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 ${[ρ]\mapsto[ρ\circφ]}$.
J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.
It is proved for Abelian groups that the Reidemeister coincidence number of two endomorphisms $ϕ$ and $ψ$ is equal to the number of coincidence points of $\whϕ$ and $\whψ$ on the unitary dual, if the Reidemeister number is finite. An affirmative answer to the bitwisted Dehn conjugacy problem for almost polycyclic groups is obtained. Finally we explain why the Reidemeister numbers are always infinite for injective endomorphisms of Baumslag-Solitar groups.
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 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…
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 …
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…
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…
This is a (mostly expository) paper on Reidemeister classes, twisted Burnside-Frobenius theory, congruences, R-infinity property and all that. It was written in 2005 and published in 2008. We post it as it was, only the bibliography data is updated. For some of the recent progress see e.g. arXiv:0903.4533, arXiv:0903.3455, arXiv:0802.2937, arXiv:0712.2601, arXiv:0704.3411, arXiv:math/0703744, arXiv:math/0606725, arXiv:math/0606764, arXiv:0805.1371 and references there.
We give a new, unexpected characterization of saturated fusion systems on a p-group S in terms of idempotents in the p-local double Burnside ring of S that satisfy a Frobenius reciprocity relation, and reformulate fusion-theoretic phenomena in the language of idempotents. Interpreting our results in stable homotopy, we answer a long-standing question on stable splittings of classifying spaces of finite groups, and generalize the Adams--Wilkerson criterion for recognizing rings of invariants in the cohomology of an elementary abelian p-group. This work is partly motivated by a conjecture of Hay…
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 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 property and study its behavior under some extensions. An affirmative answer to the twisted Dehn conjugacy problem for polycyclic-by-finite group is obtained. Some problems for the further study are indicated.
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…
