Torsion functions on moduli spaces in view of the cluster algebra
aa r X i v : . [ m a t h . G T ] N ov TORSION FUNCTIONS ON MODULI SPACES IN VIEW OF THE CLUSTERALGEBRA
TAKAHIRO KITAYAMA AND YUJI TERASHIMAA bstract . We introduce non-acyclic
PGL n ( C )-torsion of a 3-manifold with toroidal boundaryas an extension of J. Porti’s PGL ( C )-torsion, and present an explicit formula of the PGL n ( C )-torsion of a mapping torus for a surface with punctures, by using the higher Teichm¨uler theorydue to V. Fock and A. Goncharov. Our formula gives a concrete rational function which repre-sents the torsion function and comes from a concrete cluster transformation associated with themapping class.
1. I ntroduction
In the important work [P] J. Porti introduced non-acyclic
PGL ( C )-torsion of a 3-manifoldwith toroidal boundary, and began to study the torsion as a function on the moduli space of PGL ( C )-representations of the fundamental group. In particular, in the case of a mappingtorus for the once-punctured torus, he gave a concrete way to compute the torsion function, byusing trace functions.In this paper we introduce non-acyclic PGL n ( C )-torsion of a 3-manifold with toroidal bound-ary, and present an explicit formula of the PGL n ( C )-torsion of a mapping torus for a generalsurface with punctures, by using the higher Teichm ¨uler theory due to V. Fock and A. Gon-charov [FG]. See Theorems 4.1 and 4.2 for the precise statement of our main theorems. Ourformulas, with methods developed in [TY, NTY], give concrete rational functions which rep-resent the functions induced by twisted Alexander polynomials and the non-acyclic torsion oncomponents of the PGL n ( C )-character variety. The rational functions come from a concretecluster transformation [FZ1, FZ2] associated with the mapping class. Moreover, we show thatfor any pseudo-Anosov mapping class of a surface, the conjugacy class of a holonomy repre-sentation of the mapping torus is contained in the components.Other attempts to define non-acyclic PGL n ( C )-torsion and to give formulas in terms of quan-tities closely related to cluster variables should be remarked. In [MFP3] P. Menal-Ferrer andJ. Porti defines non-acyclic PGL n ( C )-torsion of a 3-manifold by another method, and showsan explicit relationship between its asymptotic behavior on n and the volume of the manifold,extending the result of M ¨uller for closed manifolds [M ¨u]. In [DG] T. Dimofte and S. Garoufa-lidis defines a series of invariants in terms of the shapes together with the gluing equations ofan ideal triangulation of a 3-manifold, and conjectures that each invariant of the series agreewith each term of the asymptotic expansion of the Kashaev invariant of the manifold. In par-ticular, its first one of the series should conjecturelly give non-acyclic PGL ( C )-torsion, andthey verify this experimentally for a large class of 3-manifolds. In [GGZ, GTZ] S. Garoufa-lidis, M. Goerner, D. P. Thurston and C. K. Zickert study moduli spaces of higher dimensional Mathematics Subject Classification.
Primary 57M27, Secondary 57Q10.
Key words and phrases. torsion invariant, cluster algebra, representation space. representations for a general 3-manifold in terms of analogous coordinates to Fock and Gon-charov’s associated to an ideal triangulation of the manifold itself. It is interesting to obtainan explicit formula of the
PGL n ( C )-torsion for a general 3-manifold, with a combination of theabove results and our method.This paper is organized as follows. In Section 2, following Fock and Goncharov [FG], wereview cluster algebras associated to an ideal triangulation of a punctured surface and then showthat the characters of geometric representations of mapping tori are described by the clustervariables. Section 3 is devoted to introduce and study non-acyclic Reidemeister torsion forhigher dimensional representations. In Section 4 we prove the main theorems, and demonstrateour theory with concrete examples. Acknowledgment.
The authors would like to thank H. Fuji, K. Nagao, Y. Yamaguchi andM. Yamazaki for valuable conversations. The authors also wishes to express their thanks to theanonymous referee for several useful comments in revising the manuscript.2. C haracter varieties and cluster algebras
Character varieties.
We begin with reviewing some of the standard facts on charactervarieties. See Lubotzky and Magid [LM] for more details.Let S be a compact connected oriented surface with m boundary circles. The group PGL n ( C )acts on the a ffi ne algebraic set Hom( π S , PGL n ( C )) by conjugation. We denote by X S , n thealgebro-geometric quotient of the action, which is called the PGL n ( C ) -character variety of π S .For a representation ρ : π S → PGL n ( C ) we write χ ρ for its image by the quotient map and callit the character of ρ . We fix representatives ˜ γ , . . . , ˜ γ m ∈ π S of the boundary circles of S . A framed representation is a pair of a representation ρ : π S → PGL n ( C ) and Borel subgroups B , . . . , B m of PGL n ( C ) such that ρ ( ˜ γ i ) ∈ B i for all i . The set e X S , n of framed representations isa closed subset of the a ffi ne algebraic set Hom( π S , PGL n ( C )) × B m , where B is the flag va-riety of PGL n ( C ) parameterizing Borel subgroups. The PGL n ( C ) acts on e X S , n by conjugation.We denote by X S , n the algebro-geometric quotient of the action, and for a framed representa-tion ( ρ, B , . . . , B m ) we write χ ( ρ, B ,..., B m ) for its image by the quotient map. Forgetting framings( B , . . . , B m ) gives a regular map e X S , n → Hom( π S , PGL n ( C )). We denoted by π : X S , n → X S , n the induced map on the quotients.The tangent space T χ ρ X S , n is identified with a subspace of the 1st twisted group cohomology H ◦ ρ ( π S ; pgl n ( C )) by the monomorphism given by d χ ρ t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = " γ d ρ t ( γ ) ρ t ( γ − ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t = , where ρ = ρ and γ ∈ π S [W]. It is easily seen that the map T ( ρ, B ,..., B m ) e X S , n → T ρ Hom( π , PGL n ( C )) is an epimorphism, and so is ( d π ) χ ( ρ, B ,..., Bm ) : T χ ( ρ, B ,..., Bm ) X S , n → T χ ρ X S , n .We denote by Γ S the mapping class group of S which is defined to be the group of isotopyclasses of orientation preserving homeomorphisms of S , where these isotopies are understoodto fix ∂ S pointwise. For ϕ ∈ Γ S we write M ϕ for the mapping torus S × [0 , / ( x , ∼ ( ϕ ( x ) , ϕ . A mapping class ϕ ∈ Γ S induces automorphisms ϕ ∗ on X S , n and X S , n by pullback ofrepresentations. For a representation ρ : π M ϕ → PGL n ( C ), χ ρ | π S is contained in the fixed pointset X ϕ ∗ S , n of ϕ ∗ : X S , n → X S , n . ORSION FUNCTIONS ON MODULI SPACES 3
Cluster algebras associated to an ideal triangulation.
We review cluster algebras for S , following [FG]. Here, in particular, we only consider y-variables . See [FZ1, FZ2] for moredetails on cluster algebras. In the following we assume that ∂ S is non-empty and that if thegenus of S is 0, then the number m of the boundary circles is greater than 3.Let Q be a quiver with the vertex set I = { , , . . . , l } and without loops and oriented 2-cycles.For i , j ∈ I we set ǫ i j : = ♯ { oriented edges from i to j } − ♯ { oriented edges from j to i } . Note that Q is uniquely determined by the skew-symmetric matrix ǫ i j . For k ∈ I the mutation µ k Q at k ∈ I is defined by the following matrix ǫ ′ i j : ǫ ′ i j = − ǫ i j if k ∈ { i , j } ,ǫ i j + | ǫ ik | ǫ kj + ǫ ik | ǫ kj | if k < { i , j } . A complex torus X Q : = ( C ∗ ) I is associated to Q . Let ( y , . . . , y l ) be the standard coordinates on the torus. For k ∈ I a rationalmap ( µ k ) ∗ : X Q → X µ k Q associated to the mutation µ k Q is defined by the following:the i th coordinate of ( µ k ) ∗ ( y , . . . , y l ) = y − i if i = k , y i (1 + y − k ) − ǫ ik if i , k and ǫ ik ≥ , y i (1 + y k ) − ǫ ik if i , k and ǫ ik ≤ . Shrinking each component of ∂ S , we get a closed surface S with marked points. A triangu-lation of S with vertices at the marked points is called an ideal triangulation of S . In this paperwe only consider an ideal triangulation without self-folded edges. Such a triangulation existsunder the above assumption on S .Let T be an ideal triangulation of S and let n be a positive integer. We identify each triangleof T with the triangle x + y + z = n , x , y , z > x = p , y = p , z = p where 0 ≤ p ≤ n isan integer. The subtriangulation T n of T is called the n-triangulation of T . The quiver Q T , n associated to T n is defined as: Q T , n : = T (1) n \ T (1) , where T (1) and T (1) n are the 1-skeletons of T and T n respectively. (See Figure 1.) The vertex set I T , n of Q T , n consists of vertices of T n except the marked points of S . The orientation of eachedge of Q T , n is provided by that of S as follows. Take a triangle ∆ of T , which is oriented as asubspace of S . Then each edge of Q T , n contained in ∆ is oriented so that the direction is parallelto one of the boundary edge of ∆ . For simplicity of notation, we set X T , n : = X Q T , n . Writing e and f for the number of edges and faces of T respectively, we have | I T , n | = ( n − e + ( n − n − f ,χ ( S ) = − e + f , e = f . T. KITAYAMA AND Y. TERASHIMA
These imply the formula dim X T , n = | I T , n | = − ( n − χ ( S ) . In the following we set l = − ( n − χ ( S ) . F igure
1. The 3-triangulation T and the quiver Q T , Fock and Goncharov [FG, Section 9] constructed a regular map ν T : X T , n → X S , n and arational map ϕ T : X S , n → X T , n such that ϕ T ◦ ν T = id . In particular, ν T is an embedding, and[FG, Theorem 9.1] implies that the images for all the triangulations cover X S , n . The regularmap ν T : X T , n → X S , n is explicitly constructed in [FG, Section 9.10], and the rational map ϕ T : X S , n → X T , n is in [FG, Section 9.3]. For ( y , . . . , y l ) ∈ X T , n we set χ ( y ,..., y l ) : = π ◦ ν T (( y , . . . , y l )) . Remark . Fock and Goncharov associated cluster algebras also to an ideal triangulation withself-folded edges. See [FG, Section 10.7] for the treatment of the case.2.3.
The mapping class group actions on cluster variables.
Here we define the action ϕ ∗ : X T , n → X T , n for each ϕ ∈ Γ S . The definition plays important role to relate the modulispace for M ϕ to cluster algebras for the fiber surface. In fact the fixed point set X ϕ ∗ T , n parameter-izing X ϕ ∗ S , n makes sense.A mapping class ϕ ∈ Γ S naturally induces a bijection ϕ ∗ : I T , n → I ϕ ( T ) , n . It defines an isomor-phism σ : X ϕ ( T ) , n → X T , n by σ ( y ϕ ∗ (1) , . . . , y ϕ ∗ ( l ) ) = ( y , . . . , y l ) . The isomorphism σ is called the labeling change of ϕ . This is essential for obtaining the genuine action ϕ ∗ : X T , n → X T , n defined later. Proposition 2.2.
For ϕ ∈ Γ S the following diagram commutes: X ϕ ( T ) , n σ −−−−−→ X T , n ν ϕ ( T ) y ν T y X S , n ϕ ∗ −−−−−→ X S , n . ORSION FUNCTIONS ON MODULI SPACES 5
Proof.
We first briefly overview the flow of the construction of the map ν T : X T , n → X S , n . Let Γ be the 1-skeleton of a dual complex of T . Replacing edges and vertices of Γ by rectanglesand hexagons respectively, we obtain a decomposition of S . The orientation of S naturallyinduces that of each edge of the decomposition. We denote by ∆ the set of oriented edges ofthe decomposition. It follows from [FG, Lemma 9.6] that X S , n can be regarded as a quotientof PGL n ( C ) ∆ . For ( y , . . . , y l ) ∈ X T , n a representative of ν T ( y , . . . , y l ) in PGL n ( C ) ∆ is explicitlygiven as in [FG, Theorem 9.2].Let e ∈ ∆ . If e is an edge of a rectangle, then let v , . . . , v q be the vertices of Q T , n on thetwo triangles sharing the edge corresponding with the rectangle. If e is an edge of a hexagon,then let v , . . . , v q be the vertices of Q T , n on the triangle corresponding with the hexagon. Itfollows from the construction [FG, Theorem 9.2] of ν T : X T , n → X S , n that for ( y , . . . , y l ) ∈ X T , n , ν T ( y , . . . , y l ) is presented by an element of PGL n ( C ) ∆ whose image of e is determined only bythe coordinates ( y v , . . . , y v q ) in ( y , . . . , y l ), and that ν ϕ ( T ) ( σ − ( y , . . . , y l )) is represented by onewhose image of ϕ ( e ) is similarly determined by ( y ϕ ( v ) , . . . , y ϕ ( v q ) ) in σ − ( y , . . . , y l ), which areequal to ( y v , . . . , y v q ) in ( y , . . . , y l ). Therefore ϕ ∗ ◦ ν ϕ ( T ) ◦ σ − ( y , . . . , y l ) = ν T ( y , . . . , y l )for any ( y , . . . , y l ) ∈ X T , n , and the lemma follows. (cid:3) Let T ′ be an ideal triangulation of S obtained from T by a flip f at an edge e . We identifyeach of two triangles sharing e as a face with the triangle x + y + z = n , x , y , z > x = e . Let v , . . . , v n − be the vertices of Q T , n on the line x =
0, and let v i , . . . , v in − i − and w i , . . . , w in − i − be these on the line x = i contained in the interiorof the two triangles for 1 ≤ i ≤ n −
2. Then the following composition of mutations change Q T , n into Q T ′ , n [FG, Proposition 10.1]: µ n − ◦ · · · ◦ µ , where µ : = µ v ◦ · · · ◦ µ v n − ,µ i : = ( µ v i ◦ · · · ◦ µ v in − i − ) ◦ ( µ w i ◦ · · · ◦ µ w in − i − ) , for 1 ≤ i ≤ n − . A rational map f ∗ : X T , n → X T ′ , n is defined as f ∗ : = ( µ n − ) ∗ ◦ · · · ◦ ( µ ) ∗ . The following commutative diagram is proved in [FG, Sections 10.5 and 10.6]: X T , n f ∗ / / ν T " " ❊❊❊❊❊❊❊❊ X T ′ , n ν T ′ | | ②②②②②②②② X S , n Definition 2.3.
We take a sequence f , . . . , f q of flips changing T into ϕ ( T ) and define a rationalmap ϕ : X T , n → X T , n as ϕ ∗ = σ ◦ ( f q ) ∗ ◦ · · · ◦ ( f ) ∗ , where σ is the labeling change of ϕ . T. KITAYAMA AND Y. TERASHIMA
The following is now a direct consequence of Proposition 2.2.
Corollary 2.4.
For ϕ ∈ Γ S the following diagram commutes: X T , n ϕ ∗ −−−−−→ X T , n ν T y ν T y X S , n ϕ ∗ −−−−−→ X S , n . Note that it follows from the above corollary that ϕ ∗ : X T , n → X T , n does not depend on thechoice of a sequence of flips.2.4. The character of a holonomy representation.
We show that the characters of geometricrepresentations of mapping tori are described by cluster variables.It is well-known that for ϕ ∈ Γ S the mapping torus M ϕ has a hyperbolic structure if and onlyif ϕ is pseudo-Anosov [Th]. Theorem 2.5.
Let ϕ ∈ Γ S be pseudo-Anosov and ρ : π M ϕ → PGL ( C ) a holonomy represen-tation of M ϕ . Then there exists y i ∈ C ∗ for i = , . . . , l such that χ ρ | π S = χ ( y ,..., y l ) .Proof. Since for any representative ˜ γ ∈ π S of a boundary circle of S a Borel subgroup con-taining ρ ( ˜ γ ) is uniquely determined, π − ( χ ρ | π S ) consists of one point χ ( ρ | π S , B ,..., B m ) . It su ffi cesto show that the rational map ϕ T : X S , → X T , is defined on the point, since, if so, then ϕ T ( χ ( ρ | π S , B ,..., B m ) ) ∈ X T , satisfies the desired condition.Let e be an edge of T and let Γ be the 1-skeleton of a dual complex of T . Write x , y , z , t for thevertices of two triangles of T sharing e so that xtz and xzy are the triangles compatible with theorientations coming from that of S . There are natural 4 (unoriented) loops γ x , γ y , γ z , γ t in Γ start-ing at a point on the dual edge of e and going around the boundary of the dual cells of the vertices x , y , z , t respectively. Take representatives ˜ γ x , ˜ γ y , ˜ γ z , ˜ γ t ∈ π S of γ x , γ y , γ z , γ t with any orienta-tions respectively, and let λ x , λ y , λ z , λ t ∈ C P be the fixed point of the M ¨obius transformations ρ ( ˜ γ x ) , ρ ( ˜ γ y ) , ρ ( ˜ γ z ) , ρ ( ˜ γ t ) respectively. Then, if defined, the coordinate y e of ϕ T ( χ ( ρ | π S , B ,..., B m ) )corresponding to the vertex on e is given by y e = ( λ x − λ t )( λ y − λ z )( λ z − λ t )( λ x − λ y ) . See [FG, Sections 9.3 and 9.5] for the definition of the coordinate functions. Since ˜ γ x , ˜ γ y , ˜ γ z , ˜ γ t are distinct nontrivial elements of the free group π S and since ρ : π M ϕ → PGL n ( C ) is faithful, ρ ( ˜ γ x ) , ρ ( ˜ γ y ) , ρ ( ˜ γ z ) , ρ ( ˜ γ t ) are non-commutative with each other, and so λ x , λ y , λ z , λ t are all distinctelements. Therefore the value y e is nonzero for each e , which implies that ϕ T : X S , → X T , isdefined on χ ( ρ | π S , B ,..., B m ) . (cid:3) Corollary 2.6.
For any pseudo-Anosov mapping class ϕ ∈ Γ S the fixed point set X ϕ ∗ T , n isnonempty.Proof. Let ι n : X T , → X T , n be the map defined as follows. For ( y , . . . , y l ) ∈ X T , , each co-ordinate of ι n ( y , . . . , y l ) corresponding to a vertex of Q T , n on an edge of T is defined to be y i corresponding to the unique vertex of Q T , on the same edge, and the other coordinates are alldefined to be 1. The commutativity of the following diagram is straightforward by the definition ORSION FUNCTIONS ON MODULI SPACES 7 of ϕ ∗ : X T , ϕ ∗ −−−−−→ X T , ι n y ι n y X T , n ϕ ∗ −−−−−→ X T , n . It follows from this commutativity, Corollary 2.4 and Theorem 2.5 that for ( y , . . . , y l ) ∈ X T , inTheorem 2.5, ι n ( y , . . . , y l ) ∈ X ϕ ∗ T , n , which proves the corollary. (cid:3)
3. T orsion functions
Reidemeister torsion.
First we review basics of Reidemeister torsion. See Milnor [Mi1]and Turaev [Tu] for more details.Let C ∗ = ( C n ∂ n −→ C n − → · · · → C ) be a finite dimensional chain complex over a commuta-tive field F , and let c = { c i } and h = { h i } be bases of C ∗ and H ∗ ( C ∗ ) respectively. Choose bases b i of Im ∂ i + for each i = , , . . . n , and take a basis b i h i b i − of C i for each i as follows. Picking alift of h i in Ker ∂ i and combining it with b i , we first obtain a basis b i h i of C i . Then picking a liftof b i − in C i and combining it with b i h i , we obtain a basis b i h i b i − of C i . The algebraic torsion τ ( C ∗ , c , h ) is defined as: τ ( C ∗ , c , h ) : = n Y i = [ b i h i b i − / c i ] ( − i + ∈ F × , where [ d ′ / d ] is the determinant of the base change matrix from d to d ′ for bases d and d ′ . If C ∗ is acyclic, then we just write τ ( C ∗ , c ). It can be easily checked that τ ( C ∗ , c , h ) does not dependon the choices of b i and b i h i b i − .The algebraic torsion τ has the following multiplicative property. Let0 → C ′∗ → C ∗ → C ′′∗ → F and let c = { c i } , c ′ = { c ′ i } , c ′′ = { c ′′ i } and h = { h i } , h ′ = { h ′ i } , h ′′ = { h ′′ i } be bases of C ∗ , C ′∗ , C ′′∗ and H ∗ ( C ∗ ) , H ∗ ( C ′∗ ) , H ∗ ( C ′′∗ ). Picking a lift of c ′′ i in C i and combining it with the image of c ′ i in C i , we obtain a basis c ′ i c ′′ i of C i . We denote by H ∗ the corresponding long exact sequence inhomology, and by d the basis of H ∗ obtained by combining h , h ′ , h ′′ . Lemma 3.1. ( [Mi1, Theorem 3. 1] ) If [ c ′ i c ′′ i / c i ] = for all i, then τ ( C ∗ , c , h ) = τ ( C ′∗ , c ′ , h ′ ) τ ( C ′′∗ , c ′′ , h ′′ ) τ ( H ∗ , d ) . In the following when we write C ∗ ( e Y , e Z ) for a CW-pair ( Y , Z ), e Y , e Z stand for the universalcover of Y and the pullback of Z by the universal covering map e Y → Y respectively. For a n -dimensional representation ρ : π Y → GL ( V ) over a commutative field F we define the twistedhomology group and the cohomology group associated to ρ as follows: H ρ i ( Y , Z ; V ) : = H i ( C ∗ ( e Y , e Z ) ⊗ Z [ π Y ] V ) , H i ρ ( Y , Z ; V ) : = H i (Hom Z [ π Y ] ( C ∗ ( e Y , e Z ) , V )) . If Z is empty, then we write H ρ i ( Y ; V ) and H i ρ ( Y ; V ) respectively. T. KITAYAMA AND Y. TERASHIMA
For a basis h of H ρ ∗ ( Y ; V ) the Reidemeister torsion τ ρ ( Y ; h ) associated to ρ and h is defined asfollows: We choose a lift ˜ e in e Y for each cell e ⊂ Y . Then τ ρ ( Y ; h ) : = τ ( C ∗ ( e Y ) ⊗ Z [ π Y ] V , h ˜ e ⊗ i e , h ) ∈ F × / ( − n det ρ ( π Y ) . If H ρ ∗ ( Y ; V ) =
0, then we drop h in the notation τ ρ ( Y ; h ). It can be easily checked that τ ρ ( Y ; h )does not depend on the choice of ˜ e and is invariant under conjugation of representations. It isknown that Reidemeister torsion is a simple homotopy invariant.Let M be a compact connected orientable 3-manifold with empty or toroidal boundary andlet ψ : π M → h t i be a homomorphism. For a representation ρ : π Y → GL n ( F ) satisfying H ψ ⊗ ρ ∗ ( Y ; F ( t ) n ) =
0, where ψ ⊗ ρ : π M → GL n ( F ( t )) is given by ψ ⊗ ρ ( γ ) = ψ ( γ ) ρ ( γ ) for γ ∈ π M ,the Reidemeister torsion τ ψ ⊗ ρ ( M ) is known by Kirk and Livingston [KL], and Kitano [K] tobe essentially equal to the twisted Alexander polynomial associated to ψ and ρ . For twistedAlexander polynomials we refer the reader to [FV].3.2. Non-acyclic Reidemeister torsion for higher dimensional representations.
We intro-duce non-acyclic Reidemeister torsion of a 3-manifold for higher dimensional representationsas a natural generalization of Porti’s torsion for a 2-dimensional representation [P].For a compact orientable manifold Y and a representation ρ : π Y → PGL n ( C ) the Killingform of pgl n ( C ) induces a non-degenerate intersection pairing:(3.1) H Ad ◦ ρ i ( Y ; pgl n ( C )) × H Ad ◦ ρ − i ( Y , ∂ Y ; pgl n ( C )) → C . Let M be a compact connected orientable 3-manifold whose boundary consists of m tori T i and let γ i ⊂ T i be a simple closed curve for each i . For a representation ρ : π M → PGL n ( C ) a homomorphism pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )), where pgl n ( C ) π T i : = { v ∈ pgl n ( C ) ; Ad ◦ ρ ( π T i ) v = v } , is defined to map v to [ ˜ γ i ⊗ v ] for v ∈ pgl n ( C ) π T i , where ˜ γ i is a lift of γ i in e M . Similarly, a homomorphism pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) is de-fined to map v to [ e T i ⊗ v ] for v ∈ pgl n ( C ) π T i , where e T i is a lift of T i in e M . We denote by ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) and ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) thedirect sums of the homomorphisms for i respectively. Definition 3.2.
A representation ρ : π M → PGL n ( C ) is called ( γ , . . . , γ m ) -regular if:(i) H Ad ◦ ρ ( M ; pgl n ( C )) = pgl n ( C ) π T i = n − i ,(iii) ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) is surjective. Remark . The above definition is equivalent to one for representations π M → S L ( C ) byPorti [P, D´efinition 3.21]. (See also [P, Proposition 3.22].)It is easily seen that if a representation ρ : π M → PGL n ( C ) is ( γ , . . . , γ m )-regular, then so isa conjugation of ρ .The following theorem strongly depends on the works of Menal-Ferrer and Porti [MFP1,MFP2]. Theorem 3.4.
Suppose that M is a hyperbolic -manifold. Let ρ : π M → PGL ( C ) be aholonomy representation and ι n : PGL ( C ) → PGL n ( C ) is the representation induced by an ir-reducible representation S L ( C ) → S L n ( C ) . Then for any γ i ⊂ T i which is not null-homologousthe composition ι n ◦ ρ : π M → PGL n ( C ) is ( γ , . . . , γ m ) -regular. ORSION FUNCTIONS ON MODULI SPACES 9
Proof.
Since Ad ◦ ι n ◦ ρ is non-commutative, H ◦ ι n ◦ ρ ( M ; pgl n ( C )) = pgl n ( C ) π M = . Now it follows from Poincar´e duality and the duality induced by the intersection pairing (3.1)that H Ad ◦ ι n ◦ ρ ( M ; pgl n ( C )) =
0, which proves the condition (i).Since ρ | π T i : π T i → is a parabolic representation for each i , it follows from [MFP1, Lemma2.1] that dim pgl n ( C ) π T i = n − i , which proves the condition (ii).We denote by X M , n and X γ i , n the PGL n ( C )-character varieties of the fundamental groups of M and γ i respectively. It follows from [MFP1, Theorem 1.1] that regular functions X M , n → C in-duced by symmetric polynomials of eigenvalues for ι n ◦ ρ ( γ i ) for all i except for the determinantgive biholomorphic local coordinates of X M , n as a m ( n − X γ i , n has a similar biholomorphic local coordinates as a ( n − T χ ι n ◦ ρ X M , n → ⊕ i T χ ι n ◦ ρ X γ i , n is an isomorphism,which implies that so is the homomorphism H ◦ ι n ◦ ρ ( M ; pgl n ( C )) → ⊕ i H ◦ ι n ◦ ρ ( γ i ; pgl n ( C )) un-der the identifications T χ ι n ◦ ρ X M , n = H ◦ ι n ◦ ρ ( M ; pgl n ( C )) and T χ ι n ◦ ρ X γ i , n = H ◦ ι n ◦ ρ ( γ i ; pgl n ( C ))for each i . (See also [MFP2, Theorem 0.3].) Now it follows from Poincar´e duality and the dual-ity induced by the intersection pairing (3.1) that the homomorphism ⊕ i H Ad ◦ ι n ◦ ρ ( γ i ; pgl n ( C )) → H Ad ◦ ι n ◦ ρ ( M ; pgl n ( C )) is an isomorphism. Sincedim pgl n ( C ) π T i = dim H Ad ◦ ι n ◦ ρ ( γ i ; pgl n ( C )) = n − , the homomorphism pgl n ( C ) π T i → H Ad ◦ ι n ◦ ρ ( γ i ; pgl n ( C )) mapping v to [ ˜ γ i ⊗ v ] for v ∈ pgl n ( C ) π T i is an isomorphism for each i . Therefore ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )), which is acomposition of the above homomorphisms, is also an isomorphism, which proves the condition(iii). (cid:3) Lemma 3.5.
If a representation ρ : π T → PGL n ( C ) satisfies that dim pgl n ( C ) π T = n − , then ( i ) dim H Ad ◦ ρ ( T ; pgl n ( C )) = dim H Ad ◦ ρ ( T ; pgl n ( C )) = n − , ( ii ) dim H Ad ◦ ρ ( T ; pgl n ( C )) = n − . Proof.
Since H Ad ◦ ρ ( T ; pgl n ( C )) is isomorphic to pgl n ( C ) π T ,dim H Ad ◦ ρ ( T ; pgl n ( C )) = dim pgl n ( C ) π T = n − . It follows from the duality induced by the intersection pairing (3.1) thatdim H Ad ◦ ρ ( T ; pgl n ( C )) = dim H Ad ◦ ρ ( T ; pgl n ( C )) = n − . Since X i = ( − i dim H Ad ◦ ρ i ( T ; pgl n ( C )) = ( n − χ ( M ) = , we havedim H Ad ◦ ρ ( T ; pgl n ( C )) = dim H Ad ◦ ρ ( T ; pgl n ( C )) + dim H Ad ◦ ρ ( T ; pgl n ( C )) = n − . (cid:3) Lemma 3.6.
If a representation ρ : π M → PGL n ( C ) is ( γ , . . . , γ m ) -regular for γ , . . . , γ m , then dim H Ad ◦ ρ ( M ; pgl n ( C )) = dim H Ad ◦ ρ ( M ; pgl n ( C )) = m ( n − . Proof.
Since ψ : ⊕ mi = pgl n ( C ) Ad ◦ ρ ( π T i ) → H Ad ◦ ρ ( M ; pgl n ( C )) is surjective, so is the homo-morphism H Ad ◦ ρ ( ∂ M ; pgl n ( C )) → H Ad ◦ ρ ( M ; pgl n ( C )). It follows from the duality inducedby the intersection pairing (3.1) that the dual homomorphism H Ad ◦ ρ ( M , ∂ M ; pgl n ( C )) → H Ad ◦ ρ ( ∂ M ; pgl n ( C )) is injective. Now the homology long exact sequence for the pair ( M , ∂ M )gives the exact sequence0 → H Ad ◦ ρ ( M , ∂ M ; pgl n ( C )) → H Ad ◦ ρ ( ∂ M ; pgl n ( C )) → H Ad ◦ ρ ( M ; pgl n ( C )) → . Hence by Lemma 3.5 (ii) we obtaindim H Ad ◦ ρ ( M ; pgl n ( C )) =
12 dim H Ad ◦ ρ ( ∂ M ; pgl n ( C )) = m X i = dim H Ad ◦ ρ ( T i ; pgl n ( C )) = m ( n − . Since X i = ( − i dim H Ad ◦ ρ i ( M ; pgl n ( C )) = ( n − χ ( M ) = , we have dim H Ad ◦ ρ ( M ; pgl n ( C )) = dim H Ad ◦ ρ ( T ; pgl n ( C )) = m ( n − . (cid:3) Lemma 3.7.
If a representation ρ : π M → PGL n ( C ) is ( γ , . . . , γ l ) -regular for γ , . . . , γ l , then ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) and ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) areisomorphisms.Proof. By Lemma 3.6dim H Ad ◦ ρ ( M ; pgl n ( C )) = dim H Ad ◦ ρ ( M ; pgl n ( C )) = dim ⊕ mi = pgl n ( C ) π T i = m ( n − . Since ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) is surjective, it is an isomorphism. Since H Ad ◦ ρ ( M ; pgl n ( C )) =
0, it follows from the duality induced by the intersection pairing (3.1)that H Ad ◦ ρ ( M , ∂ M ; pgl n ( C )) =
0. Now the homology long exact sequence for the pair ( M , ∂ M )implies that the homomorphism H ( ∂ M ; pgl n ( C )) → H Ad ◦ ρ ( M ; pgl n ( C )) is injective. Hence ψ : ⊕ mi = pgl n ( C ) π T i → H Ad ◦ ρ ( M ; pgl n ( C )) is also injective, and so it is an isomorphism. (cid:3) Definition 3.8.
For a ( γ , . . . , γ m )-regular representation ρ : π M → PGL n ( C ) we define the non-acyclic Reidemeister torsion T ( γ ,...,γ m ) ,ρ ( M ) associated to ( γ , . . . , γ m ) and ρ as follows. Wechoose a basis b i of pgl n ( C ) π T i for each i . Then T ( γ ,...,γ m ) ,ρ ( M ) = τ Ad ◦ ρ ( M ; h ∪ h ) ∈ C × / ± , where h : = h ψ ( b ) , . . . , ψ ( b m ) i , h : = h ψ ( b ) , . . . , ψ ( b m ) i . ORSION FUNCTIONS ON MODULI SPACES 11
It can be checked as follows that T ( γ ,...,γ m ) ,ρ ( M ) does not depend on the choice of b i . Let b ′ i beanother basis of pgl n ( C ) π T i for each i , and set h ′ : = h ψ ( b ′ ) , . . . , ψ ( b ′ m ) i , h ′ : = h ψ ( b ′ ) , . . . , ψ ( b ′ m ) i . Then by the definition of Reidemeister torsion we have τ Ad ◦ ρ ( M ; h ′ ∪ h ′ ) = [ h ′ / h ][ h ′ / h ] τ Ad ◦ ρ ( M ; h ∪ h ) . and an easy computation implies[ h ′ / h ] = [ h ′ / h ] = m Y i = [ b ′ i / b i ] , which shows the independence.3.3. Non-acyclic Reidemeister torsion for fibered -manifolds. We show a formula com-puting non-acyclic Reidemeister torsion of fibered 3-manifolds from the monodromy maps.The formula generalizes a homological version of [D, Main Theorem] for fibered knots and2-dimensional representations.
Theorem 3.9.
Let γ , . . . , γ m be the boundary components of S and let ϕ ∈ Γ S . For a ( γ , . . . , γ m ) -regular representation ρ : π M ϕ → PGL n ( C ) satisfying H Ad ◦ ρ ( S ; pgl n ( C )) = ,T ( γ ,...,γ m ) ,ρ ( M ϕ ) = lim t → det( t ϕ ∗ − id )( t − m ( n − , where we consider ϕ ∗ : H Ad ◦ ρ ( S ; pgl n ( C )) → H Ad ◦ ρ ( S ; pgl n ( C )) in the formula.Proof. We denote by ψ ′ : pgl n ( C ) π T i → H Ad ◦ ρ ( S ; pgl n ( C )) the factor of ψ : pgl n ( C ) π T i → H Ad ◦ ρ ( M ϕ ; pgl n ( C )). It follows from the duality induced by the intersection pairing (3.1) that H Ad ◦ ρ ( S , ∂ S ; pgl n ( C )) = H Ad ◦ ρ ( S ; pgl n ( C )) =
0. Now the homology long exact sequencefor the pair ( S , ∂ S ) implies that the homomorphism H Ad ◦ ρ ( ∂ S ; pgl n ( C )) → H Ad ◦ ρ ( S ; pgl n ( C ))is injective, and so is ψ ′ . Choose a basis b i of pgl n ( C ) π T i for each i and take a basis h = h ψ ′ ( b ) , . . . , ψ ′ ( b m ) i ∪ b of H Ad ◦ ρ ( S ; pgl n ( C )), by adding subbasis b .Take a representative of ϕ and a triangulation of S such that the representative is simplicial,and consider the following exact sequence:0 → C Ad ◦ ρ ∗ ( e S ) ⊗ pgl n ( C ) id × − ϕ ∗ × −−−−−−−→ C Ad ◦ ρ ∗ ( e S × [0 , ⊗ pgl n ( C ) → C Ad ◦ ρ ∗ ( e M ϕ ) ⊗ pgl n ( C ) → . By Lemma 3.1 τ ρ ( S × [0 , h ) = τ ρ ( S ; h ) T ( γ ,...,γ m ) ,ρ ( M ϕ ) τ ( H ∗ , d ) , where H ∗ : = (0 → H Ad ◦ ρ ( M ϕ ) → H Ad ◦ ρ ( S ) I − ϕ ∗ −−−→ H Ad ◦ ρ ( S ) → H Ad ◦ ρ ( M ϕ ) → , d : = h ∪ h ∪ h ∪ h . Since τ ρ ( S × [0 , h ) = τ ρ ( S ; h ), we have T ( γ ,...,γ m ) ,ρ ( M ϕ ) = τ ( H ∗ , d ) − . Considering the following commutative diagram of exact sequences0 −−−−−→ H Ad ◦ ρ ( ∂ M ϕ ) −−−−−→ ⊕ i H Ad ◦ ρ ( γ i ) −−−−−→ ⊕ i H Ad ◦ ρ ( γ i ) −−−−−→ H Ad ◦ ρ ( ∂ M ϕ ) y y y y −−−−−→ H Ad ◦ ρ ( M ϕ ) −−−−−→ H Ad ◦ ρ ( S ) id − ϕ ∗ −−−−−→ H Ad ◦ ρ ( S ) −−−−−→ H Ad ◦ ρ ( M ϕ ) −−−−−→ , where we omit to write the coe ffi cient pgl n ( C ), we see that the homomorphism H Ad ◦ ρ ( S ; pgl n ( C )) → H Ad ◦ ρ ( M ϕ ; pgl n ( C )) maps h ψ ′ ( b ) , . . . , ψ ′ ( b m ) i to h and that the homo-morphism H Ad ◦ ρ ( M ϕ ; pgl n ( C )) → H Ad ◦ ρ ( S ; pgl n ( C )) maps h to h ψ ′ ( b ) , . . . , ψ ′ ( b m ) i . Therefore τ ( H ∗ , d ) − = det(( id − ϕ ∗ ) : Coker ψ ′ → Coker ψ ′ ) = ± lim t → det( t ϕ ∗ − id )( t − m ( n − , which proves the theorem. (cid:3) For a later use, we recall a well-known formula of ‘twisted Alexander polynomials’ forfibered 3-manifolds. See for instance [Mi2].
Lemma 3.10.
Let ϕ ∈ Γ S and let ψ : π M ϕ → h t i be the homomorphism induced by the fibration.For a representation ρ : π M ϕ → GL n ( V ) over F , τ ψ ⊗ ρ ( M ϕ ) = det( t ϕ − id )det( t ϕ − id ) , where ϕ : H ρ ( S ; V ) → H ρ ( S ; V ) , ϕ : H ρ ( S ; V ) → H ρ ( S ; V ) are the homomorphisms inducedby ϕ . The following is a direct corollary of Theorem 3.9 and Lemma 3.10.
Corollary 3.11.
Let γ , . . . , γ m be the boundary components of S , let ϕ ∈ Γ S and let ψ : π M ϕ →h t i be the homomorphism induced by the fibration. For a ( γ , . . . , γ m ) -regular representation ρ : π M ϕ → PGL n ( C ) satisfying H Ad ◦ ρ ( S ; pgl n ( C )) = ,T ( γ ,...,γ m ) ,ρ ( M ϕ ) = lim t → τ ψ ⊗ Ad ◦ ρ ( M ϕ )( t − m ( n − , where we consider ϕ ∗ : H Ad ◦ ρ ( S ; pgl n ( C )) → H Ad ◦ ρ ( S ; pgl n ( C )) in the formula.
4. M ain theorems
Proof.
In this section we show the main theorems on torsion invariants and cluster alge-bras for surfaces.Recall that Fock and Goncharov constructed a regular map ν T : X T , n → X S , n for an idealtriangulation T of S , and that T χ ρ X S , n is identified with a subspace of H ◦ ρ ( π S ; pgl n ( C )) for arepresentation ρ : π S → PGL n ( C ). Thus ν T induces a map T ( y ,..., y l ) X T , n → H ◦ ρ ( π S ; pgl n ( C )). Lemma 4.1.
Let ( y , . . . , y l ) ∈ X T , n and ρ : π S → PGL n ( C ) a representation such that χ ( y ,..., y l ) = χ ρ . If H ◦ ρ ( π S ; pgl n ( C )) = , then the map T ( y ,..., y l ) X T , n → H ◦ ρ ( π S ; pgl n ( C )) is an isomorphism. ORSION FUNCTIONS ON MODULI SPACES 13
Proof.
Since T χ ρ X S , n embeds in H ◦ ρ ( π S ; pgl n ( C )), we have l = dim X S , n ≤ T χ ρ X S , n ≤ H ◦ ρ ( π S ; pgl n ( C )) = l , and so the inequalities are all equalities. Moreover, since for χ ( ρ, B ,..., B m ) ∈ X S , n ,( d π ) χ ( ρ, B ,..., Bm ) : T χ ( ρ, B ,..., Bm ) X S , n → T χ ρ X S , n is an epimorphism, we have l = T χ ρ X S , n ≤ T ( y ,..., y l ) X T , n = l , and the inequality is an equality, which proves the lemma. (cid:3) Now we prove the following main theorems:
Theorem 4.2.
Let ϕ ∈ Γ S . For a representation ρ : π M ϕ → PGL n ( C ) satisfyingH Ad ◦ ρ ( S ; pgl n ( C )) = and ( y , . . . , y l ) ∈ X ϕ ∗ T , n , if χ ρ | π S = χ ( y ,..., y l ) , then τ ψ ⊗ Ad ◦ ρ ( M ϕ ) = det t ∂ϕ ∗ ( y j ) ∂ y i ! − I !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) . Proof.
In the following the coe ffi cients of all the twisted homology groups and all the twistedcohomology groups are understood to be pgl n ( C ).It follows from Lemma 4.1 and Corollary 2.4 that the homomorphism ϕ ∗ : H ◦ ρ ( π S ) → H ◦ ρ ( π S ) is presented by the matrix (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) . Since H ◦ ρ ( π S ) is isomorphic to H ◦ ρ ( S )and since H ◦ ρ ( S ) is isomorphic by Poincar´e duality to the dual of H Ad ◦ ρ ( S , ∂ S ), the homo-morphism ϕ ∗ : H Ad ◦ ρ ( S , ∂ S ) → H Ad ◦ ρ ( S , ∂ S ) is presented by the transpose of (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) . Thus byLemma 3.10 we only need to show that the homomorphisms ϕ ∗ : H Ad ◦ ρ ( S ) → H Ad ◦ ρ ( S ) and ϕ ∗ : H Ad ◦ ρ ( S , ∂ S ) → H Ad ◦ ρ ( S , ∂ S ) are equivalent to each other.It follows from the duality induced by the intersection pairing (3.1) that H Ad ◦ ρ ( S , ∂ S ) = H Ad ◦ ρ ( S ) =
0. Hence the homology long exact sequence for the pair ( M , ∂ M ) gives the follow-ing commutative diagram of exact sequences:0 −−−−−→ H Ad ◦ ρ ( ∂ S ) −−−−−→ H Ad ◦ ρ ( S ) −−−−−→ H Ad ◦ ρ ( S , ∂ S ) −−−−−→ H Ad ◦ ρ ( ∂ S ) −−−−−→ (cid:13)(cid:13)(cid:13)(cid:13) ϕ ∗ y ϕ ∗ y (cid:13)(cid:13)(cid:13)(cid:13) −−−−−→ H Ad ◦ ρ ( ∂ S ) −−−−−→ H Ad ◦ ρ ( S ) −−−−−→ H Ad ◦ ρ ( S , ∂ S ) −−−−−→ H Ad ◦ ρ ( ∂ S ) −−−−−→ , where it follows again from the duality induced by the intersection pairing (3.1) that H Ad ◦ ρ ( ∂ S )is isomorphic to the dual of H Ad ◦ ρ ( ∂ S ). Now it is a simple matter to check that ϕ ∗ : H Ad ◦ ρ ( S ) → H Ad ◦ ρ ( S ) and ϕ ∗ : H Ad ◦ ρ ( S , ∂ S ) → H Ad ◦ ρ ( S , ∂ S ) are equivalent, which completes the proof. (cid:3) The proof of the following theorem is now straightforward from Corollary 3.11 and Theorem4.2.
Theorem 4.3.
Let γ , . . . , γ m be the boundary components of S , and let ϕ ∈ Γ S . For a ( γ , . . . , γ m ) -regular representation ρ : π M ϕ → PGL n ( C ) satisfying H Ad ◦ ρ ( S ; pgl n ( C )) = and ( y , . . . , y l ) ∈ X ϕ ∗ T , n , if χ ρ | π S = χ ( y ,..., y l ) , thenT ( γ ,...,γ m ) ,ρ ( M ϕ ) = lim t → det (cid:16) t (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) − I (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) ( t − m ( n − . Remark . It follows from Theorems 2.5 and 3.4 that the assumptions of the above theoremsare satisfied for a pseudo-Anosov ϕ ∈ Γ S and a holonomy representation of M ϕ .The advantage of our main theorems is that cluster variables naturally describe torsion invari-ants as functions on the moduli spaces of representations in a combinatorial way. In fact, therational function induced by the coe ffi cients of the polynomialdet t ∂ϕ ∗ ( y j ) ∂ y i ! − I ! or that by lim t → det (cid:16) t (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) − I (cid:17) ( t − m ( n − in the theorems can be algorithmically computed from the ideal triangulation T and a sequenceof flips representing ϕ , and now regarded as torsion functions on the moduli spaces. Remark . In [NTY] the cluster variables in X T , are interpreted as the shape parameters ofideal tetrahedra of M ϕ , and the volumes are also explicitly computed from the cluster variables.This is one advantage with the cluster variables to parametrize representations. For example,this is very useful for identifying the complete holonomy representation.The following question concerning the condition on the cluster variables that ensures( γ , . . . , γ m )-regularity naturally arises from Theorem 4.3: Question 4.6.
Let γ , . . . , γ m be the boundary components of S , and let ϕ ∈ Γ S . For a represen-tation ρ : π M ϕ → PGL n ( C ) and ( y , . . . , y l ) ∈ X ϕ ∗ T , n , if χ ρ | π S = χ ( y ,..., y l ) , and if lim t → det (cid:16) t (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) − I (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) ( t − m ( n − . is nonzero, then is ρ a ( γ , . . . , γ m ) -regular representation satisfying H Ad ◦ ρ ( S ; pgl n ( C )) = ? Examples.
Finally, we demonstrate our theory for ϕ = LR (the figure eight knot comple-ment) and for ϕ = LLR in the case of n = S be a one-holed torus, and we identify S with R / Z so that the marked point corre-sponds to the integral points of R . The mapping class group Γ S = S L ( Z ) is generated by thematrices L = ! , R = ! . We set ϕ = LR , and then the mapping torus M ϕ is known to be homeomorphic to the figure eightknot complement. We consider an ideal triangulation T defined by the lines x = x = y , y = x , y ) of R . Then the quiver Q T , and the coordinates( y , y , y , y , y , y , y , y ) ∈ X T , is given as in Figure 2. ORSION FUNCTIONS ON MODULI SPACES 15 y2y1 y5y4 y8y3y6 y7 F igure
2. One-holed torus S with the quiver Q T , A computation implies that L ∗ , R ∗ : X T , → X T , are described as follows: L ∗ ( y ) = (1 + y )(1 + y + y y + y y y ) y + y L ∗ ( y ) = (1 + y ) y (1 + y + y y + y y y )1 + y L ∗ ( y ) = y (1 + y ) y y (1 + y )(1 + y + y y + y y y ) L ∗ ( y ) = (1 + y + y y + y y y ) y + y + y y + y y y L ∗ ( y ) = + y y (1 + y ) y L ∗ ( y ) = (1 + y ) y y y (1 + y )(1 + y + y y + y y y ) L ∗ ( y ) = + y (1 + y ) y y L ∗ ( y ) = y (1 + y + y y + y y y )1 + y + y y + y y y R ∗ ( y ) = y (1 + y )(1 + y + y y + y y y )1 + y R ∗ ( y ) = y (1 + y )(1 + y + y y + y y y )1 + y R ∗ ( y ) = y y y (1 + y )(1 + y )(1 + y + y y + y y y ) R ∗ ( y ) = y (1 + y + y y + y y y )1 + y + y y + y y y R ∗ ( y ) = + y y y (1 + y ) R ∗ ( y ) = (1 + y ) y y y (1 + y )(1 + y + y y + y y y ) R ∗ ( y ) = + y (1 + y ) y y R ∗ ( y ) = (1 + y + y y + y y y ) y + y + y y + y y y Combining them, we compute ϕ ∗ = R ∗ ◦ L ∗ : X T , → X T , as follows: ϕ ∗ ( y ) = ( y (1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )(1 + y + y + y y + y y y + y y + y y y + y y + y y y + y y y + y y y + y y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) / ((1 + y + y + y y + y y y + y y + y y y + y y y + y y y y )(1 + y + y y + y y y )) ϕ ∗ ( y ) = ( y (1 + y + y + y y + y y y + y y + y y y + y y y + y y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y + y y + y y y + y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) / ((1 + y + y y + y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )) ϕ ∗ ( y ) = ( y y (1 + y + y + y y + y y y + y y + y y y + y y y + y y y y ) y ) / ((1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y + y y + y y y + y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) ORSION FUNCTIONS ON MODULI SPACES 17 ϕ ∗ ( y ) = ( y (1 + y + y y + y y y )(1 + y + y + y y + y y y + y y + y y y + y y + y y y + y y y + y y y + y y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) / ((1 + y + y y + y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y + y y + y y y + y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) ϕ ∗ ( y ) = ((1 + y + y y + y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )) / ( y y (1 + y + y + y y + y y y + y y + y y y + y y y + y y y y ) y ) ϕ ∗ ( y ) = ( y y y (1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )) / ((1 + y + y + y y + y y y + y y + y y y + y y y + y y y y )(1 + y + y + y y + y y y + y y + y y y + y y + y y y + y y y + y y y + y y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) ϕ ∗ ( y ) = ((1 + y + y y + y y y )(1 + y + y + y y + y y y + y y + y y y + y y y + y y y y )) / ( y y y (1 + y + y + y y + y y + y y y + y y y + y y y + y y y y )) ϕ ∗ ( y ) = ( y (1 + y + y y + y y y )(1 + y + y + y y + y y + y y y + y y y + y y y + y y y y + y y + y y y + y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y )) / ((1 + y + y y + y y y )(1 + y + y + y y + y y y + y y + y y y + y y + y y y + y y y + y y y + y y y y + y y y + y y y y + y y y y + y y y y y + y y y y y + y y y y y y ))The space of solutions of the equations ϕ ∗ ( y i ) = y i for all i parametrizes X ϕ ∗ S , and X ϕ ∗ S , . Herewe emphasize that the parametrization makes sense by using the labeling change σ and byProposition 2.2 proved in the paper.A solution is given by y = y = − − √− y = y = y = y = y = y = − + √− . This solution can be found, for example, by using the arguments in the proofs of Theorem2.5 and Corollary 2.6. First we find an element (cid:16) , − + √− , − − √− (cid:17) ∈ X ϕ ∗ T , corresponding to thecharacter of a holonomy representation of the hyperbolic manifold M ϕ as in [NTY, Section 5.1]. Then the above element of X ϕ ∗ T , is the image of the map X T , → X T , in the proof of Corollary2.6. In fact, it corresponds to the character of the composition of a holonomy representation andthe homomorphism PGL ( C ) → PGL ( C ) induced by an irreducible representation PGL ( C ) → S L ( C ).By Theorem 4.2 we obtain the twisted Alexander polynomial associated to the solution as:det t ∂ϕ ∗ ( y j ) ∂ y i ! − I !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) = ( t − ( t − t + t − t + t − t + . By Theorem 4.3 we also obtain the non-acyclic torsion associated to the solution as:lim t → det (cid:16) t (cid:16) ∂ϕ ∗ ( y j ) ∂ y i (cid:17) − I (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) ( t − = lim t → ( t − t + t − t + t − t + = − . Next, we set ϕ ′ = LLR . Similarly, we can first compute ϕ ′∗ = R ∗ ◦ L ∗ ◦ L ∗ : X T , → X T , andthe equations ϕ ′∗ ( y i ) = y i for all i defining X ϕ ′∗ T , . Then the following solution of the equationscorresponding to the character of a holonomy representation of M ϕ ′∗ is found as follows: y = y = − − √− y = y = + √− y = y = y = y = − + √− . Again by Theorems 4.2 and 4.3, we obtain the twisted Alexander polynomial and the non-acyclic torsion associated to the solution as:det t ∂ϕ ′∗ ( y j ) ∂ y i ! − I !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) = ( t − ( t + i √ t − t − i √ t + t + i √ t − t − i √ t + t + i √ t − t + , lim t → det (cid:16) t (cid:16) ∂ϕ ′∗ ( y j ) ∂ y i (cid:17) − I (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ( y ,..., y l ) = ( y ,..., y l ) ( t − = lim t → ( t + i √ t − t − i √ t + t + i √ t − t − i √ t + t + i √ t − t + = − + √− . 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