group of finite upper rank admitting an automorphism $φ$ with finite Reidemeister number $R(φ)$ (the number of $φ$-twisted conjugacy classes). We prove that such $G$ is soluble-by-finite (in other words, any residually finite group of finite upper rank, which is not soluble-by-finite, has the $R_\infty$ property). This reduction is the first step in the proof of the second main theorem of the paper: suppose, $G$ is a residually finite group of finite Prüfer rank and $φ$ is its automorphism with $R(φ)<\infty$; then $R(φ)$ is equal to the number of equivalence…
We prove that for any automorphism $ϕ$ 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 $R(ϕ)$ is infinite, i.e. these groups have the property $R_\infty$. For $\mathbb{Z}_3 \mathrm{wr} \mathbb{Z}^{2d+1}$ and $\mathbb{Z}_p \mathrm{wr} \mathbb{Z}^k$, where $p>3$ is prime, we give examples of automorphisms with finite Reidemeister numbers. So these groups do not have the property $R_\infty$. For these groups and $\mathbb{Z}_m \mathrm{wr} \mathbb{Z}$, where $m$ is relatively prime to $6$, we prove the t…
This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three well-known theorems in group theory for gyrogroups: the orbit-stabilizer theorem, the orbit decomposition theorem, and the Burnside lemma (or the Cauchy-Frobenius lemma). We then prove that under a certain condition, a gyrogroup $G$ acts transitively on the set $G/H$ of left cosets of a subgyrogroup $H$ in $G$ in a natural way. From this we prove the structure theor…
Using Frobenius normal forms of matrices over finite fields as well as the Burnside Basis Theorem, we give a direct proof of Horoševskiĭ's result that every automorphism $α$ of a finite nilpotent group has a cycle whose length coincides with $\mathrm{ord}(α)$. Also, we give two new sufficient conditions for an automorphism $α$ of an arbitrary finite group to satisfy this property, namely when $\mathrm{ord}(α)$ is a product of at most two prime powers or when $α$ has a sufficiently large cycle. This will allow us to show that the least order of a group where this property is violated is 120. Fi…
We study the numbers of involutions and their relation to Frobenius-Schur indicators in the groups $\mathrm{SO}^{\pm}(n,q)$ and $Ω^{\pm}(n,q)$. Our point of view for this study comes from two motivations. The first is the conjecture that a finite simple group $G$ is strongly real (all elements are conjugate to their inverses by an involution) if and only if it is totally orthogonal (all Frobenius-Schur indicators are 1), and we are able to show this holds for all finite simple groups $G$ other than the groups $\mathrm{Sp}(2n,q)$ with $q$ even or $Ω^{\pm}(4m,q)$ with $q$ even. We prove computat…
Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$. A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity $ω_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset of $L$. In the case that $L$ is a finite group of Lie type, we identify covers of $L$ by $c$-nilpotent subgroups, and we use these covers to constru…
