aa r X i v : . [ m a t h . G T ] M a y Profinite rigidity for twisted Alexander polynomials
Jun Ueki
Abstract
We formulate and prove a profinite rigidity theorem for the twisted Alexander polyno-mials up to several types of finite ambiguity. We also establish torsion growth formulasof the twisted homology groups in a Z -cover of a 3-manifold with use of Mahler mea-sures. We examine several examples associated to Riley’s parabolic representations oftwo-bridge knot groups and give a remark on hyperbolic volumes. Contents
1. Introduction
Recently the profinite rigidity in 3-dimensional topology is of great interest with rapid progress.Nevertheless, it is still unknown whether there exists a pair (
J, K ) of distinct prime knots withan isomorphism b π J ∼ = b π K on the profinite completions of their knot groups. In the previousarticle [Uek18] we proved that the Alexander polynomial of a knot K is determined by theisomorphism class of the profinite completion b π K of the knot group. In this article, we formulatea version of profinite rigidity for twisted Alexander polynomials and prove it up to severaltypes of finite ambiguity. In addition, we establish asymptotic formulas of the torsion growthof the twisted homology groups in a Z -cover of a 3-manifold, refining the studies of the author[Uek20] and R.Tange [Tan18]. We finally examine several examples associated to Riley’s parabolicrepresentations of two bridge knots and give a remark on hyperbolic volumes.Let π be a discrete group of finite type with a surjective homomorphism α : π ։ t Z , where t isa formal element generating Im α . Let O = O F,S denote the ring of S -integers, where S is a finiteset of maximal ideals of the ring of integers O F of a number field F , and let ρ : π → GL N ( O ) be Primary 57M27, 20E26 Secondary 20E18, 57M12, 11R06
Keywords: knot, twisted Alexander polynomial, profinite completion un Ueki a representation. Let g Fitt H i (Ker α, ρ ) denote the divisorial hull of the Fitting ideal over O [ t Z ]of the i -th Alexander module H i (Ker α, ρ ). If g Fitt H i (Ker α, ρ ) is a principal ideal, then the i -thtwisted Alexander polynomial ∆ αρ,i ( t ) is defined as a generator of this ideal, up to multiplicationby units of O [ t Z ]. (We denote by ˙= equalities up to this ambiguity. We sometime omit i and α forsimplicity.) By the universality of profinite completion, this ρ induces a continuous representation b ρ : b π → GL N ( b O ). Let β : π ′ ։ t Z and τ : π ′ → GL N ( O ) be another such pair. We say ( b α, b ρ )and ( b β, b τ ) are isomorphic if they admit isomorphisms ϕ : b π ∼ = → b π ′ and ψ : t b Z ∼ = → t b Z satisfying ψ ◦ b α = b β ◦ ϕ and b τ ◦ ϕ ∼ b ρ , where ∼ denotes the equivalence by a conjugate in GL N ( b O ). Ourmain question is the following: Question . To what extent does the isomorphism class of ( b α, b ρ ) determine ∆ αρ ( t )?We denote by Q the algebraic closure of Q in C and assume that a number field F is asubfield of Q . For each prime number p , let C p denote the p -adic completion of an algebraicclosure Q p of a p -adic numbers Q p . Such C p is known to be algebraically closed. We fix anembedding ι p : Q ֒ → C p , so that the p -adic norm is defined for any z ∈ Q . Note that the normmap Nr F/ Q : F → Q induces a map Nr F/ Q : O F,S [ t Z ] → Z S [ t Z ], where S also denotes the imageof S in Z . We say that f ( t ) , g ( t ) ∈ R [ t Z ] belong to the same Hillar class if f ( t ) ˙= u ( t ) v ( t ) and g ( t ) ˙= u ( t ) v ( t − ) holds for some u ( t ) , v ( t ) ∈ R [ t Z ]. Let A be any set. For a d -tuple ( α i ) i ∈ A d ,we denote by [ α i ] i ∈ A d / S d the equivalent class with respect to the natural action of the d -thsymmetric group S d . Now our answer to the question is stated as follows: Theorem . Let α : π ։ t Z and ρ : π → GL N ( O ) over O = O F,S be as above. Suppose that g Fitt H i (Ker α, ρ ) in O [ t Z ] is a principal ideal so that ∆ ρ ( t ) = ∆ αρ,i ( t ) is defined.(1) The isomorphism class of ( b α, b ρ ) determines the Hillar class of Nr F/ Q ∆ ρ ( t ) ∈ Z S [ t Z ] , hencethe (Euclidean/ p -adic) Mahler measure of Nr F/ Q ∆ ρ ( t ) . If in addition ∆ ρ ( t ) is reciprocal, then Nr F/ Q ∆ ρ ( t ) itself is determined.(2) Let p be a prime number such that all roots of ∆ ρ ( t ) are placed on the unit circle in C p , whichis the case for almost all p . For each root α i of ∆ ρ ( t ) , let ζ i denote the unique root of unity with | α i − ζ i | p < and suppose that this ζ i is a primitive l i -th root of unity. Put m := lcm { l i } i > ,so that we have p l i and p m . Then the isomorphism class of ( b α, b ρ ) determines the set of such p ’s and the families [ l i ] i ∈ Z d / S d , { [( α i /ζ i ) w ] i | w ∈ Z ∗ p } , and { [ ζ ui ] i | u ∈ ( Z /m Z ) ∗ } in Q d / S d .Let s be any topological generator of Im b α = t b Z . Then we have s = t v for a unit v of thePr¨uffer ring b Z = lim ←− n Z /n Z . Since the image of t in Im b α is not specified, neither is v , while t and v are independent of p . The pair ( u, w ) is the image of this v under the natural map b Z ։ Z /m Z × Z p ; v ( u, w ) . If O is a UFD (hence a PID, for O being a Dedekind domain), then ∆ ρ ( t ) is defined. If ρ is an SL -representation or more generally if ρ is conjugate to its dual, then ∆ ρ ( t ) is reciprocal([HSW10], [Hil12, Chapter 6.5]). The assertion (1) only tells about each divisor of ∆ ρ ( t ) in O [ t Z ],while (2) tells about the multiplicity. If a cyclotomic polynomial decomposes in O [ t Z ], then aninevitable ambiguity may occur (Examples 6.2). The proof of (1) is a direct extension of [Uek20]by Hillar’s result [Hil05, Theorem 1.8] on cyclic resultants. In the proof of (2), we invoke thecharacter decomposition of the Iwasawa module of a Z /m Z × Z p -cover and some local field theory.The following propositions may reduce the ambiguity in (2).2 rofinite rigidity for twisted Alexander polynomials On explicit detection of ∆ αρ ( t ), after knowing the degree and eliminating cyclotomic divisors,Hillar’s method indeed tells that Theorem 1.2 (1) is done in finite time with use of Gr¨obner basis,while Propositions 1.3, 1.4 below tell that so is (2). Proposition . Let = α ∈ Q , let k denote the Galois closure of Q ( α ) , and put d = [ k : Q ] .Then for any but finite number of prime number p satisfying d | ( p − , we have α ∈ Q p , | α | p = 1 ,and | α p − − | p < . Proposition . Let α, β ∈ Q p with | α − | p < and | β − | p < , and suppose that thereexists some w ∈ Z ∗ p with α w = β . If α and β are conjugate over Q p and are not roots of unity,then w is a root of unity. Remark . Even if g Fitt H i (Ker α, ρ ) is not a principal so that ∆ αρ,i ( t ) is not defined, the idealNr F/ Q g Fitt H i (Ker α, ρ ) is still a principal ideal of Z S [ t Z ]. If we define ∆ αρ,i ( t ) as its generator, thenthe same assertion as in Theorem 1.2 (1) holds.In any case, if π is of deficiency one (e.g., a knot group) and H (Ker α, ρ ) is a torsion O [ t Z ]-module, then the fractional ideal g Fitt H (Ker α, ρ ) / g Fitt H (Ker α, ρ ) is a principal ideal, and isgenerated by Lin–Wada’s polynomial W αρ ( t ) ∈ F ( t ) (Proposition 4.1). If ∆ αρ,i ( t ) for i = 0 , W αρ ( t ) ˙= ∆ αρ, ( t ) / ∆ αρ, ( t ) up to multiplication by units of O [ t Z ] holds.Wada’s initial definition of W ρ ( t ) may be regarded with less indeterminacy, defined up tomultiplicity by ± t ± , and coincides with the Reidemeister torsion τ ρ ( t ) ∈ F [ t ]. By Goda’s result[God17] derived from a deep theory of analytic torsion developed by M¨uller and others, we mayconclude the following. Corollary . Let ρ hol : π K → SL ( O ) denote the holonomy representation of a hyperbolicknot K over O = O F,S ⊂ C . If O = O F for an imaginary quadratic field F , then the hyperbolicvolume Vol( K ) of K is determined by the isomorphism class of b ρ hol . Fox’s formula asserts that the absolute cyclic resultants of the Alexander polynomial ∆ K ( t )of a knot K coincide with the sizes of the torsions in the cyclic covers (cf. [Web79]). Hence theasymptotic formula of the torsion growth in the Z -cover with use of Mahler measure m [GAS91]and its p -adic analogue m p [Uek20] is closely related to the profinite detection of ∆ K ( t ).Let α : π ։ t Z and ρ : π → GL N ( O ) be as before, and let X be a connected manifold with π ∼ = π ( X ). For each n ∈ N > , let X n → X denote the Z /n Z -cover in the Z -cover defined by α . Define the norm polynomial ∆ ρ ( t ) ∈ Z [ t Z ] to be a generator of the ideal Nr F/ Q g Fitt( H ( X ∞ , ρ )) ∩ Z [ t Z ].If ∆ ρ ( t ) is defined, then we have ∆ ρ ( t ) ˙= Nr F/ Q ∆ ρ ( t ) in Z S [ t Z ]. .Suppose that ρ is absolutely irreducible and non-abelian. Then Tange’s result in [Tan18] maybe refined as follows (Theorem 11.1), where we denote by Res( f ( t ) , g ( t )) ∈ O the resultant of f ( t ) , g ( t ) ∈ O [ t ].(1) There is a natural isomorphism H ( X ∞ , ρ ) / ( t n − H ( X ∞ , ρ ) ∼ = → H ( X n , ρ ).(2) Put Ψ n ( t ) = gcd(∆ ρ ( t ) , ( t n − r n = Res(∆ ρ ( t ) , ( t n − / Ψ n ( t )). Then for a boundedsequence c n = | tor H ( X ∞ , ρ ) / Ψ n ( t )) | , the equality | tor H ( X n , ρ ) | = c n | r n | Q p ∈ S | r n | p holds.(3) The asymptotic formulaslim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t ))) , lim n →∞ || tor H ( X n , ρ ) || /np = m p (∆ ρ ( t ))of torsion growth with use of Mahler measures hold, where p is any prime number.3 un Ueki This paper is organized as follows. Additional backgrounds and further questions are givenin Section 2. We prove Theorem 1.2 through Sections 3–9. In Section 3, we recall that ρ : π → GL N ( O ) induces b ρ : b π → GL N ( b O ). In Section 4, we define the twisted Alexander polynomial∆ αρ ( t ) and prove Proposition 4.1. In Section 5, we recall the continuous homology H i ( b Γ , V b ρ ) ofprofinite completions and paraphrase our question. In Section 6, we remark some propertiesof cyclotomic divisors in the profinite group ring b O [[ t b Z ]], and reduce our question to the caseswithout cyclotomic divisors. In Section 7, we recall Hillar’s result on cyclic resultant and obtaina lemma. In Section 8, we recall the norm map and prove Theorem 1.2 (1). In Section 9, weprove Theorem 1.2 (2) with use of Iwasawa modules, and give some remarks. In Section 10, weprove Propositions 1.3 and 1.4, invoking some p -adic number theory.Section 11 is devoted to refinements of results in [Uek20] and [Tan18]; Fox’s formula andtorsion growth formulas with use of Mahler measures of the twisted homology groups in a Z -cover of a 3-manifold. We recall backgrounds, state Theorem 11.1 with some remarks, and provethem. In Section 12, we examine several examples of ∆ ρ ( t ) of Riley’s parabolic representations of2-bridge knot groups from viewpoints of Theorems 1.2 and 11.1, and give remarks on topologicalentropies. In Section 13, we briefly discuss hyperbolic volume and prove Corollary 1.6.
2. Preliminary
A question of profinite rigidity is a version of finite detection problem asking to what extent atopological invariant of a 3-manifold M is determined by the isomorphism class of the profinitecompletion b π of the 3-manifold group π = π ( M ). Latest progress due to Wilton–Zalesskii,Wilkes [WZ17, WZ19, Wil18b, Wil18a, BMRS18] are on the geometries of 3-manifolds, the JSJ-decompositions, graph manifolds, and hyperbolic manifolds of a certain class. To be precise,[BMRS18] proved the profinite rigidity of certain arithmetic Kleinian groups amongst all finitelygenerated residually finite groups (the absolute profinite rigidity). In addition, Liu proved thatmapping classes of a compact surface are almost determined by its finite quotient actions, usingtwisted Reidemeister torsions and twisted Lefschetz zeta functions [Liu19]. The profinite rigidityof knots was asked also by B. Mazur from a viewpoint of the analogy between knots and primenumbers [Maz12]. We refer to [Uek18], [BF15], and [Rei18] for further background.We proved in our previous paper [Uek18] that the isomorphism class of b π K determines theAlexander polynomial of any knot K in S . Since the image of π ab K = t Z in the isomorphismclass of b π ab K = t b Z is not specified, we needed to study the completed Alexander module over thecompleted group ring b Z [[ t b Z ]] = lim ←− m,n Z /m Z [ t Z /n Z ].In this article, we extend this method to the twisted Alexander polynomial ∆ αρ ( t ) of a rep-resentation ρ : π → GL N ( O ). We remark that many important representations over C maybe regarded as those over O = O F,S (e.g., the holonomy representations of a hyperbolic knot,[CS83]). The completed ring b O is sufficiently large, while b C = { } . Hence we consider represen-tations over O .We note that Boileau–Friedl [BF15] studied a profinite rigidity of the twisted Alexanderpolynomial associated to representations over a finite field, in ordered to prove the profiniterigidity of fiberedness of knots. They assumed a “regular isomorphism” so that the image of t in b π ab K was assumed to be known. We address a slightly difficult problem here.The aim of this paper is to investigate to what extent ∆ αρ ( t ) is (abstractly) recovered fromthe isomorphism class of the continuous representation b ρ : b π → GL N ( b O ). Namely, suppose that4 rofinite rigidity for twisted Alexander polynomials π runs through all discrete groups with α : π ։ t Z . Let R denote the set of the pairs ( ρ, α ) ofthe representations ρ : π → GL N ( O ) and α , P the set of twisted Alexander polynomials ∆ αρ ( t ),and b R the set of isomorphism class of pairs ( b ρ, b α ) of the profinite completions b ρ : b π → GL N ( b O )and b α : b π ։ t b Z . Then the question is to what extent the correspondence F : R → P factorsthrough the profinite completion b : R → b R ; ( ρ, α ) the isomorphism class of ( b ρ, b α ).Here we attach further questions, which may be addressed in a future study. Question . (1) Recall that the fiberedness of a 3-manifold is a profinite property ([BF15,Theorem 1.2], [BRW17], [JZ17]). Let ρ n denote the n -th higher holonomy representation for each n > K (cf. Section 13). Then W ρ n ( t )’s are reciprocal. If K is fibered, then τ ρ n ( t )’s are monic, while the converse is conjectured [DFJ12, Por18]. If τ ρ n ( t ) ’s are monic, thenso are W ρ n ( t )’s. Is Vol( K ) determined in this case?(2) To what extent the Reidemeister torsion τ ρ ( t ) = τ ρ ⊗ α ( S − K ) is determined by theisomorphism class of b ρ ? Does the argument for (1) extends to τ ρ ( t )?(3) Does the set of all continuous representations of b π determines Vol( K )? When we studytopology of knots by using twisted Alexander polynomials, difficulty lies in how to find a repre-sentation ρ conveying topological information. In order to apply our theorems to a more rusticquestion of profinite rigidity, we need to combine another type of rigidity theorem finding agood ρ among the set of all representations, such as mentioned by Francaviglia in [Fra04]. In-deed, Vol( K ) is the maximal value of volumes if ρ runs through all geometric representation ina neighborhood of ρ hol . We may need an additional insight to obtain a similar nature over allcontinuous representation, or to detect a nice class of representations.
3. Continuous representations
Let π be a discrete group. Then the set of finite quotients { π/̟ | ̟ ✁ π of finite index } formsa directed inverse system with respect to the quotient maps. The profinite completion b π of π is the inverse limit lim ←− ̟ ✁ π π/̟ as a group and endowed the weakest topology such thatKer( b π ։ π/̟ ) is an open subgroup for every normal subgroup ̟ ✁ π of finite index. It is knownthat such an inverse system may be reconstructed from the set of isomorphism classes of finitequotients of π [RZ10, Corollary 3.2.8]. A discrete group π is said to be residually finite if forany g ∈ π there exists some normal subgroup ̟ ✁ π of finite index such that the image of g in the quotient π/̟ is non-trivial. It is equivalent to that the natural map ι : π → b π is aninjection. For any oriented connected closed 3-manifold M , the group π = π ( M ) is residuallyfinite ([Hem87]+[Per02, Per03b, Per03a]).For a number field F , let O F denote the ring of integers of F and let S be a finite set ofprime ideals of O F . The ring O F,S of S -integers of F is a Dedekind domain obtained by addingall inverse elements of primes in S to O F . For any F , there is some S such that O F,S is a UFD,hence a PID. The profinite completion b O of the ring O = O F,S is a topological ring which isdefined as the inverse limit lim ←− I O/I of the set of finite quotients
O/I as a ring and endowedwith the profinite topology. We have an inverse system { GL N ( O/I ) } I of finite groups and anisomorphism of profinite groups GL N ( b O ) ∼ = lim ←− I GL N ( O/I ).Now let ρ : π → GL N ( O ) be a representation of a discrete group. The natural map O ֒ → b O induces a representation ρ : π → GL N ( b O ). By the universality of the profinite completion [RZ10,Lemma 3.2.1], for any continuous homomorphism f : π → H to a profinite group, there is aunique continuous homomorphism b f : b π → H such that f coincides with the composite b f ◦ ι un Ueki of the natural map ι : π → b π and b f . Hence we have an induced continuous representation b ρ : b π → GL N ( b O ). In other words, we have a natural map b : Hom( π, GL N ( O )) → Hom( b π, GL N ( b O )) . If π is a residually finite group, then this map is an injection.
4. Twisted Alexander polynomials
We recall in Subsection 4.1 the twisted Alexander polynomial and prepare for the proof ofTheorem 1.2. We give in Subsection 4.2 additional information related to Remark 1.5 on Wada’sinvariant, and prove Proposition 4.1.
Let M be a finitely generated module over a commutative ring R . If R m A −→ R n ։ M → M , then the i -th Fitting ideal Fitt i M of M over R is defined as the ideal of R generated by all ( n − i )-minors of A ∈ M n,m ( R ), which is known to be independent of the choicesof a presentation. If r is the lowest i with Fitt i M = 0, then we simply write Fitt M = Fitt r M and call it the Fitting ideal of M . The divisorial hull (or the reflexive hull ) e a of an ideal a of a ring R is the intersection ofall principal ideals containing a . If R is an integrally closed Noether domain (e.g., R = O [ t Z ])and a = 0, then we have a = ∩ p a ( p ) , where p runs through all prime ideals of height one and a ( p ) denotes the localization at p [Hil12, Lemma 3.2]. If R is a UFD, then e a is a principal ideal,generated by the highest common factor of the elements of a .Now let π be a discrete group of finite type with a surjective homomorphism α : π ։ t Z anda representation ρ : π → GL N ( O ) over O = O F,S . Let V ρ denote the module O n regarded as aright π -module via the transpose of ρ . Then Γ := Ker α acts on V ρ via the restriction of ρ .Let F be a projective resolution of Z over Z [Γ]. Then the homology H i (Γ , V ρ ) with localcoefficients is defined to be the homology of the complex F ⊗ Γ V ρ . We write H i (Γ , V ρ ) = H i (Γ , ρ )for simplicity. The conjugate action of π on Γ induces the action of Im( α ) = t Z ∼ = π/ Γ on H i (Γ , ρ )[Bro94, Chapter III, Corollary 8.2].Let X be an Eilenberg–MacLane space K ( π,
1) of π (e.g., the knot exterior if π is a knotgroup), and let X ∞ → X denote the Z -cover corresponding to Γ. Then we have a naturalisomorphism H i (Γ , ρ ) ∼ = H i ( X ∞ , ρ ) of finitely generated O [ t Z ]-modules for i = 0 ,
1. We alsonote that Shapiro’s lemma [Bro94, III, Proposition 6.2] yields H i (Γ , ρ ) ∼ = H i ( π, ρ ⊗ α ) and H i ( X ∞ , ρ ) ∼ = H i ( X, ρ ⊗ α ) for the tensor representation ρ ⊗ α : π → GL N ( O [ t Z ]).If g Fitt H i (Γ , ρ ) is a principal ideal, then the i -th twisted Alexander polynomial ∆ αρ,i ( t ) is definedas a generator ∆ αρ,i ( t ) of this ideal, which is well-defined up to ˙= in O [ t Z ]. This ∆ αρ,i ( t ) is knownto be an invariant of the isomorphism class of ( ρ, α ). We sometime omit i and α . W αρ ( t )Lin–Wada’s invariant W αρ ( t ) ∈ F ( t ) was initially introduced by Lin [Lin01] and Wada [Wad94].It may be defined as the ratio of generators of g Fitt of based Alexander modules , hence definedup to ˙= over O [ t Z ]. If π is of deficiency one (e.g., a knot group), then W αρ ( t ) turns out to be apolynomial in F [ t Z ]. If a presentation of π is given, then W αρ ( t ) is explicitly calculated by usingFox differential [Wad94, KL99, DFJ12, FV11]. If ∆ αρ,i ( t ) are defined for i = 0 ,
1, then we have6 rofinite rigidity for twisted Alexander polynomials W αρ ( t ) = ∆ αρ, ( t ) / ∆ αρ, ( t ). If O is not a UFD, then g Fitt H i (Γ , ρ ) is not necessarily a principal ideal,so that ∆ αρ,i ( t ) may not be defined. Nevertheless, we have the following. Proposition . Suppose that π is of deficiency one (e.g., a knot group). Then the fractionalideal g Fitt H (Γ , ρ ) / g Fitt H (Γ , ρ ) in the ideal group of O [ t Z ] is a principal ideal generated by Wada’spolynomial W αρ ( t ) . Hillman’s theorem [Hil12, Theorem 3.12 (3)] immediately extends to the following lemma.
Lemma . Let → K → M → C → be an exact sequence of modules over a Noether ring R and suppose r = rank( C ) and s = rank( K ) . If K is a torsion module, then g Fitt r + s ( M ) = g Fitt s ( K ) g Fitt r ( C ) holds. Proof.
Since an ideals in a Noether ring R is determined by localization at every maximal ideals,we may assume that R is a local ring, so that every projective R -module is free. In order toprove the claim for the divisorial hull, it suffices to show Fitt r + s ( M ) = Fitt s ( K )Fitt r ( C ) for thelocalization at every prime ideal of hight one. If the projective dimension of K and C are less thanone, we have presentation matrixes P ( K ) ∈ M k + s,k ( R ) of rank s for K and P ( C ) ∈ M c + r,c ( R )of rank r for C , hence P ( M ) = (cid:18) P ( K ) 0 ∗ P ( C ) (cid:19) of rank r + s for M . This proves the equality forthe localization, hence the claim. Proof of Proposition 4.1.
By the proof of [SW09, Proposition 3.6] (see also [KL99, Theorem 4.1],[HLN06, Theorem 11]), we have an exact sequence0 → H (Γ , ρ ) → A → C → H (Γ , ρ ) → O [ t Z ]-modules for based Alexander modules A and C . The Fitting ideals of A and C are known to be principal ideals. Indeed, this A admits a square presentation and Fitt( A ) is aprincipal ideal generated by the numerator of Wada’s invariant.By using Lemma 4.2 twice, we obtain g Fitt H (Γ , ρ ) · Fitt C = Fitt A · g Fitt H (Γ , ρ ) of ide-als in O [ t Z ]. Therefore, we have the equality of fractional ideals g Fitt H (Γ , ρ ) / g Fitt H (Γ , ρ ) =Fitt A / Fitt C = ( W ρ ( t )).Wada’s invariant is defined for any discrete group π of finite type. Even if we do not assumethat Fitt A and Fitt C are principal ideals, we still have the equality g Fitt H (Γ , ρ ) / g Fitt H (Γ , ρ ) = g Fitt A / g Fitt C .
5. Continuous homology and profinite completions
In this section, we recall the continuous homology H i ( b Γ , V b ρ ) and paraphrase our question.Let π , α : π ։ t Z , and ρ : π → GL N ( O ) be as in Subsection 4.1. Let V b ρ denote themodule b O n regarded as a right b π -module b O n via the transpose of the continuous representation b ρ : b π → GL N ( b O ). The homology of the profinite group b Γ with coefficient being the profinitemodule V b ρ is defined as the continuous homology of the complex b F b ⊗ b Γ V b ρ , where b F is a continuousprojective resolution of b Z over the profinite group ring b Z [[ b Γ]]. If G is a finite discrete group and B a finite ring, then the continuous homology coincides with the usual group homology.Let J be an ordered set, ( π j ) j ∈ J an inverse system of profinite groups, and ( B j ) j ∈ J an inversesystem of profinite abelian groups. If π = lim ←− π j and B = lim ←− j B j , then for each i >
0, we havean isomorphism H i ( π, B ) ∼ = lim ←− j H i ( π j , B j ) [RZ10, Proposition 6.5.7].7 un Ueki Since the set of normal subgroups G of Γ of finite index and the set of ideals I of O of finiteindex are ordered sets with respect to the inclusions, by using the assertion above twice, weobtain isomorphisms H i ( b Γ , V b ρ ) ∼ = lim ←− G ✁ Γ H i (Γ /G, V b ρ ) ∼ = lim ←− G ✁ Γ lim ←− I H i (Γ /G, V ρ mod I ) . Since every H i (Γ /G, V ρ mod I ) is a quotient of a common finitely generated O [ t Z ]-module H i (Γ , ρ ),the module H i ( b Γ , V b ρ ) is a finitely generated b O [[ t b Z ]]-module, where b O [[ t b Z ]] denotes the profinitecompletion of the group ring O [ t Z ].Now recall that our aim is to reconstruct ∆ ρ ( t ) = ∆ αρ,i ( t ) from the isomorphism class of ( b ρ, b α ).For the induced map b α : b π ։ t b Z and Γ = Ker α , we have Ker b α = b Γ. The closure of g Fitt H i (Γ , ρ )in b O [[ t b Z ]] coincides with the ideal g Fitt H i ( b Γ , b ρ ) of the continuous homology. Let s be an arbitrarygenerator of t b Z . Then there exists some v ∈ b Z ∗ with t = s v . Since b O [[ t b Z ]] = b O [[ s b Z ]], the module H i ( b Γ , V b ρ ) is a finitely generated b O [[ s b Z ]]-module with g Fitt being (∆ ρ ( s v )), supposing that ∆ ρ ( t )is defined.Hence we ask to what extent ∆ ρ ( t ) ˙= g ( t ) holds for g ( t ) ∈ O [ t Z ] and v ′ ∈ b Z ∗ with (∆ ρ ( s v )) =( g ( s v ′ )). By replacing s v and v ′ /v by s and v , we may paraphrase our question as follows. Question . Let f ( t ) , g ( t ) ∈ O [ t Z ] and v ∈ b Z ∗ with an equality ( f ( t )) = ( g ( t v )) of ideals in b O [[ t b Z ]]. To what extent does f ( t ) ˙= g ( t ) hold?
6. Cyclotomic divisors
For each positive integer m , the m -th cyclotomic polynomial Φ m ( t ) is the minimal polynomialof a primitive m -th root of unity over Q . It is known that Φ m ( t ) ∈ Z [ t ], that Φ m ( t ) vanishes atevery primitive m -th root of unity, and that Q m | n Φ m ( t ) = t n − Proposition . Let O = O F,S be the ring of S -integers of a number field F and regard O [ t Z ] ⊂ b O [[ t b Z ]] . (1) Any polynomial = f ( t ) ∈ O [ t Z ] is not a zero divisor of b O [[ t b Z ]] . (2) For any cyclotomic polynomial Φ m ( t ) and a unit v ∈ b Z ∗ , the ratio Φ m ( t v ) / Φ m ( t ) is definedand is a unit of b O [[ t b Z ]] . (3) Let f ( t ) , g ( t ) ∈ O [ t Z ] and v ∈ b Z ∗ with an equality ( f ( t )) = ( g ( t v )) of ideals of b O [[ t b Z ]] .Then Φ m ( t ) | f ( t ) holds if and only if Φ m ( t ) | g ( t ) holds, and these conditions imply the equality ( f ( t ) / Φ m ( t )) = ( g ( t v ) / Φ m ( t v )) of ideals in b O [[ t b Z ]] . By these assertions, the proof of Theorem 1.2 (1) and Question 5.1 reduce to the cases withoutcyclotomic divisors. Note that if Φ m ( t ) decomposes in O [ t ], then a divisor of Φ m ( t ) may appearas a divisor of f ( t ). Theorem 1.2 (1) cannot distinguish divisors of Φ m ( t ), while (2) may tellmore information about the multiplicity. Examples . Suppose that sin 2 π/ ∈ O ⊂ C , and let ζ = ζ be a 5th root of unity. Then wehave a decomposition Φ ( t ) = φ +5 ( t ) φ − ( t ) for φ +5 ( t ) = ( t − ζ )( t − ζ ) and φ − ( t ) = ( t − ζ )( t − ζ )in O [ t ]. Since the images of φ +5 ( t ) and φ − ( t ) under O [ t Z ] ։ O [ t Z ] / ( t − ζ ) do not coincide,8 rofinite rigidity for twisted Alexander polynomials φ +5 ( t ) ˙= φ − ( t ) does not hold. However, we have the equality ( φ +5 ( t )) = ( φ − ( t v )) of ideals in b O [[ t b Z ]] for some unit v ∈ b Z . Hence we cannot distinguish these polynomials.If such an ambiguity comes from an automorphism of π , then it is not essential for ourQuestion 5.1, while multiplicity of divisors should be cared. See also Examples 12.1 (iii).
7. Cyclic resultants and Hillar’s theorem
For each positive integer n , the n -th cyclic resultant r n of a polynomial f ( t ) ∈ C [ t ] is defined asthe resultant of f ( t ) and t n −
1. We have r n = Q ζ n =1 f ( ζ ). If f ( t ) ∈ O [ t ], then we have r n ∈ O .We call | r n | the n -th cyclic resultant absolute value of f ( t ). (We consult [Uek20] for the generaldefinition with more detailed properties.)The following result due to Hillar is a generalization of Fried’s deep result [Fri88, Proposition]: Proposition . [Hil05, Theorem 1.8]. Polynomials f ( t ) , g ( t ) ∈ R [ t ] have the same sequence ofnon-zero cyclic resultant absolute values if and only if there exists u ( t ) , v ( t ) ∈ C [ t ] with u (0) = 0 and l , l ∈ Z > satisfying f ( t ) = ± t l v ( t ) u ( t − ) t deg( u ) , and g ( t ) = t l v ( t ) u ( t ) . In this article we say that f ( t ) , g ( t ) ∈ R [ t Z ] belong to the same Hillar class if there existsome u ( t ) , v ( t ) ∈ R [ t Z ] satisfying f ( t ) ˙= u ( t ) v ( t ) and g ( t ) ˙= u ( t ) v ( t − ) in R [ t Z ]. We indeed have u ( t ) , v ( t ) ∈ R [ t Z ] in Proposition 7.1, hence the following. Corollary . If f ( t ) , g ( t ) ∈ Z S [ t Z ] have the same sequence of non-zero cyclic resultant ab-solute values, then there are some u ( t ) , v ( t ) ∈ R [ t Z ] with f ( t ) ˙= u ( t ) v ( t ) and g ( t ) ˙= u ( t ) v ( t − ) ,namely, f ( t ) , g ( t ) belong to the same Hillar class. The following assertion is obtained in the proof of [Uek20, Lemma 3.6].
Proposition . Let f ( t ) , g ( t ) ∈ Z [ t ] , v ∈ b Z ∗ and suppose that they have no root on roots ofunity. If the equality ( f ( t )) = ( g ( t v )) of ideals in b Z [[ t b Z ]] holds, then they have the same sequencesof non-zero cyclic resultant absolute values. Combining these above, we obtain the following lemma.
Lemma . Let f ( t ) , g ( t ) ∈ Z [ t ] with no roots on roots of unity, and let v ∈ b Z ∗ . Then theequality ( f ( t v )) = ( g ( t )) of ideals in b Z [[ t b Z ]] implies that f ( t ) and g ( t ) belong to the same Hillarclass.
8. Norm maps
The norm map Nr F/ Q : F → Q of a finite extension F/ Q is defined by x Q σ σ ( x ) where σ runs through all embeddings σ : F ֒ → Q . If x ∈ Q , then Nr F/ Q x = x [ F : Q ] holds. For a finite set S of maximal ideals of the ring O F of integers of F , let S also denote the set of prime numbers of Z under S . Then the norm map restricts to the norm map Nr F/ Q : O F,S → Z S on the S -integers.In addition, for a Laurent polynomial f ( t ) = P i a i t i ∈ O F,S [ t Z ] and an embedding σ : F ֒ → Q ,put f σ ( t ) = P i a σi t i ∈ O F,S [ t Z ]. Then the map Nr F/ Q : O F,S [ t Z ] → Z S [ t Z ]; f ( t ) Q σ f σ ( t ) isdefined. This map coincides with the restriction of the norm map of a separable extension of afield of rational functions. Furthermore, since the preimage of each ( n, − t m ) ⊂ Z S [ t Z ] by Nr F/ Q un Ueki contains ( n, − t m ) ⊂ O F,S [ t Z ], we have an induced map Nr F/ Q : b O F,S [[ t b Z ]] → b Z S [[ t b Z ]] on theprofinite completions.Let a be an ideal of O F,S [ t Z ]. If a is a principal ideal, then Nr F/ Q a is naturally defined. Fora general a , we define Nr F/ Q a to be the ideal generated by { Nr F/ Q a | a ∈ a } . It is equivalent toput Nr F/ Q a := ( Q σ a σ ) ∩ O F,S [ t Z ]. This correspondence also extends to the ideals of b O F,S [[ t b Z ]]and b Z S [[ t b Z ]]. Proof of Theorem 1.2 (1).
Theorem 1.2 is an answer to the question asked in Section 1. Thequestion is paraphrased to Question 5.1, and reduces to the cases without cyclotomic divisors byProposition 6.1.Recall O = O F,S . Let f ( t ) , g ( t ) ∈ O [ t Z ] and v ∈ b Z ∗ with the equality ( f ( t )) = ( g ( t v )) of idealsof b O [[ t b Z ]]. By Proposition 6.1, we may assume that they have no root on roots of unity. Put F ( t ) := Nr F/ Q f ( t ), G ( t ) := Nr F/ Q g ( t ) ∈ Z S [ t Z ]. Then we have ( F ( t )) = ( G ( t v )) in b Z S [[ t b Z ]]. Nowconsider the inverse image of ideals by b Z [[ t b Z ]] ֒ → b Z S [[ t b Z ]]. We may assume that the largest p -adicnorms of coefficients of f ( t ) and g ( t ) are both 1 for every p under S , so that we have the equality( F ( t )) = ( G ( t v )) of ideals in b Z [[ t b Z ]]. Now Lemma 7.4 assures that F ( t ) and G ( t ) in Z [ t Z ] belongto the same Hillar class. If in addition f ( t ) and g ( t ) are reciprocal, then we have F ( t ) ˙= G ( t ).For any α ∈ Q and any prime number p , we have m ( t − α ) = m ( t − − α ) and m p ( t − α ) = m p ( t − − α ). Hence the Hillar class of Nr F/ Q ∆ ρ ( t ) determines the Mahler measures m (Nr F/ Q ∆ ρ ( t ))and m p (Nr F/ Q ∆ ρ ( t )).
9. Iwasawa modules
In this section, we prove Theorem 1.2 (2) by considering the character decomposition of theIwasawa module associated to a surjective homomorphism b Z ։ Z /m Z × Z p and using theIwasawa isomorphism. Basic references are [Was97] and [Uek20]. Proof of Theorem 1.2 (2).
For each prime ideal p of the ring O = O F,S , let O p denote the p -adiccompletion of O , namely, the inverse limit lim ←− n O/ p n endowed with a natural topology. TheChinese remainder theorem yields an isomorphism b O ∼ = Q p O p . Note that b O is not an integraldomain, while so is O p . We write C p [[ ]] := O p [[ ]] ⊗ O p C p . We consider the map b O [[ t b Z ]] ։ O p [[ t Z /m Z × Z p ]] ֒ → C p [[ t Z /m Z × Z p ]] ∼ = → Y ξ m =1 C p [[ t Z p ]] ∼ = → Y ξ m =1 C p [[ T ]]given by the composite of the natural surjective homomorphisms b O → O p of the coefficientrings and t b Z ։ t Z /m Z × Z p of groups, the tensor product ⊗ O p C p , the correspondence f ( t ) f ( tξ ) for each ξ with ξ m = 1, and the Iwasawa isomorphism C p [[ t Z p ]] ∼ = C p [[ T ]]; t T oneach component. For each t − α ∈ C p [ t Z /m Z × Z p ] ⊂ C p [[ t Z /m Z × Z p ]], the map C p [[ t Z /m Z × Z p ]] ։ C p [[ t Z p ]] ∼ = → C p [[ T ]] on the component of each ξ yields the correspondence ( t − α ) ( tξ − α ) =( t − α/ξ ) (1 + T − α/ξ ) = ( T − ( α/ξ − T − ( α − ξ ) /ξ ) of ideals. By the p -adic Weierstrasspreparation theorem [Was97, Theorem 7.3], for β ∈ C p , we have | β | p < T − β )is not a unit, and hence the value β is determined by the ideal ( T − β ). If α ∈ C p with | α | p = 1and | α − ζ | p <
1, then the image of ( t − α ) is nontrivial only at the component corresponding to ξ = ζ , and determines the value α/ζ −
1. 10 rofinite rigidity for twisted Alexander polynomials
Let v ∈ b Z ∗ and let ( u, w ) denote the image of v under the map b Z ։ Z /m Z × Z p . By thesame argument as above for s = t v ∈ t b Z , the image of t − α is nontrivial only for ξ = ζ u , anddetermines the values ( α/ζ ) w and ζ u .Now let ( f ( t )) be an ideal of b O [[ t b Z ]] generated by an unknown polynomial f ( t ) ∈ O [ t ],assuming that the image of t Z is not specified in t b Z . Take p so that f ( t ) is monic in C p [ t ]and every nonzero root α i ∈ C p satisfy | α i | p = 1. Suppose that every nonzero root satisfies | α mi − | p < m ∈ N > . Then the family of α mi ’s with multiplicity is determined, again viathe Iwasawa isomorphism and the p -adic Weierstrass preparation theorem.Recall the setting of Theorem 1.2 and the paraphrased Question 5.1, and note that v ∈ b Z ∗ isnot specified. By taking m as above, the families { [( α i /ζ i ) w ] | w ∈ Z ∗ p } and { [ ζ ui ] | u ∈ ( Z /m Z ) ∗ } are determined.The Iwasawa module of Z p × Z /m Z -cover decomposes into direct sum by Z /m Z -characters.The Fitting ideal of each component coincides with the ideal of C p [[ T ]] above [Was97, § Remark . Let k ′ denote the decomposition field of f ( t ) and put d ′ = [ k ′ : Q ]. Then Propo-sition 1.3 assures that for any but finite number of prime number p with d ′ | ( p − m = p −
1, namely, every nonzero root α i ∈ C p of f ( t ) satisfies | α i | p = 1 and | α m − | p < Remark . Recall that for each ( α i ) i ∈ Q d , we denote the equivalence class by permutation ofindexes by [ α i ] i = [ α , · · · , α d ] ∈ Q d / S d . For each A = [ α i ] i and n ∈ Z , we put A n = [ α ni ] i ∈ [ Q d ].If m, n ∈ Z and A, B ∈ [ Q d ] with A m = B m and A n = B n , we do not necessarily have A g = B g for g = gcm( m, n ). Indeed, let ζ = ζ be a primitive 12-th root of unity and put A = [ ζ, ζ ], B = [ ζ , ζ ]. Then we have A = B = [1 , ζ ], A = B = [ ζ , ζ ].
10. Proofs of Propositions 1.3 and 1.4
We invoke some algebraic number theory to prove Propositions 1.3 and 1.4. Although Proposition1.3 is a well known fact, we give a proof for the convenience of the reader, only assuming basicsdescribed in [Mor12] and [Uek18].
Lemma . (1) Let p be a prime number and let n ∈ N > . Let ζ i be a primitive i -th root ofunity for each i ∈ N > . Then Q p ( ζ p n − ) is the unique unramified extension of Q p of degree n ,and is a cyclic extension with the Galois group generated by the Frobenius map ζ p n − ζ pp n − .Let F p n / F p denote the residue extension of Q p ( ζ p n − ) / Q p . Then we have natural isomorphismsof Galois groups Gal( Q p ( ζ p n − ) / Q p ) ∼ = Gal( F q n / F q ) ∼ = Z /n Z . The Frobenius map corresponds tothe multiplication by p in Z /n Z . (Note that there would exist some ν < p n − with Q p ( ζ ν ) = Q p ( ζ p n − ) .)(2) A prime number p is ramified in a finite extension k/ Q if and only if p divides thediscriminant d k of k/ Q . Hence only finite number of prime numbers p are ramified in k/ Q . Proof of Proposition 1.3.
We assume that p is unramified in k/ Q , noting that only finite numberof prime numbers are ramified in k/ Q by Lemma 10.1 (2). Put k p = Q p k , and let F p n denote theresidue field of k p . Since k p / Q p is an unramified finite extension, Lemma 10.1 (1) assures thatGal( k p / Q p ) ∼ = Gal( F p n / F p ) ∼ = Z /n Z .Put l = k ∩ Q p . Then k/l is a Galois extension with Gal( k/l ) ∼ = Gal( k p / Q p ). Indeed, let p de-note the prime ideal over ( p ) defined by Q ֒ → Q p and let D <
Gal( k/ Q ) denote the decomposition11 un Ueki group of p . The action of D on k/ Q induces a continuous action of D on k p / Q p and hence a grouphomomorphism ı : D ֒ → Gal( k p / Q p ), which is an injection by k ⊂ k p . Conversely, Gal( k p / Q p )acts on k by restriction and induces a group homomorphism j : Gal( k p / Q p ) ֒ → Gal( k/ Q ). Since k is dense in k p , this map j also is an injection. By the uniqueness of extension of p -adic valu-ation, p -adic valuation of k p is stable by the action of Im( ). Hence we have Im( ) ⊂ D , and j decomposes as j : Gal( k p / Q p ) ֒ → D ֒ → Gal( k/ Q ). By definition, D ı ֒ → Gal( k p / Q p ) ֒ → D is anidentity map. Since j is an injection, ı and are inverse maps to each other, and we have anisomorphism ι : D ∼ = → Gal( k p / Q p ). Hence k D ⊂ Q p . Since k D ⊂ k , we have k D ⊂ k ∩ Q p = l . Onthe other hand, since D = Im( ), we have l ⊂ k Dp . Therefore we have k D = l and D = Gal( k/l ).Since k/l is a subextension of k/ Q , we have [ k : l ] = n and hence n | d . Recall that theFrobenius map corresponds to the multiplication by p in Z /n Z . If d | ( p − n | ( p − F q = F p , k p = Q p , and α ∈ Q p .Now let ω : F p ֒ → Q p be a Teichm¨uller lift. For all but finite number of p , we have | α | p = 1.For such p and α , the image ζ of α via Q p ։ F p ω ֒ → Q p is a ( p − | α − ζ | p < | α p − − | p < Proof of Proposition 1.4.
Let k/ Q be a finite Galois extension with α, β ∈ k , let k p denote theclosure of k in Q p , and let σ ∈ Gal( k p / Q p ) with β = α σ . Then we have σ [ k p : Q p ] = id in Gal( k p / Q p ).Since β = α σ = α w , we have α = α σ [ kp : Q p ] = α w [ kp : Q p ] . By the assumption that α is not a root ofunity, we have w [ k p : Q p ] = 1. Hence w is a root of unity.
11. Torsion growth
In this section, we refine results of [Uek20] and [Tan18]; Fox’s formula and torsion growth formulaswith use of Mahler measures of the twisted homology groups in a Z -cover of a 3-manifold. Werecall definitions and known results in § § § Let ∆ K ( t ) denote the Alexander polynomial of a knot K in S and let X n → X = S − K denote the Z /n Z -cover. Fox’s formula (cf. [Web79]) asserts that if ∆ K ( t ) does not vanish atroots of unity, then the order of the Z -torsion subgroup satisfies | tor H ( X n ) | = Y ζ n =1 | ∆ K ( ζ ) | = | Res(∆ K ( t ) , t n − | . In addition, we easily see that if we put Ψ n ( t ) = gcd(∆ K ( t ) , t n − | tor H ( X n ) | = c n | Res(∆ K ( t ) , ( t n − / Ψ n ( t )) | for a non-zero bounded sequence c n = | tor H ( X ∞ ) / (Ψ n ( t )) | .We define the Mahler measure m ( f ( t )) of a polynomial 0 = f ( t ) ∈ C [ t ] by log m ( f ( t )) = Z | z | =1 log | f ( z ) | z dz , which is naturally interpreted even if f ( t ) has roots on | z | = 1. If f ( t ) = a Q i ( t − α i ) in C [ t ], then Jensen’s formula asserts m ( f ( t )) = | a | Y | α i | > | α i | . Since the integralover | z | = 1 coincides with the limit of the mean value of f ( ζ ) for ζ n = 1, Fox’s formula yieldsthe asymptotic formula | tor H ( X n ) | /n = m (∆ K ( t )) of torsion growth [GAS91].The author proved in [Uek20] that if we replace the absolute value by the p -adic normwith | p | = p − , then a similar argument yields | tor H ( X n ) | /np = m p (∆ K ( t )). Here we define12 rofinite rigidity for twisted Alexander polynomials our p -adic Mahler measure m p ( f ( t )) of f ( t ) ∈ C p [ t ] by log m p ( f ( t )) = lim n →∞ n X ζ n =1 log | f ( ζ ) | p . If f ( t ) = a Q i ( t − α i ) in C p [ t ], then Jensen’s formula asserts m p ( f ( t )) = | a | p Y | α i | p > | α i | p . (Note thatthis m p differs from that introduced by Besser–Deninger with use of Iwasawa’s p -adic logarithmin [BD99].)Now let ρ : π K → GL N ( O ) be a knot group representation over the ring O = O F,S of S -integers of a number field F . Analogues of those results above for the twisted Alexander poly-nomial ∆ ρ ( t ) and the twisted homology groups H ( X n , ρ ) were established by Tange [Tan18].Suppose that O is a UFD so that ∆ ρ,i ( t ) are defined for i = 0 ,
1. If ρ is irreducible, namely,all residual representations are irreducible, then we have ∆ ρ, ( t ) ˙= 1 [Tan18, Corollary 3]. Sup-pose ∆ ρ, ( t ) ˙= 1, so that we have ∆ ρ ( t ) := ∆ ρ, ( t ) ˙= W ρ ( t ). Recall ∆ ρ ( t ) = Nr F/ Q ∆ ρ ( t ). TheWang exact sequence induces a natural isomorphism H ( X ∞ , ρ ) / ( t n − H ( X ∞ , ρ ) ∼ = H ( X n , ρ ).Therefore if ∆ ρ ( t ) does not vanish at roots of unity, then we have | H ( X n , ρ ) | = S | Y ζ n =1 ∆ ρ ( ζ ) | = | Res(∆ ρ ( t ) , t n − | < ∞ . Here = S indicates equality up to multiplication by S . If S = ∅ , then wehave lim n →∞ | H ( X n , ρ ) | /n = m (∆ ρ ( t )). Our aim here is to remove several assumptions above. Let O = O F,S denote the ring of S -integersof a number field F , which is not necessarily a UFD. Let π be a finite type discrete group witha surjective homomorphism α : π ։ t Z and a representation ρ : π → GL N ( O ). Let X be aconnected manifold with π ( X ) = π and let X ∞ → X and X n → X denote the Z -cover and Z /n Z cover corresponding to α and ( Z ։ Z /n Z ) ◦ α for each n ∈ N > . We define the normpolynomial ∆ ρ ( t ) ∈ Z [ t Z ] of ρ to be a generator of Nr F/ Q g Fitt H ( X ∞ , ρ ) ∩ Z [ t Z ]. If ∆ ρ ( t ) ∈ O [ t Z ]is defined, then we have ∆ ρ ( t ) ˙= Nr F/ Q ∆ ρ ( t ) in Z S [ t Z ]. Theorem . Suppose that ρ : π → GL N ( O ) is non-abelian and absolutely irreducible. Then,(1) For each n ∈ N > , the Wang exact sequence induces a natural isomorphism H ( X ∞ , ρ ) / ( t n − H ( X ∞ , ρ ) ∼ = → H ( X n , ρ ) . (2) Put Ψ n ( t ) = gcd(∆ ρ ( t ) , t n − and r n = Res(∆ ρ ( t ) , ( t n − / Ψ n ( t )) . Then the equality | tor H ( X n , ρ ) | = c n | r n | Y p ∈ S | r n | p holds, where c n is a bounded sequence defined by c n = | tor H ( X ∞ , ρ ) / Ψ n ( t ) | .(3) For any prime number p , the asymptotic formulas lim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t )) and lim n →∞ || tor H ( X n , ρ ) || /np = m p (∆ ρ ( t )) of torsion growth hold. A representation ρ is said to be non-abelian if it does not factors through the abelianizationmap π ։ π ab . If ρ is non-abelian, then ∆ ρ,i ( t ) = 0 (cf. [FKK12]). A representation ρ is saidto be absolutely irreducible if ρ ⊗ O Q is an irreducible representation. This condition is muchweaker than being irreducible in the sense of [Tan18], namely, every residual representationbeing irreducible. Nevertheless, we have important irreducible representations over O such as13 un Ueki the Holonomy representations of hyperbolic knots. An (absolutely) irreducible representation ρ : π → GL N ( O ) with N > ρ is absolutely irreducible, we still have an injective homo-morphisms p n : H ( X ∞ , ρ ) / ( t n − H ( X ∞ , ρ ) ֒ → H ( X n , ρ ) and Coker( p n ) ֒ → H ( X ∞ , ρ ).If the Dedekind domain O is a UFD (hence a PID), then Res(∆ ρ ( t ) , t n −
1) may be replacedby Nr F/ Q Res(∆ ρ ( t ) , t n − O = O F,S instead of O F have been that (i) the image of any ρ : π → GL N ( C ) may be contained in some O F,S up to conjugate, and (ii) we may take some S so that O F,S is PID and hence ∆ ρ ( t ) is defined. Theorem 11.1 removes the reason (ii). Here we regard several lemmas of general algebra described in [Tan18] and some p -adic numbertheory as basic facts and use them rather freely. Proof of Theorem 11.1 (1).
The Wang short exact sequence of twisted complex induces the fol-lowing long exact sequence · · · → H ( X ∞ , ρ ) → t n − H ( X ∞ , ρ ) → p n H ( X n , ρ ) → ∂ H ( X ∞ , ρ ) → t n − H ( X ∞ , ρ ) → · · · . Let p = 0 be any prime ideal of O , and let F ( p ) and O ( p ) denote the localizations of F and O F at p . Since ρ is absolutely irreducible, ρ is irreducible over F ( p ) . By [FKK12, Proposition A3], theFitting ideal of H ( X ∞ , ρ ) ⊗ F ( p ) over F ( p ) [ t Z ] is (1), and hence the ideal g Fitt H ( X ∞ , ρ ) ⊗ O ( p ) isgenerated by a number in O ( p ) . This implies that the map t n − H ( X ∞ , ρ ) ⊗ O ( p ) is injective,and so is t n − H ( X ∞ , ρ ). Therefore we have Coker p n ∼ = Im ∂ = 0 and an isomorphism H ( X ∞ , ρ ) / ( t n − H ( X ∞ , ρ ) ∼ = → H ( X n , ρ ). Proof of Theorem 11.1 (2).
Assume Ψ n ( t ) = 1. Put H = H ( X ∞ , ρ ) and r n = { Res( f ( t ) , t n − | f ( t ) ∈ g Fitt
H} ⊂ O . For the product of several distinct prime ideals 0 = P = Q j p j ⊂ O , let O ( P ) denote the localization at P . For each ideal a ⊂ O , we write a ( P ) = a O ( P ) . By Hillman’s Theorem[Hil12, Theorem 3.13], we have | (tor H / ( t n − H ) ⊗ O ( p ) | p = | O ( p ) / r n ( p ) | p for each p | ( p ), hence | tor H / ( t n − H| p = Q p | ( p ) | (tor H / ( t n − H ) ⊗ O ( p ) | p = Q p | ( p ) | O ( p ) / r n ( p ) | p = | O ( p ) / r n ( p ) | p = | Z S ( p ) / Nr F/ Q r n ( p ) | p = | Z S / Nr F/ Q r n | p = | Z S / Res(∆ ρ ( t ) , t n − | p = | Res(∆ ρ ( t ) , t n − | p . Sincean O -module has no p -torsion for p ∈ S , we obtain | tor H ( X n , ρ ) | = | tor H / ( t n − H| = Q p S | tor H / ( t n − H| p = Q p S | Res(∆ ρ ( t ) , t n − | p . Next, suppose that Ψ n ( t ) is not necessarily 1, and put C n := | tor H / Ψ n ( t ) | . Then ( C n ) n is abounded sequence, since { Ψ n ( t ) | n ∈ N } is a finite set. Replacing t n − t n − / Ψ n ( t ) in theargument above, we obtain the desired formula. Proof of Theorem 11.1 (3).
Note that finite number of roots of unity may be skipped in thedefinition of ( p -adic) Mahler measure. By the p -adic asymptotic formula [Uek20, Theorem 2.5]and the assumption, we have lim n →∞ | r n | /np = m p (∆ ρ ( t )) = 1 for p ∈ S . Since lim n →∞ C /nn = 1,by taking lim n →∞ • /n in the equality | tor H ( X n , ρ ) | = | r n | Q p ∈ S | r n | p × C n in (2), we obtain thedesired formula lim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t )) and lim n →∞ || tor H ( X n , ρ ) || /np = m p (∆ ρ ( t ))14 rofinite rigidity for twisted Alexander polynomials
12. Riley’s representations
We revisit examples of twisted Alexander polynomials of Riley’s parabolic representations of2-bridge knot groups exhibited in [Tan18, Section 9] and [HM10, Example 2.3].
Examples . The knot group π of a two-bridge knot K admits a standard presentation π = h a, b | aw = wb i . Let u = α be a root of Riley’s polynomial Φ K (1 , u ) ∈ Z [ u ] and let O denotethe ring of integers of F = Q ( α ). Then u = α corresponds to a parabolic irreducible representation ρ : π → SL ( O ) defined by ρ ( a ) = (cid:18) (cid:19) and ρ ( b ) = (cid:18) − u (cid:19) . If O is a UFD (hence PID),then ∆ ρ ( t ) := ∆ ρ, ( t ) ∈ O [ t Z ] is defined and satisfies ∆ ρ ( t ) = Nr F/ Q ∆ ρ ( t ). In addition, since anyresidual representation of this ρ is irreducible, [Tan18, Corollary 3.1] yields that ∆ ρ, ( t ) ˙= 1 and W ρ ( t ) ˙= ∆ ρ ( t ). If K is hyperbolic, then (a lift of) the holonomy representation is given in thisway. (cf. [Ril72, Ril84], [DHY09, Lemma 5], [HM10, Example 2.3], [Tan18, Section 9].) Let p beany prime number.(i) Let K be the trefoil 3 . Then ∆ ρ ( t ) = t + 1 = Φ ( t ) ∈ Z [ t ]. We have ∆ ρ ( t ) = Φ ( t ) ,lim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t )) = 1 and lim n →∞ | tor H ( X n , ρ ) | /np = m p (∆( t )) = 1.(ii) Let K be the figure-eight knot 4 . Then ∆ ρ ( t ) = t − t + 1 ∈ Z [ √− ], where O = Z [ √− ] is a UFD. We have ∆ ρ ( t ) = ∆( t ) , lim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t )) = (2 + √ =7 + 4 √
3, lim n →∞ | tor H ( X n , ρ ) | /np = m p (∆ ρ ( t )) = 1.(iii) Let K = 5 . Then ∆( t ) = ( t + 1)( t − √ t + 1) = Φ ( t ) φ +20 ( t ) ∈ Z [ √ ], where O = Z [ √ ] is a UFD. We have ∆ ρ ( t ) = ( t + 1) ( t − t + t − t + 1) = Φ ( t ) Φ ( t ), hencelim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t )) = 1 and lim n →∞ | tor H ( X n , ρ ) | /np = m p (∆ ρ ( t )) = 1.(iv) Let K = 5 , which is the 2-bridge knot K (3 /
7) of type (3 ,
7) in [HM10, Example 2.3(3)], so that we have π = h x, y | wx = yw i with w = xyx − y − xy , Φ K (1 , u ) = u + u + 2 u + 1,and Wada’s invariant W ρ ( t ) = (4 + α ) t − t + (4 + α ) for ρ ⊗ C corresponding to a root u = α of Φ K (1 , u ). PARI/GP [The18] tells that the class number of F = Q ( α ) is 1, hence O F = Z [ α ] is a PID. (In addition, the discriminant is −
23, hence only p = 23 is ramified in F/ Q .) Thus ∆ ρ ( t ) is defined and coincides with W ρ ( t ). (Even if O is not a UFD, we still have∆ ρ ( t ) = Nr F/ Q W ρ ( t ).) We have ∆ ρ ( t ) = Nr F/ Q ∆ ρ ( t ) = Q Φ(1 ,u )=0 ((4 t − t + 4) + u ( t + 1)) =(4 t − t + 4) − t − t + 4) ( t + 1) + 2(4 t − t + 4)( t + 1) + ( t + 1) = 25 t − t +219 t − t + 219 t − t + 25. By Theorem 11.1, we have lim n →∞ | tor H ( X n , ρ ) | /n = m (∆ ρ ( t ))and lim n →∞ | tor H ( X n , ρ ) | /np = m p (∆ ρ ( t )) = 1.Theorem 1.2 (1) assures that the isomorphism class of b ρ determines ∆ ρ ( t ) of (i)–(iv), hence∆ ρ ( t ) of (i) and (ii). Indeed, ∆ ρ ( t ) in (i) and (ii) are recovered from the image by Nr F/ Q , only bynoting that they should belong to O [ t Z ]. In (i) for instance, even if we take a larger field F so that t ± √− ∈ O [ t Z ], we may still say ∆ ρ ( t ) = ( t ± √− hence ∆ ρ ( t ) = t + 1 by Theorem 1.2 (2).In (iii), we cannot distinguish divisors φ ± = t − ±√ t + 1 ∈ Z [ √ ] of Φ ( t ). This ambiguityis inevitable, since these two factors correspond to each other by t t v for some v ∈ b Z ∗ . If thereis some automorphism τ of π such that ∆ ρ ◦ τ ( t ) = ( t + 1)( t − −√ t + 1) = Φ ( t ) φ − ( t ) holds,then this ambiguity is not essential. In (iv), ∆ ρ ( t ) gives three candidates for ∆ ρ ( t ) ∈ Z [ α ][ t Z ].Since the holonomy representation corresponds to both of non-real roots of u + u + 2 u + 1, we15 un Ueki may eliminate one candidate and determine ∆ ρ ( t ) up to complex conjugates.Here we attach remarks on the topological entropy h and its purely p -adic analogue ~ p of thesolenoidal dynamical system defined as the dual of the meridian action on the twisted Alexandermodule.The polynomial ∆ ρ ( t ) in Example 12.1 (iv) coincides with ∆ ρ Φ ( t ) in [SW09, Example 4.5]associated to the total representation ρ Φ : π → GL ( Z ) of the Riley polynomial Φ K (1 , u ) = u + u + 2 u + 1. The topological entropy is given by the Mahler measure as h = log m (∆ ρ ( t )).If ∆ ρ ( t ) does not vanish on | z | p = 1 (this may happen only if it is not monic), then Besser–Deninger’s purely p -adic log Mahler measure m p is defined with use of Iwasawa’s p -adic logarithm,which is different from our log m p but still satisfies the Jensen formula ([BD99], see also [Uek20]).By [Kat19, Theorem 1.4], which extends [Den09], the purely p -adic periodic entropy is given by ~ p = m p (Nr F/ Q ∆ ρ ( t )).
13. Hyperbolic volumes
An optimistic conjecture of L¨uck implies the profinite rigidity of hyperbolic volumes (cf. [Rei18,Section 7]). One may ask if the hyperbolic volume Vol( K ) of a hyperbolic knot K is determinedby the isomorphism class of the profinite completion b ρ hol of the holonomy representation ρ hol : π K → SL ( O ) ⊂ SL ( C ). In this section, we prove Corollary 1.6 to partially answer this question.Recall that Wada’s initial definition of W ρ ( t ) in [Wad94] may be regarded with less indeter-minacy, so that it is defined up to multiplication by ± t ± , and coincides with the Reidemeistertorsion τ ρ ( t ) = τ ρ ⊗ α ( S − K ) ∈ F [ t Z ].Let ρ n denote the composite of ρ hol and the symmetric representation SL ( O ) → SL n ( O )for each n >
1. Based on the theory of analytic torsions due to M¨uller, Menal-Ferrer, Porti[M¨ul93, M¨ul12, MFP14] and the relationship among several torsions proved Kitano and Yam-aguchi [Kit96, Yam08], Goda proved the following formula. (Recently it was announced thatB´enard–Dubois–Heusener–Porti [BDHP19] replaced t = 1 by t = ζ for any ζ ∈ C with | ζ | = 1.) Proposition . For each m > , put A m ( t ) = τ ρ m ( t ) /τ ρ ( t ) and A m +1 ( t ) = τ ρ m +1 ( t ) /τ ρ ( t ) . Then lim m →∞ log | A m +1 (1) | (2 m + 1) = lim m →∞ log | A m (1) | (2 m ) = Vol( K )4 π holds. By Remark 1.5, if the modulus | u | of any u ∈ O is determined by | Nr F/ Q u | , then b ρ determinesthe hyperbolic volume. The knots in Examples 12.1 (ii) and (iv) are hyperbolic, while (i) and(iii) are not. In the case of the figure eight knot (ii), since u ∈ O satisfies Nr F/ Q u = | u | , thevolume Vol( K ) is determined by b ρ . (Due to Boileau–Friedl [BF15] and Bridson–Reid [BR15], thefigure eight knot has the profinite rigidity amongst 3-manifold groups. See also [BRW17].)In general, if the numbers r and 2 r of real and complex infinite places of a number field F satisfy r + r >
1, then O = O F contains a unit u with the modulus | u | 6 = 1. The field F in (iv)is a complex cubic field and satisfies r + r >
1. Hence the modulus | ∆ ρ (1) | cannot be recoveredfrom the norm Nr F/ Q ∆ ρ (1). Note in addition that O = Z does not occur, since a subgroup ofSL ( Z ) is a Fuchsian group, while a hyperbolic knot group is a Kleinian group which is notFuchsian (cf. [MR03]). At this moment by Theorem 1.2 (1) and Remark 1.5, we may concludeCorollary 1.6: If O = O F for an imaginary quadratic field F , then the hyperbolic volume Vol( K )of K is determined by the isomorphism class of b ρ hol . Further approaches are attached in Question2.1. 16 rofinite rigidity for twisted Alexander polynomials Acknowledgments
I would like to express my gratitude to L´eo B´enard, Michel Boileau, Frank Calegari, YuichiHirano, Teruhisa Kadokami, Takenori Kataoka, Tomoki Mihara, Yasushi Mizusawa, Alan Reid,Ryoto Tange, Anastasiia Tsvietkova, Yoshikazu Yamaguchi, people I met at CIRM in Luminyand at GAU in G¨ottingen, and anonymous referees of the previous article and this article forfruitful conversations. This work was partially supported by JSPS KAKENHI Grant NumberJP19K14538.
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