Twisted Lefschetz Numbers of Infra-Solvmanifolds and Algebraic Groups

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G, i ϕ ∗ M ( a ovaiod(resp. solvmanifold a hat n[4,Nmz showed Nomizu [14], In . pefruafrtwisted for formula mple zfie on oml for formula point fixed tz M,E E .W en h twisted the define We ). r-ovaiod which fra-solvmanifolds : ) o mohmap smooth a For Γ). thmgnosspaces homogeneous ct mrhs fany of omorphism hnw have we then , ehre hncompu- than harder re eddt i group Lie a to tended rlzto flin- of eralization M stepl-akof pull-back the is ϕ x ) . ) sadiffeomorphism a is → . M A M M yusing by a discrete a has econsider We . fschetz easmooth a be p twisted up, f Γ : E → te . 2 HISASHI KASUYA that the de Rham cohomology of a nilmanifold G/Γ for a discrete presentation (G, Γ) is isomorphic to the Lie algebra cohomology of the Lie algebra g of G. Hence, considering the induced homomorphism T∗ : g → g, taking a matrix presentation A of T∗ : g → g, we can get the ”linearization formula” L(ϕ) = det(I − A) where L(ϕ) is the ordinary Lefschetz number. We are interested in getting a ”linearization formula” for solvmanifolds. But there are many difficulties: • In general, a solvmanifold M does not have a discrete presentation (G, Γ). • In general, for a simply connected solvable Lie group with a cocompact discrete subgroup Γ, a homomorphism Γ → Γ may not extend to a Lie group homomorphism G → G. • In general, the de Rham cohomology of a solvmanifold G/Γ for a discrete presentation (G, Γ) is not isomorphic to the Lie algebra cohomology of the Lie algebra g of G. Assuming some conditions, we can get a ”linearization formula” (see for examples [12] and [10]). However on general solvmanifolds, it seems to be difficult to get a ”linearization formula”. In this paper we consider the more general setting. An infra-solvmanifold is a manifold of the form G/∆, where G is a simply connected solvable Lie group, and ∆ is a torsion-free subgroup of Aut(G) ⋉ G such that the closure of h(∆) in Aut(G) is compact where h : Aut(G) ⋉ G → Aut(G) is the projection. An infra-solvmanifold is a generalization of a solvmanifold. See [2] and [4] for properties of infra-solvmanifolds. The purpose of this paper is to give a ”linearization formula” for any diffeomorphism of any infra-solvmanifold by using the twisted Lefschetz number and algebraic groups. Let M be an infra-solvmanifold with the fundamental group Γ and ϕ : M → M a diffeomorphism. Denote by f : Γ → Γ the homomorphism induced by ϕ. Then, by methods of algebraic groups given in [16, 2], we can say that f : Γ → Γ induces an explicit isomorphism on a certain Lie algebra uΓ. For a matrix presentation A of such isomorphism, we will prove that by taking some E and Ξ, the formula L(ϕ, E, Ξ) = det(I − A) holds.

2. Algebraic groups and representations

An algebraic group G is a Zariski-closed subgroup of GLn(C). We denote by U(G) the unipotent radical of G. If U(G) is trivial, an algebraic group G is called reductive. Denote by Rep(G) the category of ﬁnite-dimensional rational representations Vρ of G. Vρ ∈ Rep(G) says Vρ is a C-vector space with a rational representation ρ : G → GL(Vρ). For Vρ ∈ Rep(G), we denote by Vρ the C-vector space Vρ with the trivial representation.

3. Representations of reductive algebraic groups

Let G be a reductive algebraic group and Vρ ∈ Rep(G). We assume that we have an algebraic group isomorphism F : G → G and a linear isomorphism Φ : Vρ → Vρ satisfying ρ(g) ◦ Φ=Φ ◦ ρ(F (g)) for every g ∈ G. TWISTED LEFSCHETZ NUMBERS OF INFRA-SOLVMANIFOLDS 3

We take a finite set Λρ = {Vα} of irreducible representations of G such that we have an irreducible decomposition Vρ = Λρ Eα and Eα is a non-trivial irreducible component of Vρ corresponding to an irreducibleL Vα ∈ Rep(G). Denote Eα◦F = Φ(Eα). Then we have the irreducible decomposition Vρ = Λρ Eα◦F . Consider Vα◦F ∈ Rep(G) as the vector space Vα with the representation αL◦ F . Then, Eα◦F is a non-trivial irreducible component of Vρ corresponding to an irreducible Vα◦F ∈ Rep(G). Comparing two irreducible decomposition Vρ = Λρ Eα = Λρ Eα◦F , for each α ∈ Λρ, we have a ∼ unique β ∈ Λρ such that we haveL an isomorphismL ψα : Vα◦F = Vβ. ∗ ∗ Let us consider the linear map Φα : Vρ ⊗ Vα ⊗ Vα → Vρ ⊗ Vα◦F ⊗ Vα◦F defined by ∗ Φα(fα ⊗ vα)=Φ ◦ fα ⊗ vα for fα ∈ Vρ ⊗ Vα , vα ∈ Vα (we should notice Vα = Vα◦F ). ∗ G Then this linear map is G-equivariant and hence we can restrict Φα : (Vρ ⊗Vα ) ⊗Vα → ∗ G −1 ∗ ∗ ∗ (Vρ ⊗ Vα◦F ) ⊗ Vα◦F . Consider the isomorphism (ψα ) ⊗ ψα : Vα◦F ⊗ Vα◦F → Vβ ⊗ Vβ. −1 ∗ ∗ G ∗ G Then we have the linear map ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vα ) ⊗ Vα → (Vρ ⊗ Vβ ) ⊗ Vβ. ∗ G We consider Vφ ∈ Rep(G) as Vφ = Λρ (Vρ ⊗ Vα ) ⊗ Vα. Then we have the linear −1 ∗ LG G map Λρ ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vφ) → (Vρ ⊗ Vφ) . L G ∼ Lemma 3.1. There exists an linear isomorphism (Vρ ⊗ Vφ) = Vρ which identifies the linear map −1 ∗ G G with . Λρ ((ψα ) ⊗ ψα) ◦ Φα : (Vρ ⊗ Vφ) → (Vρ ⊗ Vφ) Φ : Vρ → Vρ L Proof. By [17, Theorem 27.3.6], we have the isomorphism ∗ G (Vρ ⊗ Vα ) ⊗ Vα ∋ fα ⊗ vα 7→ f(vα) ∈ Eα. G ∼ Thus we have the isomorphism (Vρ ⊗ Vφ) = Λρ Eα = Vρ. Since we have −1 ∗ L −1 ((ψα ) ⊗ ψα) ◦ Φα(fα ⊗ vα)=Φ ◦ fα ◦ ψα ⊗ ψα(vα), the lemma follows from this isomorphism.

4. Rational cohomology of algebraic groups Let G be an algebraic group. In this section, we denote U = U(G) Let F : G → G be an isomorphism of algebraic groups. Consider the extension (1) 1 / U / G / G/U / 1 .

It is known that this extension splits ([13]). The restriction FU : U →U is also an isomorphism and F induces an isomorphism F¯ : G/U → G/U. ∗ ∗ C For Vφ ∈ Rep(G), we deﬁne the rational cohomology H (G,Vφ) = ExtG( ,Vφ) as ∗ ∗ ∗ in [7] and [9]. Consider the induced map H (F ) : H (G,Vφ) → H (G,Vφ◦F ). Let ∗ Ψ : Vφ◦F → Vφ be an isomorphism in Rep(G). We consider the linear map Ψ ◦ H (F ) : ∗ ∗ H (G,Vφ) → H (G,Vφ). It is known that we have a natural isomorphism ∗ ∼ ∗ G/U ∼ ∗ G/U H (G,Vφ) = H (U,Vφ) = H (u,Vφ) induced by the extension (1) and the exponential map u → U (see [7],[5]) where u ∗ is the Lie algebra of U and H (u,Vφ) is the Lie algebra cohomology and we regard ∗ ∗ H (U,Vφ),H (u,Vφ) ∈ Rep(G/U) via the adjoint action. This isomorphism identiﬁes ∗ ∗ ∗ ∗ the linear map Ψ ◦ H (F ) : H (G,Vφ) → H (G,Vφ) with the linear map Ψ ◦ H (FU∗) : ∗ G/U ∗ G/U H (u,Vφ) → H (u,Vφ) where FU∗ : u → u is the derivation of FU : U→U. Denote by H∗(u) the Lie algebra cohomology with values in the 1-dimensional trivial representation. We apply the arguments in the last section to the objects: • The reductive algebraic group G/U. 4 HISASHI KASUYA

• The representation H∗(u) ∈ Rep(G/U). • The isomorphism F¯ : G/U → G/U. ∗ ∗ ∗ • The linear isomorphism H (FU∗) : H (u) → H (u). −1 ∗ Then, taking Vφ ∈ Rep(G/U) and Ψ = Λρ ((ψα ) ⊗ ψα) as in the last section and extending Vφ ∈ Rep(G) by the extensionL (1), the isomorphism ∗ G/U ∗ G/U ∼ ∗ H (u,Vφ) = (H (u) ⊗ Vφ) = H (u) ∗ ∗ G/U ∗ G/U in Lemma 3.1 identiﬁes the linear map Ψ ◦ H (FU∗) : H (u,Vφ) → H (u,Vφ) ∗ ∗ ∗ with the linear map H (FU∗) : H (u) → H (u). This implies the following:

Proposition 4.1. Let A be a matrix presentation of the linear map FU∗ : u → u. Then we have i ∗ i (−1) trace(Ψ ◦ H (F ))|H (G,Vφ) = det(I − A). Xi 5. Group cohomology of torsion-free virtually polycyclic groups Lemma 5.1. ([16, Lemma 4.36.]) Let Γ be a torsion-free virtually polycyclic group. For a finite-dimensional representation ρ :Γ → GL(Vρ) on a complex vector space Vρ, denote by G the Zariski-closure of ρ(Γ) in GL(Vρ). Then we have dim U(G) ≤ rankΓ. By this lemma, we consider the following definition. Definition 5.2. Let Γ be a torsion-free virtually polycyclic group. For a real algebraic group G, a representation ρ : Γ → G is called full if the image ρ(Γ) is Zariski-dense in G and we have dim U(G) = rankΓ. We can compute the group cohomology of torsion-free virtually polycyclic groups by using rational cohomology of algebraic groups. Let Rep(Γ) be the category of ∗ representations Vρ of Γ. For Vρ ∈ Rep(Γ), we define the group cohomology H (Γ,Vρ)= ∗ C ExtΓ( ,Vρ) as in [3]. Theorem 5.3 ([8]). Let Γ be a torsion-free virtually polycyclic group. We suppose that we have an algebraic group G and an injection Γ ⊂ G which is a full representation. Let V be a rational G-module. Then the inclusion Γ ⊂ G induces a cohomology isomorphism ∗ ∼ ∗ H (G,Vρ) = H (Γ,Vρ). This isomorphism was discovered by Baues for the case of trivial coefficients see the sentence after [2, Theorem 1.8]. For any torsion-free virtually polycyclic group, we have a good full representation.

Definition 5.4. We call an algebraic group AΓ an algebraic hull of Γ if there exists an injective group homomorphism ψ :Γ →AΓ so that • ψ :Γ →AΓ is a full representation. • ZAΓ (UΓ) ⊂UΓ where ZAΓ (UΓ) is the centralizer of the unipotent radical UΓ of AΓ. Theorem 5.5. ([2, Theorem A.1]) There exists an algebraic hull of Γ and an algebraic hull of Γ is unique up to algebraic group isomorphism. Let Γ be a torsion-free virtually polycyclic group and f : Γ → Γ an isomorphism. Take an algebraic hull ψ : Γ → AΓ. Then, f extends uniquely to an isomorphism F : AΓ →AΓ by [16, Lemma 4.41]. In fact, we obtain F by the following way. Define Φf : Γ → AΓ ×AΓ as Φf (γ) = (ψ(γ), ψ ◦ f(γ)). Let Gf be the real Zariski- closure of Φf (Γ) in AΓ ×AΓ. For the projections p1 : AΓ ×AΓ ∋ (a, b) 7→ (a) ∈ AΓ TWISTED LEFSCHETZ NUMBERS OF INFRA-SOLVMANIFOLDS 5 and p2 : AΓ ×AΓ ∋ (a, b) 7→ (b) ∈ AΓ, we consider the restriction πi : Gf → AΓ of pi −1 on Gf for i = 1, 2. Then π1 is an isomorphism and desired F is π2 ◦ π1 . For the isomorphism F : AΓ →AΓ, by Theorem 5.3, for Vφ ∈ Rep(AΓ), the induced map on the cohomology ∗ ∗ ∗ H (f) : H (Γ,Vφ) → H (Γ,Vφ◦f ) is identified with the map ∗ ∗ ∗ H (F ) : H (AΓ,Vφ) → H (AΓ,Vφ◦F ).

We apply the arguments in the last section to the algebraic group AΓ. Take Vφ ∈ Rep(AΓ/UΓ) and Ψ for Proposition 4.1. The isomorphisms ∗ ∼ ∗ ∼ ∗ G/UΓ ∗ G/UΓ ∼ ∗ H (Γ,Vφ) = H (AΓ,Vφ) = H (uΓ,Vφ) = (H (uΓ) ⊗ Vφ) = H (uΓ) ∗ ∗ ∗ ∗ ∗ identify the linear map Ψ ◦ H (f) : H (Γ,Vφ) → H (Γ,Vφ) with H (FUΓ∗) : H (uΓ) → ∗ H (uΓ) where uΓ is the Lie algebra of UΓ.

Proposition 5.6. Let A be a matrix presentation of the linear map FU∗ : uΓ → uΓ. Then we have i ∗ i (−1) trace(Ψ ◦ H (f))|H (Γ,Vφ) = det(I − A). Xi 6. Main result We give a ”linearization formula” for any diffeomorphism of any infra-solvmanifold by using the twisted Lefschetz number and algebraic groups. It is known that the fundamental group of every infra-solvmanifold is a torsion-free virtually polycyclic group (see [4]). Let M be an infra-solvmanifold with the fundamental group Γ and ϕ : M → M a diffeomorphism. Denote by f : Γ → Γ the homomorphism induced by ϕ. For the universal covering M˜ , we write M = M/˜ Γ. Then, ϕ is described as a diffeomorphimϕ ˜ : M˜ → M˜ so that for x ∈ M˜ and g ∈ Γ,ϕ ˜(gx)= f(g)ϕ ˜(x). For Vφ ∈ Rep(Γ), we define ∗ the flat bundle Eφ = (M˜ × Vφ)/Γ. Then ϕ Eφ = Eφ◦f . By the standard theory (see ∗ ∗ ∗ ∗ [16, Lemma 7.4] for instance), the induced map H (ϕ) : H (M, Eφ) → H (M, ϕ Eφ) ∗ ∗ ∗ is identified with H (f) : H (Γ,Vφ) → H (Γ,Vφ◦f ). Take Vφ ∈ Rep(AΓ/UΓ) and an isomorphism Ψ : Vφ◦F → Vφ as in the last section. Considering Ψ : Vφ◦f → Vφ as ∗ an isomorphism in Rep(Γ), Ψ corresponds to an isomorphism Ξ : ϕ Eφ → Eφ of flat bundles. Thus, by Proposition 5.6, we obtain the following result.

Theorem 6.1. Let A be a matrix presentation of FUΓ∗ : u → u. Then we have

L(ϕ, Eφ, Ξ) = det(I − A). Remark 1. We have a good presentation of the universal covering M˜ of any infra- solvmanifold M. For a torsion-free virtually polycyclic group Γ with an algebraic hull ψ :Γ →AΓ, since the extension (1) splits, we have a splitting AΓ = AΓ/UΓ ⋉ UΓ. We ′ can take AΓ defined over R and ψ (Γ) contained in the R-rational points AΓ(R) of AΓ. By an injection ψ :Γ →AΓ(R), we have the Γ-action on UΓ(R) by the splitting. In [2], Baues proved that the quotient space UΓ(R)/Γ is a compact aspherical manifold with the universal covering UΓ(R) and every infra-solvmanifold M with the fundamental group Γ is diffeomorphic to UΓ(R)/Γ. Thus, we can take M˜ = UΓ(R). We notice that as a Lie group, UΓ(R) is simply connected nilpotent Lie group and by the exponential map, UΓ(R) is diffeomorphic to the Euclidean space of dimension dim UΓ. 6 HISASHI KASUYA

Remark 2. If M is a nilmanifold, then Γ is a nilpotent and we have AΓ = UΓ (see [16]) and the above quotient space UΓ(R)/Γ is a quotient of a simply connected nilpotent Lie group UΓ(R) by a discrete subgroup Γ. In this case, the flat bundle Eφ is trivial and so Theorem 6.1 is the usual linearization formula for a nilmanifold as we explained in Introduction. cos πt − sin πt Example 1. Let G = R⋉ R2 such that λ(t)= for t ∈ R. Then the λ sin πt cos πt 2 subgroup Γ = Z⋉λZ is a cocompact discrete subgroup in G. Consider the solvmanifold 3 M = G/Γ. We have the explicit algebraic hull ψ :Γ →AΓ where AΓ = hτi ⋉C such 1 0 0 that τ =  0 −1 0  and ψ(t, u, v) = (τ t, t, u, v) for (t, u, v) ∈ Γ. We consider 0 0 −1   −1 0 0 the automorphism f : Γ → Γ defined by the matrix A =  0 a b  such that 0 c d   a b ∈ SL(2, Z). This f can not be extended to an automorphism of the Lie c d group G. We can extend f to the automorphism F : AΓ → AΓ such that F (τ) = τ 3 and F on UΓ = C is defined by the matrix A. We can compute C (0 ≤ i ≤ 3) Hi(M, C)= Hi(Γ, C)= Hi(C3)hτi = 0 (other i) ∗ i and we have H (f)|Hi(Γ,C) = (−1) . Consider a diffeomorphism ϕ : M → M inducing f on the fundamental group Γ (e.g. ϕ is the map on M = R3/Γ defined by the matrix A where we consider G = R3 as a smooth manifold). Then the ordinary Lefschetz number is L(ϕ) = 4. In contrast, for the above Eφ and Ξ, by Theorem 6.1, we have

L(ϕ, Eφ, Ξ) = det(I − A) = 4 − 2(a + d).

7. Application We consider a solvmanifold G/Γ for a discrete presentation (G, Γ) and a diffeomorphism ϕ : G/Γ → G/Γ. Let g be the Lie algebra of G. We suppose that the induced map f : Γ → Γ can be extended to a Lie group homomorphism T : G → G. Consider ∗ ∗ the derivation T∗ : g → g. If an isomorphism H (G/Γ, R) =∼ H (g, R) holds, then we have the equation L(ϕ) = det(I − B) where B is a matrix presentation of T∗ : g → g. But in general an isomorphism H∗(G/Γ, R) =∼ H∗(g, R) does not hold. Thus the equation L(ϕ) = det(I − B) does not hold. Even if an isomorphism H∗(G/Γ, R) =∼ H∗(g) does not hold, it is expected that the number det(I − B) gives some information for fixed points. As an application of Theorem 6.1, we will answer such expectation. A solvmanifold G/Γ is an infra-solvmanifold with the fundamental group Γ. For the same notation as in Theorem 6.1, we have: Theorem 7.1. L(ϕ, Eφ, Ξ) = det(I − B). Proof. Let G be a simply connected solvable Lie group. For a finite dimensional representation ρ : G → GL(Vρ), taking the Zariski-closure G of ρ(G), f we have dim U(G) ≤ dim G ([16, Lemma 4.36.]). For an algebraic group G and a representation TWISTED LEFSCHETZ NUMBERS OF INFRA-SOLVMANIFOLDS 7

ρ : G → G, we say that ρ is a full representation if the image ρ(G) is Zariski-dense in G and we have dim U = dim G. We call an algebraic group AG an algebraic hull of G if there exists an injective Lie ′ group homomorphism ψ : G →AG so that: ′ • ψ : G →AG is a full representation.

• ZAG (UG) ⊂UG where UG is the unipotent radical of AG. As similar to virtually polycyclic case, there exists an algebraic hull AG of G and an algebraic hull of G is unique up to algebraic group isomorphism (see [16, Proposition ′ 4.40, Lemma 4.41]). Moreover, we can take AG defined over R and ψ (G) contained in the R-rational points AG(R) of AG. ′ Take a real algebraic hull ψ : G → AG of G such that AG is defined over R and ′ ψ (G) ⊂ AG(R). Then, by the similar arguments on virtually polycyclic groups, for an isomorphism T : G → G of Lie groups, T extends uniquely to an isomorphism ′ F : AG →AG. It is known that the composition Q = q ◦ ψ : G →UG(R) as a smooth map is a diffeomorphism (see [2, Proposition 2.3]). Hence, for the derivation FUG∗ : R R −1 UG( ) → UG( ), we have F∗ = Q∗T∗Q∗ . This implies that for matrix presentations A and B of F∗ and T∗ respectively, we have det(I − A) = det(I − B). We suppose that G has a cocompact discrete subgroup Γ and T : G → G is an ′ extension of f : Γ → Γ. It is known that the Zariski-closure of ψ (Γ) in AG is the algebraic hull AΓ of a torsion-free polycyclic group Γ and UΓ = UG (see [16, Proof of Theorem 4.34]). Thus we can say that the restriction F : AΓ → AΓ is a unique extension of f :Γ → Γ. Now we assume that f : Γ → Γ is induced by a diffeomorphism ϕ : G/Γ → G/Γ. Then, by Theorem 6.1, we have

L(ϕ, Eφ, Ξ) = det(I − A). Hence the theorem follows.

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(Hisashi Kasuya) Department of Mathematics, Graduate School of Science, Osaka Uni- versity, Osaka, Japan. E-mail address: [email protected]