Twisted Burnside-Frobenius theory for endomorphisms of polycyclic groups

TL;DR

This paper extends the Burnside-Frobenius theory to endomorphisms of polycyclic groups, establishing a correspondence between Reidemeister numbers and fixed points of induced maps on the unitary dual, with applications to Gauss congruences.

Contribution

It proves a twisted Burnside-Frobenius theorem for endomorphisms of polycyclic groups, linking Reidemeister numbers to fixed points on the dual space, and explores finiteness conditions.

Findings

Reidemeister number equals fixed points of induced dual map for certain groups.

Established Gauss congruences for Reidemeister numbers.

Identified examples with finite Reidemeister numbers among groups with R_infinity property.

Abstract

Let $R(\phi)$ be the number of $\phi$-conjugacy (or Reidemeister) classes of an endomorphism $\phi$ of a group $G$. We prove for several classes of groups (including polycyclic) that the number $R(\phi)$ is equal to the number of fixed points of the induced map of an appropriate subspace of the unitary dual space $\widehat G$, when $R(\phi)<\infty$. Applying the result to iterations of $\phi$ 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 $R_\infty$ property.

Equations

$$g\sim xg\phi (x^{-1}).$$

$$xy\phi (x^{-1})g=x(yg)g^{-1}\phi (x^{-1})g=x(yg)({\tau }_{g^{-1}}\circ \phi )(x^{-1}).$$

$$\rho \sim \rho \circ {\phi }^n\sim \cdots \sim \rho \circ {\phi }^{kn}\sim \dots$$

$$\mu(d)=\left\{\begin{array}[]{ll}1&{\rm if}\ d=1,\\ (-1)^{k}&{\rm if}\ d\ {\rm is\ a\ product\ of}\ k\ {\rm distinct\ primes,}\\ 0&{\rm if}\ d\ {\rm is\ not\ square-free.}\end{array}\right.$$

$$\sum _{d\mid n}\mu (d)\cdot R({\varphi }^{n/d})\equiv 0\mathrm{mod}n$$

$$\sum _{d|n}\mu (d)R({\phi }^{n/d})=P_n\equiv 0\mathrm{mod}n,$$

$$x\phi (x^{-1})y\phi (y^{-1})=(xy)\phi ((xy{)}^{-1}),(x\phi (x^{-1}){)}^{-1}=x^{-1}\phi (x),$$

$$xg\phi (x^{-1})(yg\phi (y^{-1}){)}^{-1}=(x{y}^{-1})\phi ((x^{-1}y)).$$

$$\mathbb{C}[G]\cong \underset{[\rho ]\in \hat{G}}{⨁}\mathrm{End}{V}_{\rho },\rho :G\to U(V_{\rho }),$$

$$R(\phi )=\sum _{[\rho ]\in \hat{G}}\dim T_{\rho },T_{\rho }:=\{a\in \mathrm{End}{V}_{\rho }|a=\rho (g)a\rho (\phi (g^{-1})\text{mbox}forallg\in G\}.$$

$$R({\varphi})=\sum\limits_{[\rho]\in\widehat{G}}\left\{\begin{array}[]{l}1,\mbox{ if }\rho\sim\rho\circ{\varphi}\\ 0,\mbox{ if }\rho\not\sim\rho\circ{\varphi}\end{array}\right.=\mbox{ number of }\widehat{{\varphi}}\mbox{-{f}-points}.$$

$$T_{S,\rho }:g\mapsto \mathrm{Tr}(S\circ \rho (g)),$$

$$\rho (\phi (x))S=S\rho (x)\text{mbox}foranyx\in G.$$

$$T_{S,\rho }(xg\phi (x^{-1}))=\mathrm{Tr}(S\rho (xg\phi (x^{-1})))=\mathrm{Tr}(\rho (\phi (x))S\rho (g)\rho (\phi (x^{-1}))=\mathrm{Tr}(S\rho (g)).$$

$$\mathrm{Tr}(D\rho (a))=\mathrm{Tr}(D\rho (xa\phi (x^{-1})))=\mathrm{Tr}(\rho (\phi (x^{-1}))D\rho (x)\rho (a)).$$

$$yg\phi (y^{-1})zh{z}^{-1}=y(gz)z^{-1}\phi (y^{-1})zh{z}^{-1}=y(gz)({\tau }_{z^{-1}}\circ \phi )(y^{-1})h{z}^{-1}$$

$$=x(gz)({\tau }_{z^{-1}}\circ \phi )(x^{-1})z^{-1}=xg\phi (x^{-1})$$

$$e\cdot h=xe({\tau }_g\circ \phi )(x^{-1}).$$

$$\phi ({\tau }_g(h_0))=\phi (g)\phi (h_0)(\phi (g){)}^{-1}={\tau }_{\phi (g)}(\phi (h_0)),h_0,\phi (h_0)\in H_0,$$

$$H\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

Full text

In contrast with the case of automorphisms, studied previously, we have a plenty of examples having the above finiteness condition, even among groups with $R_{\infty}$ property.

The work of A.F. is partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

The work of E.T. is partially supported by the Russian Foundation for Basic Research under grant 16-01-00357.

Introduction

The Reidemeister number or ${\varphi}$-conjugacy number of an endomorphism ${\varphi}$ of a group $G$ is the number of its Reidemeister or ${\varphi}$-conjugacy classes, defined by the equivalence

[TABLE]

The interest in twisted conjugacy relations has its origins, in particular, in the Nielsen-Reidemeister fixed point theory (see, e.g. [29, 30, 6]), in Arthur-Selberg theory (see, e.g. [42, 1]), Algebraic Geometry (see, e.g. [27]), and Galois cohomology (see, e.g. [41]). In representation theory twisted conjugacy probably occurs first in [23] (see, e.g. [44, 37]).

An important problem in the field is to identify the Reidemeister numbers with numbers of fixed points on an appropriate space in a way respecting iterations. This opens possibility of obtaining congruences for Reidemeister numbers and other important information.

For the role of the above “appropriate space” typically some versions of unitary dual can be taken. This desired construction is called the twisted Burnside-Frobenius theory (TBFT), because in the case of a finite group and identity automorphism we arrive to the classical Burnside-Frobenius theorem on enumerating of (usual) conjugacy classes.

In the present paper we prove TBFT for endomorphisms of any polycyclic group (Theorem 6.1) and Gauss congruences for Reidemeister numbers.

In the case of automorphism this problem was solved for polycyclic-by-finite groups in [13, 21]. Preliminary and related results, examples and counter-examples can be found in [8, 5, 6, 12, 14, 9, 11, 45, 46].

For endomorphisms of polycyclic fundamental groups of infra-solvmanifolds of type (R) the Gauss congruences were obtained in [19].

The importance of obtaining the present results namely for endomorphisms is justified by a plenty of examples (in contrast with the case of automorphisms) (see, in particular, 1.1, 1.4, and 6.4 below).

A related fact is that, in contrast with TBFT for automorphisms, TBFT for endomorphisms is weakly connected with the theory of $R_{\infty}$-groups (see e.g. Example 6.4). A group is called $R_{\infty}$ if any its automorphism has infinite Reidemeister number. This was the subject of an intensive recent research and for many groups this property was established, see the following partial bibliography and the literature therein: [7, 33, 15, 16, 10, 2, 25, 17, 36, 34, 18, 26, 40, 4, 20, 28]. In some situations the property $R_{\infty}$ has some direct topological consequences (see e.g. [26]).

Decision problems for twisted conjugacy classes of endomorphisms of polycyclic groups were studied in [39].

The paper is organized in the following way.

In Section 1 we show how drastically the theory of twisted conjugacy classes for endomorphisms differs from the theory for automorphisms.

In Section 2 we introduce and investigate a dual object for a pair $(G,{\varphi})$.

In Section 3 we prove Gauss congruences for Reidemeister numbers.

In Section 4 we discuss proof of twisted Burnside–Frobenius theorem for endomorphisms of any finite group.

Section 5 is technical.

In Section 6 we proof the twisted Burnside–Frobenius theorem for endomorphisms of polycyclic groups. We finish the paper with a series of examples of $R_{\infty}$ groups admitting endomorphisms with finite Reidemeister numbers .

Acknowledgement: This work is a part of our joint research project at the Max-Planck Institute for Mathematics (Bonn) and the most part of the results were obtained there in Spring 2017.

The work of A.F. is partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.

The work of E.T. is partially supported by the Russian Foundation for Basic Research under grant 16-01-00357.

1. Preliminaries

First of all, let us make the following observation, showing how drastically the Reidemeister numbers world for endomorphisms differs from the Reidemeister world for automorphisms.

Proposition 1.1**.**

For any group $G$ there exists an endomorphism ${\varphi}:G\to G$ with $R({\varphi})<\infty$, namely $R({\varphi})=1$.

Proof.

Take ${\varphi}$ to be the trivial map ${\varphi}(g)=e$ for any $g\in G$. ∎

This observation can be enforced.

Proposition 1.2**.**

Suppose, ${\varphi}:G\to G$ is an endomorphism and $K:=\mathop{\rm Ker}\nolimits{\varphi}$. Then all Reidemeister classes are some unions of $K$-cosets.

Proof.

Let $g_{1}$ and $g_{2}$ be in a $K$-coset, i.e. $g_{1}g_{2}^{-1}=k\in K$. Then $g_{1}=kg_{2}=kg_{2}{\varphi}(k^{-1})$. ∎

Using Lemma 1.9 we immediately obtain

Corollary 1.3**.**

The map $p_{\varphi}:G\to G/K$ gives a bijection of Reidemeister classes.

Corollary 1.4**.**

Any endomorphism with finite image has a finite Reidemeister number.

Definition 1.5**.**

Denote by $\widehat{G}$ the unitary dual of $G$, by $\widehat{G}_{f}$ the part of the unitary dual formed by irreducible finite-dimensional representations, and by $\widehat{G}_{ff}$ the part of $\widehat{G}_{f}$ formed by finite representations, i.e. representations that factorize through a finite group.

Definition 1.6**.**

Let us call o.t. commutant the operator theoretical commutant of a set $D$ of bounded operators on a Hilbert space, i.e. the subset of the algebra of all bounded operators on this space, formed by all operators that commute with all elements of $D$. Denote it $D^{\bigstar}$.

Lemma 1.7**.**

A representation is irreducible if and only if the o.t. commutant of the set of representing operators is formed just by scalar operators [31, Theorem 2, p. 114].

In particular, if $\rho$ is irreducible and ${\varphi}$ is an epimorphism, $\rho\circ{\varphi}$ is also irreducible.

Lemma 1.8**.**

If representations $\pi$ and $\rho$ of $G$ are equivalent, then $\pi\circ{\varphi}$ and $\rho\circ{\varphi}$ are equivalent for any endomorphism ${\varphi}:G\to G$.

Proof.

Indeed: use the same intertwining operator. ∎

The following statement is well known

Lemma 1.9**.**

Suppose, ${\varphi}:G\to G$ is an endomorphism and $H\subset G$ is a normal ${\varphi}$-invariant subgroup, then $p:G\to G/H$ induces an epimorphism of Reidemeister classes.

Proof.

Indeed, suppose, $p(g^{\prime})=p(\widetilde{g})p(g)p({\varphi}(\widetilde{g}^{-1}))$. Then it is equal to $p(\widetilde{g}g{\varphi}(\widetilde{g}^{-1}))$. ∎

Also, we need the following

Lemma 1.10**.**

Any Reidemeister class of ${\varphi}$ is ${\varphi}$-invariant.

Proof.

Indeed, ${\varphi}(x)=x^{-1}x{\varphi}(x)$. ∎

The following fact can be extracted from [24, Prop. 1.6].

Lemma 1.11**.**

In the above situation $R({\varphi}|_{H})\leqslant R({\varphi})\cdot|C({\varphi}_{G/H})|$ where $C({\varphi}_{G/H})$ is the fixed point subgroup for the induced map ${\varphi}_{G/H}:G/H\to G/H$.

The following statement is well known in the field.

Lemma 1.12**.**

A right shift by $g\in G$ maps bijectively Reidemeister classes of ${\varphi}$ onto Reidemeister classes of $\tau_{g^{-1}}\circ{\varphi}$, where $\tau_{g}$ is the inner automorphism: $\tau_{g}(x)=gxg^{-1}$.

Proof.

This follows immediately from the equality

[TABLE]

∎

2. Dual object for a pair $(G,{\varphi})$

For an automorphism or at least epimorphism ${\varphi}$ and an irreducible representation $\rho$, the representation $\rho\circ{\varphi}$ is always irreducible. For a general endomorphism and a finite-dimensional $\rho$ one can only decompose the representation into irreducible components. So, one obtains a sort of multi-valued mapping and the corresponding dynamical system is not defined. To avoid this problem we will act as follows.

Definition 2.1**.**

We will call a representation class $[\rho]$ a $\widehat{{\varphi}}$-f-point, if $\rho$ is equivalent to $\rho\circ{\varphi}$ (we avoid to say that it is a fixed point, because we can not define the corresponding dynamical system).

Definition 2.2**.**

An element $[\rho]\in\widehat{G}$ (respectively, in $\widehat{G}_{f}$ or $\widehat{G}_{ff}$) is called ${\varphi}$-irreducible if $\rho\circ{\varphi}^{n}$ is irreducible for any $n=0,1,2,\dots$.

Denote the corresponding subspaces of $\widehat{G}$ (resp., $\widehat{G}_{f}$ or $\widehat{G}_{ff}$) by $\widehat{G}^{\varphi}$ (resp., $\widehat{G}^{\varphi}_{f}$ or $\widehat{G}^{\varphi}_{ff}$).

In some important cases these subspaces coincide with the entire spaces:

Proposition 2.3**.**

1. (1)

If $G$ is abelian, $\widehat{G}=\widehat{G}_{f}=\widehat{G}^{\varphi}=\widehat{G}^{\varphi}_{f}$. 2. (2)

If ${\varphi}$ is an epimorphism, $\widehat{G}=\widehat{G}^{\varphi}$ and $\widehat{G}_{f}=\widehat{G}^{\varphi}_{f}$

Proof.

The first statement immediately follows from the fact that a representation of an abelian group is irreducible if and only if it is 1-dimensional.

The second one follows from Lemma 1.7 keeping in mind that ${\varphi}^{n}$ is an epimorphism if and only if ${\varphi}$ is. ∎

Evidently continuous (w.r.t. the topology of weak containment) maps $\widehat{{\varphi}^{n}}:[\rho]\mapsto[\rho\circ{\varphi}^{n}]$ are defined for these subspaces $\widehat{G}^{\varphi}$, $\widehat{G}_{f}^{\varphi}$, and $\widehat{G}_{ff}^{\varphi}$ (generally not a homeomorphism!) and $\widehat{{\varphi}^{n}}=(\widehat{{\varphi}})^{n}$. Thus we obtain a dynamical system as the corresponding action of the semigroup $\mathbb{N}_{0}=\{0,1,2,\dots\}$ (we will reserve $\mathbb{N}$ for $\{1,2,\dots\}$).

The key observation is the following one:

Lemma 2.4**.**

Let $[\rho]\in\widehat{G}$ be an $\widehat{{\varphi}}^{n}$-f-point for some $n\geq 1$, i.e. the representations $\rho$ and $\rho\circ{\varphi}^{n}$ are equivalent. Then $[\rho]\in\widehat{G}^{\varphi}$.

Proof.

By Lemma 1.8 we obtain

[TABLE]

In particular, all they are irreducible. Now consider an arbitrary $m$ and choose $k$ such that $m\leq kn$. Then $\mathop{\rm Im}\nolimits\rho\circ{\varphi}^{m}\supseteq\mathop{\rm Im}\nolimits\rho\circ{\varphi}^{kn}$ and $(\mathop{\rm Im}\nolimits\rho\circ{\varphi}^{m})^{\bigstar}\subseteq(\mathop{\rm Im}\nolimits\rho\circ{\varphi}^{kn})^{\bigstar}$. It remains to apply Lemma 1.7 and conclude that $\rho\circ{\varphi}^{m}$ is irreducible. ∎

Corollary 2.5**.**

So, we have no dynamical system generated by ${\varphi}$ on $\widehat{G}$ (resp, $\widehat{G}_{f}$, or $\widehat{G}_{ff}$) generally, but we have the well-defined notion of a $\widehat{{\varphi}}^{n}$-f-point.

The corresponding well-defined dynamical system exists on $\widehat{G}^{\varphi}$ (resp, $\widehat{G}^{\varphi}_{f}$, or $\widehat{G}^{\varphi}_{ff}$) and its $n$-periodic points are exactly $\widehat{{\varphi}}^{n}$-f-points.

Definition 2.6**.**

Denote the number of $\widehat{{\varphi}}^{n}$-f-points by $\mathbf{F}(\widehat{{\varphi}}^{n})$.

3. TBFT implies congruences

Definition 3.1**.**

We say that TBFT (resp., TBF$T_{f}$, TBF$T_{ff}$) takes place for an endomorphism ${\varphi}:G\to G$ and its iterations, if $R({\varphi}^{n})<\infty$ and $R({\varphi}^{n})$ coincides with the number of $\widehat{{\varphi}}^{n}$-f-points in $\widehat{G}$ (resp., in $\widehat{G}_{f}$, $\widehat{G}_{ff}$) for all $n\in\mathbb{N}$.

Similarly, one can give a definition for a single endomorphism (without iterations).

Definition 3.2**.**

Denote by $\mu(d)$, $d\in{\mathbb{N}}$, be the Möbius function, i.e.

[TABLE]

Theorem 3.3**.**

Suppose, TBFT (resp., TBFTf or TBFTff) takes place for an endomorphism ${\varphi}:G\to G$ and its iterations. In particular, $R({\varphi}^{n})<\infty$ for any $n$. Then one has the following Gauss congruences for Reidemeister numbers:

[TABLE]

for any $n$.

Proof.

This follows from 2.5 and the general theory of congruences for periodic points (cf. [43, 47]).

More precisely, let $P_{n}$ be the number of periodic points of least period $n$ of the dynamical system of 2.5. Then $R({\varphi}^{n})=\mathbf{F}(\widehat{{\varphi}}^{n})=\sum\limits_{d|n}P_{d}$. By the Möbius inversion formula,

[TABLE]

since each orbit brings to $P_{n}$ just $n$ points. ∎

Remark 3.4**.**

In [22] it is shown that TBFTf and TBFTff are equivalent for finitely generated groups (see also [46]).

4. TBFT for endomorphisms of abelian and finite groups

For these classes all irreducible representations are finite-dimensional. That is why TBFT is the same as TBFTf. In [8] using Pontryagin duality the following statement was proved.

Proposition 4.1**.**

TBFTf holds for any endomorphism of an abelian group.

For polycyclic groups below we need also

Proposition 4.2**.**

TBFTff holds for any endomorphism of an abelian group.

Proof.

Indeed, in an abelian group

[TABLE]

[TABLE]

This shows that the Reidemeister class of $e$ is a subgroup $H$, and the other classes are $H$-cosets. Being a Reidemeister class, $H$ is ${\varphi}$-invariant (see Lemma 1.10 above) and the factorization $p:G\to G/H$ gives a bijection of Reidemeister classes. The induced action on $G/H$ is trivial, as well as on $\widehat{G/H}$. The fixed representations $\rho\circ p$, where $\rho$ runs over $\widehat{G/H}$, are desired finite representations. ∎

Theorem 4.3** (cf. [8]).**

Let ${\varphi}:G\to G$ be an endomorphism of a finite group $G$. Then the Reidemeister number $R({\varphi})$ coincides with the number of $\widehat{{\varphi}}$-f-points on $\widehat{G}$, i.e. TBFT is true in this situation.

Proof.

Let us note that $R({\varphi})$ is equal to the dimension of the space of ${\varphi}$-class functions (i.e. those functions that are constant on Reidemeister classes). They can be also described as fixed elements of the action $a\mapsto ga{\varphi}(g^{-1})$ on the group algebra ${\mathbb{C}}[G]$. For the latter algebra we have the Peter-Weyl decomposition

[TABLE]

which respects the left and right $G$-actions. Hence,

[TABLE]

Thus, if $0\neq a\in T_{\rho}$, $a$ is an intertwining operator between the irreducible representation $\rho$ and some representation $\rho\circ{\varphi}$. This implies that $\rho$ is equivalent to some (irreducible) subrepresentation $\pi$ of $\rho\circ{\varphi}$ (cf. [35, VI, p.57]). Hence, $\dim\rho=\dim\pi$, while $\dim\rho=\dim\rho\circ{\varphi}$. Thus, $\pi=\rho\circ{\varphi}$, and is irreducible. In this situation $\dim T_{\rho}=1$ by the Schur lemma. Evidently, vice versa, if $\rho\sim\rho\circ{\varphi}$ then $\dim T_{\rho}=1$. Hence,

[TABLE]

∎

5. Technical lemmas

We will need the following

Lemma 5.1**.**

Let $\rho$ be a finite representation. It is a $\widehat{{\varphi}}$-f-point if and only if there exists a non-zero ${\varphi}$ class function being a matrix coefficient of $\rho$.

In this situation this function is unique up to scaling and is defined by the formula

[TABLE]

where $S$ is an intertwining operator between $\rho$ and $\rho\circ{\varphi}$:

[TABLE]

In particular, TBFTff is true for ${\varphi}$ if and only if the above matrix coefficients form a base of the space of ${\varphi}$-class functions.

Proof.

First, let us note that (1) defines a class function:

[TABLE]

If $S\neq 0$, then $\rho(a)=S^{*}$ for some $a\in\ell^{1}(G)$, and $\operatorname{Tr}(SS^{*})\neq 0$. Thus, the ${\varphi}$-class function is non-zero. On the other hand, any matrix coefficient of $\rho$, i.e. a functional $T:\mathop{\rm End}\nolimits(V_{\rho})\to{\mathbb{C}}$ has the form $g\mapsto\operatorname{Tr}(D\rho(g))$ for some fixed matrix $D\neq 0$. If it is a ${\varphi}$-class function, then for any $g\in G$, or similarly, $a\in\ell^{1}(G)$,

[TABLE]

Since $\rho(a)$ runs over all the matrix algebra, this implies $D=\rho({\varphi}(x^{-1}))D\rho(x)$, or $\rho({\varphi}(x))D=D\rho(x)$, i.e. $D$ is the desired non-zero intertwining operator.

The uniqueness up to scaling follows now from the explicit formula and the Shur lemma.

The last statement follows from linear independence of matrix coefficients of non-equivalent representations. ∎

Lemma 5.2**.**

A group $G$ satisfies TBFTff for an endomorphism ${\varphi}:G\to G$ if and only if there exists a ${\varphi}$-equivariant factorization of $G$ onto a finite group $F$, such that Reidemeister classes on $G$ map onto distinct classes on $F$.

Proof.

Let $\rho_{1}$,…$\rho_{k}$ be all finite $\widehat{{\varphi}}$-f-representations, $F_{1}$, …$F_{k}$, $F_{i}=\rho_{i}(G)$ the corresponding finite groups. Suppose, $g\in\mathop{\rm Ker}\nolimits\rho_{i}=:K_{i}$. Then $\rho({\varphi}(g))=S\rho(g)S^{-1}=SS^{-1}=e$. Hence, $K_{i}$ is a normal ${\varphi}$-invariant subgroup. Define $K:=\cap K_{i}$. It is still a normal ${\varphi}$-invariant subgroup of finite index. Let $F:=G/K$. By Lemma 1.9 the Reidemeister number of the induced map ${\varphi}_{F}$ satisfies $R({\varphi}_{F})\leq R({\varphi})$. On the other hand, $\rho_{i}$ define disjoint irreducible representations of $F$ corresponding to $\widehat{{\varphi}}_{F}$-f-points on $\widehat{F}$. By Lemma 5.1 this implies $R({\varphi}_{F})\geq R({\varphi})$. Thus, $G\to F$ gives a bijection of Reidemeister classes.

The opposite statement follows from Lemma 5.1 and Theorem 4.3. Indeed, if $p:G\to F$ is the mentioned epimorphism with $r=R({\varphi})=R({\varphi}_{F})$, then by Theorem 4.3 we have exactly $r$ pairwise non-equivalent irreducible unitary $\widehat{{\varphi}_{F}}$-f-representations $\rho_{1},\dots,\rho_{r}$ of $F$. Then $\rho_{i}\circ p$ play the same role for $G$. Finally $G$ can not have “additional” $\widehat{{\varphi}}$-f-representations by linear independence in Lemma 5.1. ∎

We can enforce this statement.

Lemma 5.3**.**

Let ${\varphi}:G\to G$ be an endomorphism with $R({\varphi})<\infty$. Suppose there exists a (not necessarily equivariant) factorization $p$ of $G$ onto a finite group $F$, such that Reidemeister classes on $G$ map bijectively onto some disjoint subsets (not necessarily classes) on $F$. Then $G$ has TBFTff for ${\varphi}$.

Proof.

For $F$, characteristic functions of any sets are matrix coefficients of some (finite) representations. Thus, characteristic functions of Reidemeister classes on $G$ are matrix coefficients of some finite representations (coming from $F$). Lemma 5.1 completes the proof. ∎

Definition 5.4**.**

We will say under the supposition of Lemma 5.3 that $p$ separates Reidemeister classes. Similarly, we will say that $p$ separates some classes if it maps them to disjoint subsets.

Lemma 5.5**.**

Let ${\varphi}$ be an endomorphism of a group $G$ with $R({\varphi})<\infty$. Then $G$ has TBFTff for ${\varphi}$ if and only if the right shifts of all Reidemeister classes form a finite number of subsets of $G$.

Proof.

First, let us note, that if we have a bijection for Reidemeister classes of ${\varphi}$, then we have a bijection for Reidemeister classes of any $\tau_{g}\circ{\varphi}$ by Lemma 1.12.

Now the “only if” direction is evident, because these sets are pre-images of some sets in finite $F$.

Suppose, there are finitely many shifts. This means, that the stabilizers under right shifts of any Reidemeister class of ${\varphi}$ have finite index. Since $R({\varphi})<\infty$, their intersection is a subgroup $H$ in $G$ of finite index. By Lemma 1.12, its elements stabilize Reidemeister classes of any $\tau_{g}\circ{\varphi}$.

Suppose, $h\in H$, then

[TABLE]

[TABLE]

for some $x$. Thus, $H$ is normal (and ${\varphi}$-invariant, if ${\varphi}$ is an automorphism). The projection $G\to G/H=:F$ maps Reidemeister classes of ${\varphi}$ bijectively onto disjoint sets in $F$. Indeed, if $h\in H$, then for any $g$,

[TABLE]

This means, that $H$ entirely is in the Reidemeister class of $e$ of any $\tau_{g}\circ{\varphi}$. By Lemma 1.12 this means that all Reidemeister classes of $\phi$ are formed by $H$-cosets and we are done.

It remains to apply Lemma 5.3. ∎

Lemma 5.6**.**

Suppose, $G$ has only finitely many inner automorphisms and an endomorphism ${\varphi}:G\to G$ with $R({\varphi})<\infty$. Then $G$ has TBFTff for ${\varphi}$.

Proof.

By Lemma 1.12, the number of right shifts of Reidemeister classes is not more than $R(\phi)\cdot|Inn(G)|$. It remains to apply Lemma 5.5. ∎

Lemma 5.7**.**

Let ${\varphi}:A\to A$ be an endomorphism of a finitely generated abelian group $A$ with $R({\varphi})<\infty$. Then the number of fixed elements $C({\varphi})$ on $A$ is finite.

Proof.

The torsion subgroup is finite and totally invariant. Factorizing we reduce the problem to the case of ${\mathbb{Z}}^{n}$ by Lemma 1.9. In this case we will show that ${\varphi}$ has only the trivial fixed element [math]. By [8] in this case $\det(\operatorname{Id}-{\varphi})\neq 0$, considered as $n\times n$ integer matrix. Diagonalising this matrix by left and right multiplication by unimodular matrix (as it was done e.g. in [3]) we see that it can not have (non-zero) eigenvector with eigenvalue zero, i.e. there is no non-trivial ${\varphi}$-fixed point. ∎

6. TBFTff for endomorphisms of polyciclic groups

Consider a polycyclic group $G$. Its (finite) derived series is formed by fully invariant subgroups $G_{i}$ with abelian quotients $A_{i}=G_{i}/G_{i+1}$. The key difference from a general finitely generated solvable group is the following: all $G_{i}$ and $A_{i}$ are finitely generated (see [38] for details). Let $G_{n}\neq\{e\}$, $G_{n+1}=\{e\}$.

We will argue by induction. For the basis of this induction let us observe that for the abelian group $G_{n}$ and any its endomorphism with finite Reidemeister number we have TBFTff by Prop. 4.1. Now suppose by induction that same is true for $G_{i+1}$ and prove it for $G_{i}$. Denote $G_{i+1}=:H$, $G_{i}=:{\Gamma}$, $A_{i}=:A$.

Consider an endomorphism ${\varphi}:{\Gamma}\to{\Gamma}$ with $R({\varphi})<\infty$ and induced endomorphisms ${\varphi}_{H}:H\to H$ and ${\varphi}_{A}:A\to A$. Then by Lemma 1.9 $R({\varphi}_{A})<\infty$ and by Lemma 5.7 the number of fixed elements of ${\varphi}_{A}$ on $A$ is finite. Thus, by Lemma 1.11 $R({\varphi}_{H})<\infty$. Let $H_{0}\subset H$ be a normal ${\varphi}$-invariant subgroup of finite index such that $H\to H/H_{0}$ gives a bijection on Reidemeister classes (see Lemma 5.2). This means that classes of ${\varphi}_{H}$ are some unions of $H_{0}$ cosets. Consider automorphisms $\tau_{g}:H\to H$, $\tau_{g}(h)=ghg^{-1}$, $g\in{\Gamma}$, and define $H_{1}:=\cap_{g\in{\Gamma}}\tau_{g}(H_{0})$. All subgroups in the intersection have the same finite index in $H$. Since $H$ is finitely generated, $H_{1}$ also has a finite index. By construction, $H_{1}$ is normal in ${\Gamma}$. Also,

[TABLE]

i.e. ${\varphi}(\tau_{g}(H_{0}))\subset\tau_{{\varphi}(g)}(H_{0})$. Hence, $H_{1}$ is ${\varphi}$-invariant and $H_{1}\subset H_{0}$, thus $H_{1}$ can play the same role as $H_{0}$ with an additional property of being normal in ${\Gamma}$. In particular, classes of ${\varphi}_{H}$ are some unions of $H_{1}$ cosets. The same is true for intersections of Reidemeister classes of ${\varphi}$ with $H$ (because they are unions of some classes of ${\varphi}_{H}$). This means that ${\Gamma}\to{\Gamma}/H_{1}$ separates these classes, i.e. the classes which map on the Reidemeister class of $e\in A$ under ${\Gamma}\to A$. Similarly we can find subgroups $H_{2},\dots H_{R({\varphi}_{A})}$ which separate classes over other classes (i.e. which are mapped onto other classes) of $A$. For this purpose, suppose $g\in{\Gamma}$ is over some other class of $A$. Then the classes of $\tau_{g}\circ{\varphi}$ (with the same finite Reidemeister number) that intersect with $H$ are obtained from the classes under consideration by a shift by $g$ (see Lemma 1.12). Then a group $H_{2}=H^{g}_{1}$ constructed for $\tau_{g}\circ{\varphi}|_{H}$ in the same way as $H_{1}$ for ${\varphi}_{H}$, will separate these classes. Moreover, it will be $\tau_{g}\circ{\varphi}$-invariant for this specific $g$ and normal in ${\Gamma}$, i.e. $\tau_{g^{\prime}}$-invariant for any $g^{\prime}\in{\Gamma}$. Taking $g^{\prime}=g^{-1}$ we see that $H^{g}_{1}$ is ${\varphi}$-invariant. Similarly we define the other $H_{i}$ choosing $g$ over other Reidemeister classes of ${\varphi}_{A}$. Taking the (finite !) intersection of $H_{1}$, $H_{2}$,… $H_{R({\varphi}_{A})}$ we obtain a ${\varphi}$-invariant subgroup of finite index $H^{\prime}\subset H$, which is normal in ${\Gamma}$, and $p:{\Gamma}\to{\Gamma}/H^{\prime}={\Gamma}^{\prime}$ gives a bijection on Reidemeister classes; $F:=p(H)=H/H^{\prime}\subset{\Gamma}^{\prime}$ is a finite normal subgroup and ${\Gamma}^{\prime}/F\cong A$:

[TABLE]

Thus, ${\Gamma}^{\prime}$ is finitely generated finite-by-abelian group, and it has finitely many inner automorphisms. Lemma 5.6 completes the proof of the following statement.

Theorem 6.1**.**

Let ${\varphi}:G\to G$ be an endomorphism of a polycyclic group with $R({\varphi})<\infty$. Then TBFTff is true for ${\varphi}$.

Remark 6.2**.**

The results of this section can be extended to some virtually polycyclic groups (under supposition of some polycyclic subgroup to be ${\varphi}$-invariant).

More precisely the following statement can be proved by word-by-word rewriting of the above argument.

Theorem 6.3**.**

Let $G$ be an almost polycyclic group admitting a fully invariant polycyclic subgroup of finite index. Then TBFTff is true for any endomorphism ${\varphi}:G\to G$ with $R(\phi)<\infty$.

Let us illustrate this theorem by the following

Example 6.4**.**

In [28] seven series of almost polycyclic (polycyclic-by-finite) groups with the property $R_{\infty}$ (any automorphism has infinite Reidemeister number) were found. They have a fully invariant polycyclic subgroup of finite index ([28, Remark 2.1], [32]). Thus, they are covered by Theorem 6.3.

On the other hand, they evidently have endomorphisms with finite Reidemeister number (see Prop. 1.1 and Cor. 1.4).