A dichotomy between rationality and a natural boundary for Reidemeister type zeta functions

Max-Planck-Institut für Mathematik Bonn

A dichotomy between rationality and a natural boundary for Reidemeister type zeta functions

by

Wojciech Bondarewicz Alexander Fel’shtyn

Malwina Zietek

Max-Planck-Institut für Mathematik Preprint Series 2023 (5)

Date of submission: February 15, 2023

A dichotomy between rationality and a natural boundary for Reidemeister type zeta

functions

by

Wojciech Bondarewicz Alexander Fel’shtyn

Malwina Zietek

Max-Planck-Institut für Mathematik Vivatsgasse 7

53111 Bonn Germany

Instytut Matematyki Uniwersytet Szczecinski ul. Wielkopolska 15 70-451 Szczecin Poland

MPIM 23-5

A DICHOTOMY BETWEEN RATIONALITY AND A NATURAL BOUNDARY FOR REIDEMEISTER TYPE ZETA FUNCTIONS

WOJCIECH BONDAREWICZ, ALEXANDER FEL’SHTYN AND MALWINA ZIETEK

ABSTRACT. We prove a dichotomy between rationality and a natural boundary for the analytic behavior of the Reidemeister zeta function for endomorphisms of groupsZ^dp,whereZpthe group of p-adic integers. We also prove the rationality of the coincidence Reidemeister zeta function for tame endomorphisms pairs of finitely generated torsion-free nilpotent groups, based on a weak commutativity condition.

0. INTRODUCTION

Let G be a group and ϕ : G → G an endomorphism. Two elements α, β ∈ G are said to be ϕ-conjugate or twisted conjugate iff there exists g ∈ G with β = gαϕ(g⁻¹). We shall write {x}_ϕ for the ϕ-conjugacy or twisted conjugacyclass of the elementx ∈G. The number ofϕ-conjugacy classes is called the Reidemeister number of an endomorphism ϕ and is denoted byR(ϕ). Ifϕis the identity map then theϕ-conjugacy classes are the usual conjugacy classes in the groupG. We call the endomorphismsϕ tameif the Reidemeister numbersR(ϕⁿ)are finite for alln ∈N. Taking a dynamical point of view, we consider the iterates of a tame endomorphism ϕ, and we may define following [11] a Reidemeister zeta function ofϕas a power series:

R_ϕ(z) = exp

∞

X

n=1

R(ϕⁿ) n zⁿ

! ,

where z denotes a complex variable. The following problem was investi- gated [13]: for which groups and endomorphisms is the Reidemeister zeta function a rational function? Is this zeta function an algebraic function?

In [11, 13, 19, 14, 12], the rationality of the Reidemeister zeta function R_ϕ(z)was proven in the following cases: the group is finitely generated and

2010Mathematics Subject Classification. Primary 37C25; 37C30; 22D10; Secondary 20E45; 54H20; 55M20.

Key words and phrases. Twisted conjugacy class; Reidemeister number; Reidemeister zeta function; unitary dual.

The work is funded by the Narodowe Centrum Nauki of Poland (NCN) (grant No.2016/23/G/ST1/04280 (Beethoven 2)).

1

an endomorphism is eventually commutative; the group is finite; the group is a direct sum of a finite group and a finitely generated free abelian group;

the group is finitely generated, nilpotent and torsion-free. In [29] the ratio- nality of the Reidemeister zeta function was proven for endomorphisms of fundamental groups of infra-nilmanifolds under some sufficient conditions.

Recently, the rationality of the Reidemeister zeta function was proven for endomorphisms of fundamental groups of infra-nilmanifolds [6]; for endo- morphisms of fundamental groups of infra-solvmanifolds of type (R) [16];

for automorphisms of crystallographic groups with diagonal holonomyZ2

and for automorphisms of almost-crystallographic groups up to dimension 3 [7]; for the right shifts of a non-finitely generated, non-abelian torsion groupsG=⊕i∈ZF_i,F_i ∼=F andF is a finite non-abelian group [28].

LetGbe a group andϕ, ψ:G →Gtwo endomorphisms. Two elements α, β ∈Gare said to be(ϕ, ψ)−conjugateiff there existsg ∈Gwith

β =ψ(g)αϕ(g⁻¹).

The number of (ϕ, ψ)-conjugacy classes is called the Reidemeister coin- cidence number of endomorphisms ϕ and ψ, denoted by R(ϕ, ψ). If ψ is the identity map then the(ϕ, id)-conjugacy classes are theϕ- conjugacy classes in the groupGandR(ϕ, id) = R(ϕ). The Reidemeister coincidence numberR(ϕ, ψ)has useful applications in Nielsen coincidence theory. We call the pair (ϕ, ψ) of endomorphisms tame if the Reidemeister numbers R(ϕⁿ, ψⁿ)are finite for alln ∈ N. For such a tame pair of endomorphisms we define following [15] thecoincidence Reidemeister zeta function

R_ϕ,ψ(z) = exp

∞

X

n=1

R(ϕⁿ, ψⁿ) n zⁿ

! .

Ifψ is the identity map thenR_ϕ,id(z) = R_ϕ(z). In the theory of dynam- ical systems, the coincidence Reidemeister zeta function counts the syn- chronisation points of two maps, i.e. the points whose orbits intersect under simultaneous iteration of two endomorphisms; see [23], for instance.

In [17], in analogy to works of Bell, Miles, Ward [1] and Byszewski, Cornelissen [2, §5] about Artin–Mazur zeta function, the P´olya–Carlson dichotomy between rationality and a natural boundary for analytic behavior of the coincidence Reidemeister zeta function was proven for tame pair of commuting automorphisms of non-finitely generated torsion-free abelian groups that are subgroups ofQ^d, d≥1.

In [15] P´olya–Carlson dichotomy was proven for coincidence Reidemeis- ter zeta function of tame pair of endomorphisms of non-finitely generated torsion-free nilpotent groups of finite Pr¨ufer rank by means of profinite completion techniques.

In this paper we prove a dichotomy between rationality and a natural boundary for the Reidemeister zeta function of endomorphisms of the groupsZ^dp, d ≥ 1, whereZ^p, p-prime, is the additive group ofp-adic inte- gers.

We also prove the rationality of the coincidence Reidemeister zeta func- tion for tame endomorphisms pairs of finitely generated torsion-free nilpo- tent groups, based on a weak commutativity condition .

Acknowledgments. This work was supported by the grant Beethoven 2 of the Narodowe Centrum Nauk of Poland(NCN), grant No.

2016/23/G/ST1/04280. The second author is indebted to the Max-Planck- Institute for Mathematics(Bonn) for the support and hospitality and the pos- sibility of the present research during his visit there.

1. P ´OLYA–CARLSON DICHOTOMY FOR THEREIDEMEISTER ZETA FUNCTION OF ENDOMORPHISMS OF THE GROUPSZ^dp

In this section we prove a P´olya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the Reidemeister zeta function for endomorphisms of groupsZ^dp, d ≥ 1, whereZ^p, p-prime, de- notes the additive group of p-adic integers. The groupZ^p is the most basic infinite pro-p group, it is totally disconnected, compact, abelian, torsion- free group.The field of p-adic numbers is denoted by Qp and the p-adic absolute value (as well as its unique extension to the algebraic closureQp) by|·|_p.

We remind the definition of a natural boundary ( see [26], sec. 6.2).

Definition 1.1. Suppose that an analytic functionF is defined somehow in a regionDof the complex plane. If there is no point of the boundary∂Dof Dover whichF can be analytically continued, then∂Dis called anatural boundaryforF .

We need the following statement Lemma 1.2. (cf. [1]) Let Z(z) = P∞

n=1R(ϕⁿ)zⁿ. If R_ϕ(z) is rational then Z(z) is rational. If R_ϕ(z) has an analytic continuation beyond its circle of convergence, then so does Z(z) too. In particular, the existence of a natural boundary at the circle of convergence for Z(z) implies the existence of a natural boundary forR_ϕ(z).

Proof. This follows from the fact thatZ(z) = z·R_ϕ(z)^′/R_ϕ(z). □

One of the important links between the arithmetic properties of the coef- ficients of a complex power series and its analytic behaviour is given by the P´olya–Carlson theorem [26].

P´olya–Carlson Theorem. A power series with integer coefficients and ra- dius of convergence1 is either rational or has the unit circle as a natural boundary.

Lemma 1.3. End(Zp) =Zp for abelian groupZp.

Proof. Let ϕ ∈ End(Zp). We have pⁿϕ(x) = ϕ(pⁿx). Then ϕ(pⁿZp) ⊂ pⁿZp, so ϕ is continuous. For every x ∈ Zp there exists a sequence of integersx_nconverging tox. Then

ϕ(x) = limϕ(xn) = limxnϕ(1) =ϕ(1)x,

soϕis a multiplication byϕ(1). □

Letϕ ∈End(Zp), thenϕ(x) = ax, wherea ∈Zp. We haveϕⁿ(x) = aⁿx.

By definition,

y∼_ϕx⇔ ∃b ∈Z^p :y=b+x−ab=x+b(1−a)⇔y≡x(mod(1−a)).

This implies thatR(ϕ) =|Zp/(1−a)Zp|. But

(1−a)Zp =p^vp(1−a)Zp =|1−a|⁻¹_p Zp,

so we can writeR(ϕ) = |1−a|⁻¹_p =|a−1|⁻¹_p and, more generally, R(ϕⁿ) = |1−aⁿ|⁻¹_p =|aⁿ−1|⁻¹_p ,for alln∈N.

Now consider a groupZ^dp, d≥2. It follows easily from Lemma 1.3, that End(Z^dp) = M_d(Zp). For any matrixA ∈ M_d(Zp)there exists a diagonal matrix D ∈ Md(Z^p) and unimodular matrices E, F ∈ Md(Z^p) such that D=EAF.

Lemma 1.4. For endomorphismϕ_p :Z^dp →Z^dpwe have R(ϕ_p) = #Coker(1−ϕ_p) =|det(Φ_p−Id)|⁻¹_p , whereΦ_p is a matrix ofϕ_p.

Proof. Let matrices D, E, F ∈ M_d(Zp)be such that D = E(Id−Φ_p)F, whereD= (a_i)is diagonal matrix,a_i ∈ Zp,1≤i ≤d, and matricesE, F are unimodular. Then we have

R(ϕ_p) = #Coker(1−ϕ_p) =|Z^dp : (Id−Φ_p)Z^dp|=|Z^dp :DZ^dp|=

=|Zp :a₁Zp| · |Zp :a₂Zp| ·...· |Zp :a_dZp|=|a₁|⁻¹_p ·...· |a_d|⁻¹_p =

=|det(D)|⁻¹_p =|det(Id−Φp)|⁻¹_p =|det(Φp−Id)|⁻¹_p .

□

In order to handle the sequenceR(ϕⁿ) = |aⁿ−1|⁻¹_p , n∈Nmore easily, we need a way to evaluate expressions of the form|aⁿ−1|_p when|a|_p = 1.

The following technical lemma is useful.

Lemma 1.5. (cf.[21, Lemma 4.9],[1, Lemma 2]) LetK_vbe a

non-archimedean local field and supposex ∈K_v has|x|_v = 1and infinite multiplicative order. Let p > 0 be the characteristic of the residue field F_v and γ ∈ N the multiplicative order of the image of x in F_v . Then

|xⁿ −1|_v = 1 whenever (γ, n) = 1 and γ ̸= 1. Furthermore, there are constants 0 < C < 1 and r₀ ≥ 0 such that whenever n = kγp^r with (p, k) = 1andr > r₀, then|xⁿ−1|_v =C|p|^r_v if char(K_v) = 0.

Now we prove a P´olya–Carlson dichotomy between rationality and a nat- ural boundary for the analytic behaviour of the Reidemeister zeta function for endomorphisms of groupsZ^dp, d ≥1.

Theorem 1.6. Letϕ_p :Z^dp →Z^dp be a tame endomorphism and

λ₁, λ₂, ...λ_d ∈ Qp be the eigenvalues of Φ_p, including multiplicities. Then the Reidemeister zeta functionR_ϕp(z)is either a rational function or it has the unit circle as a natural boundary. Furthermore, the latter occurs if and only if|λ_i|_p = 1for somei∈ {1, . . . , d}.

Proof. Firstly, we consider separately the case of the group Zp as it illus- trates some ideas needed for the proof of the dichotomy in general case whend ≥1. Lemma 1.3 yieldsϕ_p(x) = ax, wherea∈Zp. Hence|a|_p ⩽1.

Then the Reidemeister numbers R(ϕⁿ_p) = |aⁿ − 1|⁻¹_p , for all n ∈ N. If

|a|p < 1, then R(ϕⁿ_p) = |aⁿ− 1|⁻¹_p = 1, for all n ∈ N. Hence the ra- dius of convergence ofR_ϕp(z)equals 1 and the Reidemeister zeta function R_ϕp(z) = _1−z¹ is a rational function.

From now on, we shall writea(n)<< b(n)if there is a constantcinde- pendent ofn for whicha(n) < c·b(n). When|a|_p = 1, we show that the radius of convergence ofRϕp(z)equals 1 by deriving the bound

(1) 1

n << |aⁿ−1|_p ⩽1

Upper bound in (1) follows from the definition of the p-adic norm. We may suppose that |aⁿ −1|_p < 1. Let F denote the smallest field which containsQp and is both algebraically closed and complete with respect to

| · |_p . The p-adic logarithmlog_p is defined as log_p(1 +z) =

∞

X

n=1

(−1)ⁿ⁺¹ n zⁿ,

and converges for allz ∈F such that|z|p <1.Settingz =aⁿ−1we get log_p(aⁿ) = (aⁿ−1)−(aⁿ−1)²

2 + (aⁿ−1)³ 3 −...

and so|log_p(aⁿ)|_p ⩽|aⁿ−1|_p. We always have 1

n <<|nlog_p(a)|_p =|log_p(aⁿ)|_p, so this establishes (1).

The bound (1) implies by Cauchy - Hadamard formula that the radius of convergence of R_ϕp(z) equals 1. Hence it remains to show that the Reidemeister zeta function R_ϕp(z) is irrational if |a|_p = 1. ThenR_ϕp(z) has the unit circle as a natural boundary by the Lemma 1.2 and by the P´olya–Carlson Theorem. For a contradiction, assume that Reidemeister zeta function R_ϕp(z) is rational. Then Lemma 1.2 implies that the func- tion Z_p(z) = P∞

n=1R(ϕⁿ_p)zⁿ is rational also. Hence the sequence R(ϕⁿ_p) satisfies a linear recurrence relation. Definen = γp^r, where integer con- stantr ⩾0. Applying Lemma 1.5, we see that

R(ϕ^kn_p ) =R(ϕⁿ_p)

wheneverk is coprime ton. Hence the sequence R(ϕⁿ_p)assumes infinitely many values infinitely often, and so it cannot satisfy a linear recurrence by a result of Myerson and van der Poorten [24, Prop. 2], giving a contradiction.

Now we consider the general case of a tame endomorphism ϕ_p : Z^dp → Z^dp, d ≥1. According to the Lemma 1.4 we have

R(ϕⁿ_p) = #Coker(1−ϕⁿ_p) =|det(Φⁿ_p −Id)|⁻¹_p =

d

Y

i=1

|λⁿ_i −1|⁻¹_p ,

whereΦ_pis a matrix ofϕ_p andλ₁, λ₂, ...λ_d∈Qp are the eigenvalues ofΦ_p, including multiplicities. The polynomialQd

i=1(X−λ_i)has coefficients in Zp; in particular,|λ_i|_p ⩽1for everyi∈ {1, . . . , d}(see [15]). If|λ_i|_p <1, for everyi∈ {1, . . . , d}thenR(ϕⁿ_p) =Qd

i=1|λⁿ_i −1|⁻¹_p = 1,for alln ∈N. Hence the radius of convergence ofR_ϕp(z)equals 1 and the Reidemeister zeta function R_ϕp(z) = _1−z¹ is a rational function. If |λ_i|_p = 1, for some i∈ {1, . . . , d}then the bound (1) implies the bound

(2) 1

n^d << R(ϕⁿ_p) =

d

Y

i=1

|λⁿ_i −1|⁻¹_p ⩽1.

Hence the radius of convergence ofRϕp(z)equals 1 by the

Cauchy– Hadamard formula and the bound (2). Now for the proof of the theorem it remains to show that the Reidemeister zeta function R_ϕp(z)

is irrational if |λi|p = 1 for some i ∈ {1, . . . , d}. Then Rϕp(z) has the unit circle as a natural boundary by the Lemma 1.2 and by the P´olya–

Carlson Theorem. For a contradiction, assume that Reidemeister zeta func- tionR_ϕp(z)is rational. Then Lemma 1.2 implies that the functionZ_p(z) = P∞

n=1R(ϕⁿ_p)zⁿis rational also. Hence the sequenceR(ϕⁿ_p)satifies a linear recurrence relation. Definen =γp^r, where integer constantr ⩾0. Apply- ing Lemma 1.5, we see thatR(ϕ^kn_p ) = R(ϕⁿ_p)wheneverkis coprime ton.

Hence the sequenceR(ϕⁿ_p)assumes infinitely many values infinitely often, and so it cannot satisfy a linear recurrence by a result of Myerson and van der Poorten [24, Prop. 2], giving a contradiction.

□ 2. THE RATIONALITY OF THE COINCIDENCEREIDEMEISTER ZETA

FUNCTION FOR ENDOMORPHISMS OF FINITELY GENERATED TORSION-FREE NILPOTENT GROUPS

Example 2.1. ([15], Example 1.3) Let G= Zbe the infinite cyclic group, written additively, and let

ϕ: Z→Z, x7→d_φx and ψ: Z→Z, x7→d_ψx

for d_ϕ, d_ψ ∈ Z. The coincidence Reidemeister number R(ϕ, ψ) of endo- morphisms ϕ, ψ of an Abelian group G coincides with the cardinality of the quotient group Coker (ϕ−ψ) = G/Im(ϕ−ψ) (orCoker (ψ −ϕ) = G/Im(ψ−ϕ)). Hence we have

R(ϕⁿ, ψⁿ) =

(|d_ψⁿ−d_ϕⁿ| ifd_ϕⁿ ̸=d_ψⁿ,

∞ otherwise.

Consequently,(ϕ, ψ)is tame precisely when|d_ϕ| ̸=|d_ψ|and, in this case, R_ϕ,ψ(z) = 1−d₂z

1−d1z whered₁ = max{|d_ϕ|,|d_ψ|}andd₂ = d_ϕd_ψ d1

. This simple example (or at least special cases of it) are known. The aim of the current section is to generalise this example to finitely generated torsion- free nilpotent groups. LetGbe a finitely generated group andϕ, ψ :G→G two endomorphisms.

Lemma 2.2. Let ϕ, ψ : G → G are two automorphisms. Two elements x, y of G are ψ⁻¹ϕ-conjugate if and only if elements ψ(x) and ψ(y) are (ψ, ϕ)-conjugate. Therefore the Reidemeister numberR(ψ⁻¹ϕ)is equal to R(ϕ, ψ). For a tame pair of commuting automorphisms ϕ, ψ : G→ Gthe coicidence Reidemeister zeta functionR_ϕ,ψ(z)is equal to the Reidemeister zeta functionR_ψ⁻¹_ϕ(z).

Proof. If x and y are ψ⁻¹ϕ-conjugate, then there is a g ∈ G such that x = gyψ⁻¹ϕ(g⁻¹). This implies ψ(x) = ψ(g)ψ(y)ϕ(g⁻¹). So ψ(x) and ψ(y) are(ϕ, ψ)-conjugate. The converse statement follows if we move in

opposite direction in previous implications. □

We assumeXto be a connected, compact polyhedron andf :X →Xto be a continuous map. The Lefschetz zeta function of a discrete dynamical systemfⁿis defined asL_f(z) := exp

P∞ n=1

L(fⁿ) n zⁿ

,where L(fⁿ) :=

dimX

X

k=0

(−1)^ktr h

f_∗kⁿ :Hk(X;Q)→Hk(X;Q) i

is the Lefschetz number of the iteratefⁿoff. The Lefschetz zeta function is a rational function ofzand is given by the formula:

L_f(z) =

dimX

Y

k=0

det I−f∗k.z(−1)^k+1

.

In this section we consider finitely generated torsion-free nilpotent group Γ. It is well known [20] that such groupΓis a uniform discrete subgroup of a simply connected nilpotent Lie groupG(uniform means that the coset spaceG/Γis compact). The coset spaceM =G/Γis called a nilmanifold.

Since Γ = π1(M) and M is a K(Γ,1), every endomorphism ϕ : Γ → Γ can be realized by a selfmap f : M → M such that f∗ = ϕ and thus R(f) = R(ϕ). Any endomorphismϕ : Γ → Γ can be uniquely extended to an endomorphism F : G → G. Let F˜ : ˜G → G˜ be the corresponding Lie algebra endomorphism induced fromF and letSpectr( ˜F)be the set of eigenvalues ofF˜.

Lemma 2.3. (cf. Theorem 23 of[12]and Theorem 5 of[14]) Letϕ: Γ→Γ be a tame endomorphism of a finitely generated torsion free nilpotent group.

Then the Reidemeister zeta functionR_ϕ(z) = R_f(z)is a rational function and is equal to

(3) R_ϕ(z) =R_f(z) = L_f((−1)^pz)^(−1)r,

wherepthe number ofµ∈Spectr( ˜F)such thatµ <−1, andrthe number of real eigenvalues ofF˜whose absolute value is>1.

Every pair of automorphismsϕ, ψ : Γ → Γof finitely generated torsion free nilpotent groupΓcan be realized by a pair of homeomorphismsf, g : M → M such that f∗ = ϕ , g∗ = ψ and thus R(g⁻¹f) = R(ψ⁻¹ϕ) = R(ϕ, ψ). An automorphism ψ⁻¹ϕ : Γ → Γ can be uniquely extended to an automorphismL : G → G. LetL˜ : ˜G → G˜ be the corresponding Lie algebra automorphism induced fromL.

Lemma 2.2 and Lemma 2.3 imply the following

Theorem 2.4. Letϕ, ψ : Γ → Γbe a tame pair of commuting automor- phisms of a finitely generated torsion-free nilpotent groupΓ. Then the co- incidence Reidemeister zeta functionR_ϕ,ψ(z)is a rational function and is equal to

(4) Rϕ,ψ(z) =R_ψ⁻¹_ϕ(z) =R_g⁻¹_f(z) = L_g⁻¹_f((−1)^pz)^(−1)r,

wherepthe number ofµ∈Spectr( ˜L)such thatµ <−1, andrthe number of real eigenvalues ofL˜whose absolute value is>1.

For an arbitrary group G, we can define the k-fold commutator group γ_k(G) inductively as γ₁(G) := G and γ_k+1(G) := [G, γ_k(G)]. Let G be a group. For a subgroup H ⩽ G, we define the isolator √^G

H ofH in G as: √^G

H = {g ∈ G|gⁿ ∈ Hfor somen ∈ N}.Note that the isolator of a subgroupH ⩽ G doesn’t have to be a subgroup in general. For example, the isolator of the trivial group is the set of torsion elements of G.

Lemma 2.5. (see[5], Lemma 1.1.2 and Lemma 1.1.4) Let G be a group.

Then

(i) for allk ∈N, p^G

γ_k(G)is a fully characteristic subgroup ofG, (ii) for allk ∈N, the factorG/p^G

γ_k(G)is torsion-free, (iii) for allk, l∈N, the commutator[p^G

γk(G), p^G

γl(G)]≤ p^G

γk+l(G), (iv) for allk, l∈Nsuch thatk ⩾lifM := p^G

γ_l(G), then

G/Mp

γ_k(G/M) = p^G

γ_k(G)/M.

We define the adapted lower central series of a groupGas G= p^G

γ₁(G)⩾ p^G

γ₂(G)⩾...⩾ p^G

γ_k(G)⩾..., whereγ_k(G)is thek-th commutator ofG.

The adapted lower central series will terminate if and only if G is a torsion-free, nilpotent group. Moreover, all factors p^G

γ_k(G)/p^G

γ_k+1(G) are torsion-free.

We are particularly interested in the case whereGis a finitely generated, torsion-free, nilpotent group. In this case the factors of the adapted lower central series are finitely generated, torsion-free, abelian groups, i.e. for all k ∈Nwe have that

pG

γ_k(G)/p^G

γ_k+1(G)∼=Z^dk,for somed_k∈N.

Let N be a normal subgroup of a group G and ϕ, ψ ∈ End(G) with ϕ(N) ⊆ N, ψ(N) ⊆ N. We denote the restriction ofϕ toN byϕ|N, ψ to N byψ|_N and the induced endomorphisms on the quotientG/N byϕ^′, ψ^′ respectively. We then get the following commutative diagrams with exact

rows:

(5)

1 N G G/N 1

1 N G G/N 1

i

ϕ|_N,^ψ|N

p

ϕ,^ψ ϕ^′,^ψ′

i p

Note that both i and p induce functionsˆi,pˆon the set of Reidemeister classes so that the sequence

R[ϕ|_N, ψ|_N] ^ˆi R[ϕ, ψ] ^pˆ R[ϕ^′, ψ^′] 0

is exact, i.e. pˆis surjective and pˆ⁻¹[1] = im(ˆi), where 1 is the identity element ofG/N (see also [18]).

Lemma 2.6. IfR(ϕ|_N, ψ|_N)<∞, R(ϕ^′, ψ^′)<∞andN ⊆Z(G), then R(ϕ, ψ)⩽R(ϕ|_N, ψ|_N)R(ϕ^′, ψ^′).

Proof. LetR(ϕ|_N, ψ|_N) = {[n₁]_ϕ|N,ψ|N, ...,[n_R(ϕ|N,ψ|N)]_ϕ|N,ψ|N}and R(ϕ^′, ψ^′) = {[g₁N]_ϕ^′_,ψ^′, ...,[g_R(ϕ^′_,ψ^′₎N]_ϕ^′_,ψ^′}be the

(ϕ|_N, ψ|_N)–Reidemeister classes and(ϕ^′, ψ^′)–Reidemeister classes respec- tively. Let us takeg ∈ G. ThengN ∈[g_iN]_ϕ^′_,ψ^′ for somei, so there exists hN ∈G/N such that

gN =ψ^′(hN)g_iN ϕ^′(hN)⁻¹ =ψ^′(h)g_iϕ^′(h)⁻¹N.

It follows that there exists n ∈ N such that g = ψ^′(h)giϕ^′(h)⁻¹n. In turnn ∈ [nj]ϕ|_N,ψ|_N for some j, hence there exists m ∈ N such that n = ψ|_N(m)n_jϕ|_N(m)⁻¹. Since n, m, n_j, ψ|_N(m), ϕ|_N(m) ∈ N ⊂ Z(G), it follows that

g = [ψ(hm)](g_in_j)[ϕ(hm)⁻¹],

i.eg ∈[g_in_j]_ϕ,ψ. Since this is true for arbitraryg ∈Gwe obtain that R(ϕ, ψ)⩽R(ϕ|_N, ψ|_N)R(ϕ^′, ψ^′).

□ Theorem 2.7. LetN be a finitely generated, torsion-free, nilpotent group and

N = ^Np

γ₁(N)⩾ p^N

γ₂(N)⩾...⩾ p^N

γ_c(N)⩾ ^Np

γ_c+1(N) = 1 be an adapted lower central series ofN. Suppose that R(ϕ, ψ) < ∞and R(ϕ_k, ψ_k) < ∞ for a pair ϕ, ψ of endomorphisms of N and for every

pairϕk, ψkof induced endomorphisms on the finitely generated torsion-free abelian factors

Np

γ_k(N)/p^N

γ_k+1(N)∼=Z^dk, d_k ∈N, 1≤k ≤c, then

R(ϕ, ψ) =

c

Y

k=1

R(ϕ_k, ψ_k).

Proof. We will prove the product formula for the coincidence Reidemeister numbers by induction on the length of an adapted lower central series. Let us denote p^N

γ_k(N)asN_k. Ifc = 1, the result follows trivially. Letc > 1 and assume the product formula holds for a central series of lengthc−1.

Letϕ, ψ∈End(N), thenϕ(Nc)⊆Nc, ψ(Nc)⊆Ncand hence we have the following commutative diagram of short exact sequences:

1 N_c N N/N_c 1

1 N_c N N/N_c 1

i

ϕc,ψc

p

ϕ,ψ ϕ^′,ψ^′

i p ,

whereϕ_c, ψ_care induced endomorphisms on theN_c. The quotientN/N_cis a finitely generated, nilpotent group with a adapted central series

N/N_c=N₁/N_c⩾N₂/N_c⩾...⩾Nc−1/N_c⩾N_c/N_c= 1 of lengthc−1.

Every factor of this series is of the form

(N_k/N_c)/(N_k+1/N_c)∼=N_k/N_k+1by the third isomorphism theorem, hence it is also torsion-free. Moreover, because of this natural isomorphism we know that for every induced pair of endomorphisms(ϕ^′_k, ψ^′_k)on

(N_k/N_c)/(N_k+1/N_c)it is true thatR(ϕ^′_k, ψ_k^′) =R(ϕ_k, ψ_k).

The assumptions of the theorem imply thatR(ϕ^′, ψ^′)<∞and R(ϕc, ψc)<∞.

Moreover, let[g1Nc]ϕ^′,ψ^′, ...,[gnNc]ϕ^′,ψ^′ be the(ϕ^′, ψ^′)-Reidemeister classes and[c₁]_ϕc,ψc, ...,[c_m]_ϕc,ψc - the(ϕ^′_c, ψ_c^′)-Reidemeister classes. Since N_c⊆Z(N), by Lemma 2.6 we obtain thatR(ϕ, ψ)⩽R(ϕ_c, ψ_c)R(ϕ^′, ψ^′).

To prove the opposite inequality it suffices to prove that every class [c_ig_j]_ϕ,ψ represents a different(ϕ, ψ)-Reidemeister class. Then we obtain

R(ϕ, ψ) =R(ϕ_c, ψ_c)R(ϕ^′, ψ^′)

and then the theorem follows from the induction hypothesis.

Suppose, that there exists someh∈N such thatcigj =ψ(h)cagbϕ(h)⁻¹. Then by taking the projection toN/N_cwe find that

g_jN_c=p(c_ig_j) =p(ψ(h)c_ag_bϕ(h)⁻¹) =ψ^′(hN_c)(g_bN_c)ϕ^′(hN_c)⁻¹, hence[gjNc]ϕ^′,ψ^′ = [gbNc]ϕ^′,ψ^′. Now assume that

cigj = ψ(h)cagjϕ(h)⁻¹.Ifh ∈ Nc ⊆Z(N), thencigj =ψ(h)caϕ(h)⁻¹gj

and consequently[c_i]_ϕc,ψc = [c_a]_ϕc,ψc, so let us assume thath /∈N_cand that N_kis the smallest group in the central series which containsh. Thenc_ig_j = ψ(h)c_ag_jϕ(h)⁻¹ ⇔g_jc_i =ψ(h)c_ag_jϕ(h)⁻¹ ⇔c_i =g⁻¹_j ψ(h)c_ag_jϕ(h)⁻¹

⇔ci =g_j⁻¹ψ(h)gjϕ(h)⁻¹ca ⇔cic⁻¹_a =g_j⁻¹ψ(h)gjϕ(h)⁻¹and therefore c_ic⁻¹_a N_k+1 =g⁻¹_j ψ(h)g_jϕ(h)⁻¹N_k+1

= [gj, ψ(h)⁻¹](ψ(h)ϕ(h)⁻¹)Nk+1

Asc_ic⁻¹_a ∈N_c⊆N_k+1and[g_j, ψ(h)⁻¹]∈N_k+1, we find that(ϕ_k)(hN_k+1)

= (ψ_k)(hN_k+1).That means that the set of coincidence points Coin(ϕ^′_k, ψ_k^′)

̸={1}, which implies thatR(ϕ^′, ψ^′) = ∞and this contradicts assumption.

□ Theorem 2.8. Let ϕ, ψ: N → N be a tame pair of endomorphisms of a finitely generated torsion-free nilpotent group N. Let cdenote the nilpo- tency class of N and, for 1 ⩽ k ⩽ c, let ϕ_k, ψ_k: G_k → G_k, 1 ≤ k ≤ c, denote the tame pairs of induced endomorphisms of the finitely generated torsion-free abelian factor groupsG_k =N_k/N_k+1 =

= ^Np

γk(N)/^Np

γk+1(N) ∼= Z^dk, for some dk ∈ N, that arise from an adapted lower central series ofN. Then the following hold.

(1)For eachn∈N, R(ϕⁿ, ψⁿ) =

c

Y

k=1

R(ϕ_kⁿ, ψ_kⁿ) forn ∈N.

(2)For1⩽k⩽c, let

ϕ_k,Q, ψ_k,Q:G_k,Q →G_k,Q

denote the extensions ofϕ_k, ψ_kto the divisible hullG_k,Q =Q⊗_ZG_k ∼=Q^dk of G_k. Suppose that each pair of endomorphisms ϕ_k,Q, ψ_k,Q is simultane- ously triangularisable. Letξ_k,1, . . . , ξ_k,dkandη_k,1, . . . , η_k,dkbe the eigenval- ues ofϕk,Q andψk,Qin the fieldC, including multiplicities, ordered so that, forn ∈N, the eigenvalues ofϕ_k,ⁿ_Q−ψ_k,ⁿ_Q areξ_k,1ⁿ −η_k,1ⁿ , . . . , ξ_k,dⁿ

k −η_k,dⁿ

k. Then for eachn∈N,

(6) R(ϕ_kⁿ, ψ_kⁿ) =

dk

Y

i=1

|ξ_k,iⁿ −η_k,iⁿ|;

(3) Moreover, suppose that|ξ_k,i| ̸= |η_k,i| for1 ⩽ k ⩽ c, 1 ⩽ i ⩽ d_k. If

ϕ, ψis a tame pair of endomorphisms ofN andϕk,Q, ψk,Q, 1⩽ k ⩽ c, are simultaneously triangularisable pairs of endomorphisms ofG_k,Q, then the coincidence Reidemeister zeta functionR_ϕ,ψ(z)is a rational function.

Proof. The coincidence Reidemeister number R(ϕ, ψ) of automorphisms ϕ, ψ of an Abelian group Gcoincides with the cardinality of the quotient groupCoker (ϕ−ψ) = G/Im(ϕ−ψ)(orCoker (ψ−ϕ) =G/Im(ψ−ϕ)).

For1 ⩽ k ⩽ c, let tame pairs ϕ_k, ψ_k: G_k → G_k of induced endomor- phisms of the finitely generated torsion-free abelian factor groups G_k = N_k/N_k+1 ∼= Z^dk, are represented by integer matrices A_k, B_k ∈ M_dk(Z) associated to them respectively. There is a diagonal integer matrix Ck = diag(c1, ..., cd_k) such thatCk = Mk(Ak−Bk)Nk; where Mk andNk are unimodular matrices. Now we havedetC_k = det(A_k−B_k)and the order of the cokernel ofϕ_k−ψ_kis the order of the groupZ/c₁Z⊕...⊕Z/c_dkZ. Thus the order of the cokernel ofϕ_k−ψ_kis|Coker (ϕ_k−ψ_k)|=|c₁···c_dk|=

|detC_k|=|det(ϕ_k−ψ_k)|.

Then for eachn∈Nand1⩽k ⩽c,R(ϕ_kⁿ, ψ_kⁿ) =|Coker (ϕ_k−ψ_k)|=

|det(ϕ_k−ψ_k)|=|det(ϕ_k,Q−ψ_k,Q)|=Qdk

i=1|ξ_k,iⁿ −η_k,iⁿ|.

Now we will prove the rationality ofR_ϕ,ψ(z). We open up the absolute values in the productR(ϕ_kⁿ, ψ_kⁿ) =Qdk

i=1|ξ_k,iⁿ −η_k,iⁿ|,1⩽k ⩽c. Complex eigenvaluesξ_k,iin the spectrum ofϕ_k,Q, respectivelyη_k,iin the spectrum of ψ_k,Q, appear in pairs with their complex conjugateξ_k,i, respectivelyη_k,i.

Moreover, such pairs can be lined up with one another in a simultane- ous triangularisation as follows. Writeϕ_k,C, ψ_k,Cfor the induced endomor- phisms of the C-vector space V = C ⊗_Q G ∼= C^dk. If v ∈ V is, at the same time, an eigenvector of ϕ_k,C with complex eigenvalue ξ_k,dk and an eigenvector ofψ_k,Cwith eigenvalueη_k,dk, then there isw ∈ V such thatw is, at the same time, an eigenvector of ϕ_k,C with eigenvalue ξ_k,dk ̸= ξ_k,dk and an eigenvector of ψ_k,C with eigenvalue η_k,dk, possibly equal to η_k,dk. Thus we can start our complete flag of{ϕ, ψ}-invariant subspaces ofV with {0} ⊂ ⟨v⟩ ⊂ ⟨v, w⟩, and proceed withV /⟨v, w⟩by induction to produce the rest of the flag in the same way, treating complex eigenvalues ofψk,Cin the same way as they appear. If at least one ofξ_k,i, η_k,iis complex so that these eigenvalues ofϕ_Qandψ_Qare paired with eigenvaluesξ_k,j =ξ_k,i, η_k,j =η_k,i, for suitablej ̸=i, as discussed above, we see that

ξ_k,iⁿ −η_k,iⁿ

ξ_k,jⁿ −η_k,jⁿ =

ξ_k,iⁿ −η_k,iⁿ

2 = ξ_k,iⁿ −η_k,iⁿ

·(ξ_k,iⁿ−η_k,iⁿ . If ξ_k,i and η_k,i are both real eigenvalues ofϕ_Q and ψ_Q, not paired up with another pair of eigenvalues, then exactly as in Example 2.1 above we have

|ξ_k,iⁿ −η_k,iⁿ| = δ_1,k,iⁿ −δ_2,k,iⁿ , where δ_1,k,i = max{|ξ_k,i|,|η_k,i|}and δ_2,k,i =

ξk,i·η_k,i

δ1,k,i . Hence we can expand each productR(ϕ_kⁿ, ψ_kⁿ),1⩽k ⩽cusing an

appropriate symmetric polynomial, to obtain for the Reidemeister numbers R(ϕⁿ, ψⁿ)an expression of the form

(7) R(ϕⁿ, ψⁿ) =

c

Y

k=1

R(ϕ_kⁿ, ψ_kⁿ) =

c

Y

k=1 dk

Y

i=1

|ξ_k,iⁿ −η_k,iⁿ|=X

j∈J

c_jw_jⁿ, whereJ is a finite index set,c_j ∈ {−1,1}and{w_j | j ∈ J} ⊆ C ∖{0}.

Consequently, the coincidence Reidemeister zeta function can be written as R_ϕ,ψ(z) = exp

∞

X

n=1

R(ϕⁿ, ψⁿ) n zⁿ

!

= exp X

j∈J

c_j

∞

X

n=1

(w_jz)ⁿ n

! . and it follows immediately thatR_ϕ,ψ(z) = Q

j∈J(1−w_jz)^−cj is a rational function.

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WOJCIECHBONDAREWICZ, INSTYTUTMATEMATYKI, UNIWERSYTETSZCZECIN-

SKI,UL. WIELKOPOLSKA15, 70-451 SZCZECIN, POLAND

Email address:wojciech.bondarewicz@usz.edu.pl

ALEXANDERFEL’SHTYN, INSTYTUTMATEMATYKI, UNIWERSYTETSZCZECINSKI,

UL. WIELKOPOLSKA15, 70-451 SZCZECIN, POLAND

Email address:alexander.felshtyn@usz.edu.pl

MALWINA ZIETEK, INSTYTUT MATEMATYKI, UNIWERSYTET SZCZECINSKI, UL. WIELKOPOLSKA15, 70-451 SZCZECIN, POLAND

Email address:malwina.zietek@gmail.com