Rigidity Results for Solvable Arithmetic Groups

and Applications

Fritz Grunewald Mathematisches Institut Heinrich Heine Universität D-40225 Düsseldorf e-mail: fritz@math.uni-duesseldorf.de

Vladimir Platonov Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L3G1 e-mail: vplatonov@math.uwaterloo.ca

Table of contents

1. Introduction

2. Rigidity Results

3. Embedding Results

4. Automorphism Groups

5. Non-arithmetic Polycyclic Groups

1. Introduction

The main aim of this paper is to describe solvable arithmetic groups for which natural rigidty properties hold and to apply these rigidity results. We study the automorphism groups of solvable arithmetic groups, prove arithmeticity results for their finite extensions and construct a new class of non-arithmetic polycyclic groups. In fact we work with the more general class of finite extensions of solvable arithmetic groups and obtain rigidity results in this case. This also is a new point of view since all previous rigidity results where related to connected groups. A ℚ-defined linear algebraic group G is a subgroup G ≤ GL(n, ℂ) (n ∈ ℕ) which is also an affine algebraic set defined by polynomials with rational coefficients in the natural coordinates of GL(n, ℂ). If R is a subring of ℂ we put G(R) := G∩GL(n, R). We have G = G(ℂ). Let G be a ℚ-defined linear algebraic group. A subgroup Γ ≤ G is called an arithmetic subgroup of G if Γ is commensurable with G(ℤ). An abstract group Δ is called arithmetic if it is isomorphic to an arithmetic subgroup of a ℚ-defined linear algebraic group. Two subgroups Γ_{1}, Γ_{2} of GL(n, ℂ) are called commensurable if their intersection Γ_{1} ∩Γ_{2} has finite index both in Γ_{1} and Γ_{2}. These definitions are taken from [P-R]. A more narrow definition is used in [B1], [S]. In these papers the group Γ is called arithmetic if it is isomorphic to a subgroup of G(ℚ) which is commensurable to G(ℤ). The relation between the two concepts is explained in [G-P1] and [G-P2]. We consider here the case of finite extensions of solvable linear algebraic groups. The arithmetic subgroups of them are known to be finite extension groups of polycyclic groups. The first question we study is whether an isomorphism between arithmetic subgroups extend to ℚ-defined rational isomorphisms between the corresponding linear algebraic groups. These questions are traditionally called rigidity problems. Well known results of Margulis [Ma], Mostow [Mo] and Prasad [Pr] establish a positive answer to the rigidity problems for arithmetic subgroups of ℚ-simple linear algebraic groups of adjoint type under a condition on their real rank. Algebraic groups with radical are analyzed in [G-P2]. The solvable case is studied in [M-P]. The following example shows that in general there will not be any relation between the isomorphisms of arithmetic groups and the rational homomorphisms of the algebraic groups containing them. Let K be an algebraic number field and O its ring of integers. We write G_{m} for the multiplicative group considered as an algebraic group defined over K. Let R_{K/ℚ}(G_{m}) be the restriction of scalars aplied to G_{m}. This is a linear algebraic group defined over ℚ. We have a natural identification R_{K/ℚ}(G_{m})(ℤ) = O^{∗} where O^{∗} stands for the group of units of O. Dirichlet’s unit theorem shows that R_{K/ℚ}(ℤ) contains a subgroup of finite index Γ with Γ ≅ ℤ^{r} for appropriate r ∈ ℕ ∪ {0}. If K is neither ℚ nor an imaginary quadratic number field then r ≥ 1. The group Γ is also isomorphic to an arithmetic subgroup in G_{a}^{r} where G_{a} denotes the additive group. Since R_{K/ℚ}(G_{m}) is a torus and G_{a}^{r} a unipotent group there is no non-trivial rational homomorphism between them. In this paper we shall show that rigidity results still hold under a natural condition on the solvable-by-finite linear algebraic groups under consideration. Before we can state our results some notation is required. We write Z(G) for the center of a group G, U(G) for the unipotent radical and G^{∘} for the connected component of the identity of a linear algebraic group G. We call a linear algebraic group G solvable-by-finite if G^{∘} is solvable. The linear algebraic group G acts by conjugation on U(G) and G is called reduced if the kernel of this action is contained in U(G) (see Definition 2.4 for more details). If G is reduced then we have Z(G) ≤ Z(G^{∘}) ≤ U(G). Our rigidity theorem for solvable-by-finite reduced groups is:

Theorem A: Let G_{1}, G_{2} be reduced solvable-by-finite linear algebraic groups which are defined over ℚ. Let Γ_{1} ≤ G_{1}(ℚ), Γ_{2} ≤ G_{2}(ℚ) be Zariski-dense arithmetic subgroups and φ : Γ_{1} → Γ_{2} be a group isomorphism. Then there is a rational ℚ-defined isomorphism Φ : G_{1} → G_{2} and a map u : Γ_{1} → Z(G_{2}^{∘})(ℚ) such that

(1.1)  Φ(γ) = u(γ) · φ(γ)

for all γ ∈ Γ_{1}. The map u satisfies u(γ_{1}γ_{2}) = u(γ_{1})φ(γ_{1})u(γ_{2})φ(γ_{1})^{−1} for all γ_{1}, γ_{2} ∈ Γ_{1} and u(γ) = 1 for all γ ∈ Γ_{1} ∩ U(G_{1}).

If G_{1}, G_{2} are connected and if the centers of G_{1}, G_{2} are trivial then G_{1}, G_{2} are reduced and Theorem A is contained in [M-P]. If G_{1}, G_{2} are unipotent that is G_{1} = U(G_{1}), G_{2} = U(G_{2}) then Theorem A amounts to a classical result of Malcev, see [Mc]. Theorem A is also a sharper version of a rigidity result for more general linear algebraic groups contained in [G-P2]. The reducedness condition in Theorem A cannot be dropped, this is shown by the examples above. Before we can explain our further results we have to introduce some notation. A dgroup is a linear algebraic subgroup which consists of semi-simple elements. Let G be a solvable-by-finite linear algebraic group defined over ℚ and S ≤ G be a maximal d-subgroup of G which is also defined over ℚ. We then have a semi-direct product decomposition G = U(G) · S ≅ U(G) ⋊ S. This is proved in Proposition 2.1. It follows that the quotient G/U(G) is a d-group. Let π_{U} : G → G/U(G) be the quotient homomorphism. If Γ ≤ G(ℚ) is a Zariski-dense arithmetic subgroup then M(Γ) := π_{U}(Γ) ∩ (G/U(G))^{∘} is a finitely generated abelian group and F (Γ) := π_{U}(Γ)/M(Γ) is a finite group. The finite group F (Γ) acts by conjugation on M(Γ). Assume now that G is reduced then the finite group F (Γ) also acts by conjugation on the vector space Z(G^{∘})(ℚ) ≤ U(G) and stabilizes the subspace Z(G)(ℚ). We call Γ separated in G if

(1.2)  Hom_{F (Γ)}(M(Γ), Z(G^{∘})/Z(G)(ℚ)) = {0}.

Here Hom_{F (Γ)} stands for the vector space of F (Γ)-equivariant homomorphisms. Clearly if G is connected then every arithmetic subgroup is separated. We consider the case of a connected group G as a very important special case in the results which will follow. For more details on the definition of separatedness see Definition 2.8. We prove the following version of Theorem A.

Theorem A’: Let G_{1}, G_{2} be reduced solvable-by-finite linear algebraic groups which are defined over ℚ. Let Γ_{1} ≤ G_{1}(ℚ), Γ_{2} ≤ G_{2}(ℚ) be Zariski-dense arithmetic subgroups and φ : Γ_{1} → Γ_{2} be a group isomorphism. Assume further that Γ_{2} is separated in G_{2}. Then there is a rational ℚ-defined isomorphism Φ : G_{1} → G_{2} and a homomorphism u : Γ_{1} → Z(G_{2})(ℚ) such that Φ(γ) = u(γ) · φ(γ) for all γ ∈ Γ_{1}. The homomorphism u satisfies u(γ) = 1 for all γ ∈ Γ_{1} ∩ U(G_{1}).

The concept of separatedness arises naturally as is shown in the following theorem.

Theorem B: Let G be a reduced solvable-by-finite linear algebraic groups which is defined over ℚ. Let Γ ≤ G(ℚ) be a Zariski-dense arithmetic subgroup. Suppose that for every automorphism φ of Γ there is a rational ℚ-defined automorphism Φ of G and a homomorphism u : Γ → Z(G)(ℚ) which satisfies u(γ) = 1 for all γ ∈ Γ ∩ U(G) such that Φ(γ) = u(γ) · φ(γ) for all γ ∈ Γ. Then Γ is separated in G.

We go on to prove the following uniform rigidity version of our rigidity result.

Theorem C: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is seperated in G. Let 𝔄𝔲𝔱_{ℚ}(G) be the group of ℚ-defined rational automorphisms of G and Aut(Γ) the automorphism group of the abstract group Γ. Then there is a homomorphism

(1.3)  D : Aut(Γ) → 𝔄𝔲𝔱_{ℚ}(G),  φ ↦ D_{φ}

and also a map

(1.4)  u : Aut(Γ) → Hom(Γ, Z(G)(ℚ)),  φ ↦ u_{φ}

with u_{φ}(γ) = 1 for all φ ∈ Aut(Γ) and γ ∈ Γ∩U(G) such that D_{φ}(γ) = u_{φ}(γ)·φ(γ) for every φ ∈ Aut(Γ) and γ ∈ Γ. The homomorphism D and and the map u are uniquely determined by these properties. The map u of (1.4) furthermore satisfies:

(1.5)  u_{φ∘ψ}(γ) = D_{φ}(u_{ψ}(γ)) · u_{φ}(ψ(γ))

for all φ, ψ ∈ Aut(Γ) and γ ∈ Γ.

Again Theorem C is a sharper version of a uniform rigidity result for more general linear algebraic groups contained in [G-P2].

As our results are mainly for the class of arithmetic subgroups of reduced solvable-by-finite ℚ-defined linear algebraic groups, it is important to note that this class does not differ too much from the class of all arithmetic subgroups of solvable ℚ-defined linear algebraic groups. In Section 3 we prove for example that each of the latter groups has a subgroup of finite index which embeds as a Zariski-dense arithmetic subgroup into G(ℚ) where G is a solvable reduced ℚ-defined linear algebraic group. Theorem 3.4 even contains an abstract characterisation of the arithmetic subgroups of reduced solvable ℚ-defined linear groups.

Rigidity results like the above have numerous applications. We shall describe three of them. Firstly Theorem B implies:

Theorem D: Let Γ be a solvable arithmetic group. Then every finite extension of Γ an arithmetic group.

This result is a corollary of a corresponding result from [G-P2] for more general arithmetic groups Γ. We include a proof in Section 3 since it is much simpler than that of the general case. As a second application we are able to give a precise description of the automorphism group of certain solvable-by-finite arithmetic groups. This extends results of Malcev on the automorphism group of an arithmetic subgroup of a unipotent ℚ-defined linear algebraic group. We prove:

Theorem E: Let Γ ≤ G(ℚ) be a Zariski-dense arithmetic subgroup of a reduced solvable-by-finite ℚ-defined linear algebraic group G which is separated in G. Then Aut(Γ) is an arithmetic group. In particular Aut(Γ) is an arithmetic group for any Zariski-dense arithmetic subgroup of G(ℚ) where G is a connected reduced and ℚdefined linear algebraic group.

If Γ is a torsion-free finitely generated nilpotent group then both Γ and Aut(Γ) are arithmetic groups in a very natural way (see [13], chapter 6). Our proof of Theorem E (which is described in Section 4) is similarly canonical. The characterisation of arithmetic subgroups of reduced solvable linear algebraic groups already mentioned yields the following abstract version of Theorem E.

Theorem E’: Let Γ be a solvable arithmetic group with torsion-free Fitting subgroup F(Γ). If Γ/F(Γ) is nilpotent then Aut(Γ) is an arithmetic group.

Let G be a reduced solvable linear algebraic group defined over ℚ and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup. Let Aut_{alg}(Γ) be the subgroup of Aut(Γ) consisting of those elements which are the restriction of an element from 𝔄𝔲𝔱_{ℚ}(G). We also can compare Aut(Γ) with Aut_{alg}(Γ). To do this we call an element φ ∈ Aut(Γ) F-central if φ(γ)γ^{−1} ∈ Z(Γ) for all γ ∈ Γ and if φ is the identity when restricted to the Fitting subgroup F(Γ). We define

(1.6)  Aut_{0}(Γ) := { φ ∈ Aut(Γ)  φ is F-central }.

The group Aut_{0}(Γ) is abelian and normal in Aut(Γ). It can be identified with the group of homomorphisms from Γ/F(Γ) to Z(Γ). We have:

Theorem F: Let G be a reduced solvable linear algebraic group defined over ℚ and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is separated in G. Then the subgroup Aut_{0}(Γ) · Aut_{alg}(Γ) is of finite index in Aut(Γ).

If G is a ℚ-defined unipotent linear algebraic group and Γ ≤ G(ℚ) is an arithmetic subgroup then Aut_{alg}(Γ) = Aut(Γ). Theorem F is an extension of this classical result. Theorem F follows directly from Theorem 4.7 where a further decompositon Aut_{alg}(Γ) is obtained.

As a further application of our techniques we construct new polycyclic groups which are not arithmetic. Notice that, as already mentioned, every arithmetic subgroup Γ of a solvable ℚ-defined linear algebraic group is a polycyclic group. But not every polycyclic group is arithmetic. The first counterexamples are contained in [P3]. Further examples can be found in [Se], chapter 11. The examples of non-arithmetic polycyclic groups Γ constructed in [P3] and [Se] do not have a subgroup of finite index which is a split extension of a nilpotent group by an abelian group. This in fact immediately implies that Γ is not arithmetic. We propose a new construction of non-arithmetic polycyclic groups in Section 5. Our examples have split extensions of abelian groups by abelian groups as subgroups of arbitrarily high index. This shows that they give new types of groups not commensurable with any previous example. Also it turns out to be much more difficult to prove that they are not arithmetic. A group Γ is called polycyclic if it has a (finite) decomposition series which has only cyclic factors. If Γ is a polycyclic group then the Hirsch number h(Γ) is defined as the number of factors isomorphic to ℤ in a decomposition series for Γ. We prove the following surprising theorem.

Theorem G: Let h be a natural number with h ≥ 4. Then there are non-arithmetic polycyclic groups of Hirsch number h which are split extensions of ℤ^{h−1} by an infinite cyclic group.

Since it is known that every torsion-free finitely generated nilpotent group is arithmetic and because of Theorem D our examples are in an appropriate sense minimal. Chapter 11 of [Se] contains an example of a polycyclic group of Hirsch number 5 which is not arithmetic. The author then poses the question whether there are such examples with Hirsch number < 5. We prove:

Theorem H: The minimal Hirsch number of a non-arithmetic polycyclic group is 4.

A short exposition of the main results of this paper is contained in [G-P3], [G-P4]. The results and concepts from the theory of linear algebraic groups that we use in this paper can be found in the book [P-R] those on polycyclic groups are contained in [Se].

Acknowledgements: We thank Armand Borel and Dan Segal for advice on the subject. The first version of this paper was prepared while the second named author was visiting the Heinrich-Heine-Universität at Düsseldorf. He was partially supported by grants from the Deutsche Forschungsgemeinschaft and from NSERC. The paper was completed when the second author was visiting the Max-Planck-Institute of Mathematics in Bonn.

2. Rigidity Results

In this section we investigate prove the rigidity results contained in Theorems A, A’, C. We also prove Theorem B. We start by establishing some facts about linear algebraic groups which will be needed in the sequel. We write U(G) for the unipotent radical of a linear algebraic group G. The unipotent group U(G) is ℚ-defined if G has this property. Note that U(G) = U(G^{∘}) where G^{∘} stands for the connected component of the identity in G. We say that G is a d-group if it consists of semi-simple elements only. An algebraic subgroup H ≤ G is called a d-subgroup if H is a d-group. A torus is a connected commutative d-group. Note that a solvable-by-finite d-group is a finite extension of a torus. The following result is contained in section 5 of [B-S]. For connected groups G it is a well known result. We give here a different and simpler proof for the general case which is of independent interest.

Proposition 2.1: Let G be a solvable-by-finite ℚ-defined linear algebraic group. Then there is a ℚ-defined d-subgroup S of G such that

(2.1)  G = U(G) · S ≅ U(G) ⋊ S.

The algebraic group S is a finite extension of a ℚ-defined torus which is maximal in G^{∘}. Also S is a maximal d-subgroup of G. Every maximal ℚ-defined d-subgroup of G is conjugate to S by an element of U(G)(ℚ) and every maximal d-subgroup of G is conjugate to S by an element of U(G).

Proof (Existence): We consider now the case of an arbitrary solvable-by-finite group G. We shall first reduce the question of the existence of a decomposition (2.1) to the case when G^{∘} = U(G). Let π : G → G/U(G) be the quotient homomorphism. We may choose a ℚ-defined torus T ≤ G^{∘} such that G^{∘} = U(G) · T. Since G/U(G) is a finite extension of π(T) it is a d-group. Let N_{G}(T) be the normalizer of T in G. This is a ℚ-defined algebraic subgoup of G, it is also solvable by finite. Suppose we have a decomposition N_{G}(T) = U(N_{G}(T)) · Ŝ where Ŝ is a ℚ-defined d-group. Since G/U(G) is a d-group we have π(U(N_{G}(T))) = {1} and U(N_{G}(T)) ≤ U(G). If g ∈ G then gT g^{−1} is a maximal torus in G^{∘} hence there is a u ∈ U(G) with gT g^{−1} = uT u^{−1}. This shows that G = U(G) · N_{G}(T) and also G = U(G) · Ŝ. Hence it suffices to establish (2.1) for N_{G}(T). Note that G^{∘} ∩ N_{G}(T) = U(N_{G}(T)) · T. This shows that N_{G}(T) = U(N_{G}(T)) · T has finite index in N_{G}(T) and also that the quotient group H := N_{G}(T)/T is unipotent by finite. Let p : N_{G}(T) → H be the quotient homomorphism and suppose we know (2.1) for H. This means there is a finite ℚdefined algebraic subgroup S̃ ≤ H such that H = U(H) · S̃. Consider Ŝ := p^{−1}(S̃). Since Ŝ is a finite extension of T it is a d-group. Since the kernel of p consists of semi-simple elements we get the decomposition N_{G}(T) = U(N_{G}(T)) · Ŝ. Hence it is enough to prove (2.1) for an H which is a finite extension of a unipotent group.

Let now H be a ℚ-defined linear algebraic group with H^{∘} = U(H). Suppose we know (2.1) for all H so that U(H) is abelian. We then do induction on the nilpotency class of U(H). Let π : H → H/Z(U(H)) =: H_{0} be the quotient homomorphism where Z(U(H)) is the center of U(H). The ℚ-defined linear algebraic group H_{0} again has the property that H_{0}^{∘} = U(H_{0}) but the nilpotency class of U(H_{0}) is one less then that of U(H). By induction there is a ℚ-defined finite subgroup F_{0} ≤ H_{0} such that H_{0} = U(H_{0}) · F_{0}. Put H_{1} := π^{−1}(F_{0}). The ℚ-defined linear algebraic group H_{1} has the property that H_{1}^{∘} = U(H_{1}) and U(H_{1}) is abelian. Hence there is a ℚ-defined finite subgroup F_{1} ≤ H_{1} such that H_{1} = U(H_{1}) · F_{1}. It is straightforward to see that H = U(H) · F_{1}. Hence it is enough to prove (2.1) for an H which is a finite extension of an abelian unipotent group.

Let now H be a ℚ-defined linear algebraic group with H^{∘} = U(H) and where U(H) is abelian. The finite group H/H^{∘} acts through conjugation on U(H). Since this action is rational and also ℚ-defined the centralizer V (H) of H/H^{∘} in U(H) is a ℚ-defined algebraic subgroup of U(H). Suppose we have established (2.1) in the two cases that V (H) = {1} and V (H) = U(H). We then consider the ℚ-defined linear algebraic group H_{1} := H/V (H), notice that V (H) is always normal in H. Let π : H → H_{1} be the quotient homomorphism. We have V (H_{1}) = {1} hence there is a ℚ-defined finite subgroup F_{1} ≤ H_{1} with H_{1} = U(H_{1}) · F_{1}. Put H_{2} := π^{−1}(F_{1}). The linear algebraic group H_{2} satisfies H_{2}^{∘} = U(H_{2}) and V (H_{2}) = U(H_{2}). Hence there is a ℚ-defined finite subgroup F ≤ H_{2} with H_{2} = U(H_{2}) · F. It is easy to see that also H = U(H) · F holds.

Let now H be a ℚ-defined linear algebraic group with H^{∘} = U(H) and where U(H) is abelian. Let \overline{ℚ} be the algebraic closusure of ℚ in ℂ. Consider the quotient homomorphism H(\overline{ℚ}) → H/H^{∘}(\overline{ℚ}) = H/H^{∘}. Since H/H^{∘}(\overline{ℚ}) is finite and H^{∘}(\overline{ℚ}) is a vector space over \overline{ℚ} this homomorphism splits that there is a finite subgroup F ≤ H(\overline{ℚ}) which maps bijectively onto H/H^{∘} and which satisfies H(\overline{ℚ}) = U(H)(\overline{ℚ}) · F. We shall show that there is the choice of a ℚ-defined finite group F ≤ H(\overline{ℚ}) with these properties. Let us first show that two finite subgroups F_{1}, F_{2} ≤ H(\overline{ℚ}) with the properties just indicated are conjugate by an element of U(H)(\overline{ℚ}). Given f ∈ F_{1} there is a unique u(f) ∈ U(H)(\overline{ℚ}) such that u(f)f ∈ F_{2}. The map u : F_{1} → U(H)(\overline{ℚ}) satisfies u(f_{1}f_{2}) = u(f_{1})f_{1}u(f_{2})f_{1}^{−1} for all f_{1}, f_{2} ∈ F_{1}. Hence it is a derivation of the finite group F_{1} into the \overline{ℚ}-vector space U(H)(\overline{ℚ}) which is made into an F_{1}-module through conjugation. It is well known that u is inner, that is there is a v ∈ U(H)(\overline{ℚ}) such that u(f) = v^{−1}fvf^{−1} for all f ∈ F_{1}. We conclude that u(f)f = v^{−1}fv for all f ∈ F_{1}. This implies that v^{−1}F_{1}v = F_{2}. We fix now a finite group F ≤ H(\overline{ℚ}) with the properties indicated above. Let 𝔾 be the Galois group of \overline{ℚ} over ℚ. If σ is an element of 𝔾 that σ(F) is another finite group with the above properties. Hence there is a u_{σ} ∈ U(H)(\overline{ℚ}) with σ(F) = u_{σ}F u_{σ}^{−1}. If V (H) = U(H) it follows that σ(F) = F and F is ℚ-defined. This already establishes the second of the two cases left open in the previous paragraph. We shall proceed under the further hypothesis that V (H) = \{1\}. Then u_{σ} is uniquely determined by σ ∈ 𝔾. The map u : 𝔾 → U(H)(\overline{ℚ}) satisfies then u_{στ} = u_{σ}σ(u_{τ}) for all σ, τ ∈ 𝔾. Hence it represents an element of the Galois cohomology set H^{1}(ℚ, U(H)). It is well known that H^{1}(ℚ, U(H)) consists of the base point only (U(H) is a unipotent group). Hence there is a v ∈ U(H)(\overline{ℚ}) such that u_{σ} = v^{−1}σ(v) for all σ ∈ 𝔾. A simple computation then shows that S := v^{−1}F v satisfies σ(S) = S for all σ ∈ 𝔾. Hence the finite group S is ℚ-defined. This establishes the first case left open in the previous paragraph and altogether the existence of a decomposition (2.1) in general. It is evident that the S just constructed is a maximal ℚ-defined d-subgroup of G. It is also clear that S^{∘} = G^{∘} ∩ S is a maximal ℚ-defined torus in G^{∘}. It remains to prove the statement about the conjugacy of all ℚ-defined maximal d-subgroups of G. We postpone this until after the next lemma which is an important step in the proof.

Lemma 2.2: Let G be a solvable-by-finite linear algebraic group which is defined over ℚ. Assume that F ≤ G is a ℚ-defined finite subgroup of G. Then there is a ℚ-defined maximal torus T ≤ G^{∘} which is normalized by F.

Proof: Suppose we have proved the lemma under the assumption that the unipotent radical U(G) is abelian. We now prove the general result by induction on the nilpotency class of U(G). Let Z := Z(U(G)) be the center of the unipotent radical of G. This is a ℚ-defined normal algebraic subgroup of G. Let π : G → H := G/Z be the quotient homomorphism. Then π(F) is a ℚ-defined finite subgroup of H. By induction there is a ℚ-defined maximal torus T_{1} ≤ H^{∘} which is normalized by π(F). Then T_{1} · π(F) is a ℚ-defined d-subgroup of H. Define H_{0} := π^{−1}(T_{1} · π(F)). The algebraic group H_{0} is ℚ-defined and its unipotent radical U(H_{0}) is abelian. We in fact have U(H_{0}) = Z. We also have F ≤ H_{0}. By assumption we find a ℚ-defined maximal torus T_{0} ≤ H_{0} which is normalized by F. Since T_{0} ∩ G^{∘} has finite index in T_{0} and since T_{0} is connected we find that T_{0} ≤ G^{∘}. It is easy to see that T_{0} is a maximal ℚ-defined torus in G^{∘}. In fact the dimension of any maximal torus in G^{∘} is equal to the dimension of a maximal in H^{∘} and the dimension of T_{0} is equal to that of T_{1}. We proceed under the hypothesis that U(G) is abelian. We fix a maximal ℚ-defined torus T ≤ G^{∘}. Let N_{U}(T) be the normalizer of T in U(G). This is a ℚ-defined algebraic subgroup of U(G). We shall show that N_{U}(T) is normal in G. Let u ∈ N_{U}(T). By the result in [P1] mentioned already above There is for every g ∈ G a u(g) ∈ U(G) such that gT g^{−1} = u(g)T u(g)^{−1} (see [P1]). We then have

(2.2) \begin{aligned}gug^{−1}T gu^{−1}g^{−1} & = guu(g^{−1})T u(g^{−1})^{−1}u^{−1}g^{−1} = gu(g^{−1})uT u^{−1}u(g^{−1})^{−1}g^{−1} \\ & = gg^{−1}T gg^{−1} = T. \end{aligned}

It follows that N_{U}(T) is normal in G. By an argument similar to that in the beginning of this prove (compare G with G/N_{U}(T)) the result (under the assumption that U(G) is abelian) follows from the two particular cases: Case 1: U(G) = N_{U}(T). In this case U(G) centralizes T and G^{∘} has a unique ℚ-defined maximal torus T. Of course then F has to normalize T. Case 2: N_{U}(T) = {1}. We write the multiplication in U(G) in this case additively. Note that for every element f ∈ F = F (\overline{ℚ}) there is unique u(f) ∈ U(G)(\overline{ℚ}) with fT f^{−1} = u(f)T u(f)^{−1}. We make U(G)(\overline{ℚ}) into an F module through conjugation. The function u : F → U(G)(\overline{ℚ}) satisfies u(f_{1}f_{2}) = u(f_{1})+f_{1}u(f_{2})f_{1}^{−1} for all f_{1}, f_{2} ∈ F. Hence u is a derivation of F into the F -module U(G)(\overline{ℚ}). Since F is finite it is an inner derivation. Put

(2.3)  v := \frac{1}{|F |} ∑_{f∈F} u(f)

A standard computation shows that u(f) = −v + fvf^{−1} for all f ∈ F. We infer that F normalizes v^{−1}T v. By definition we have v ∈ U(G)(\overline{ℚ}). We show now that v ∈ U(G)(ℚ). Let 𝔾 be the Galois group of \overline{ℚ} over ℚ. Since u(f) is uniquely determined by f ∈ F and since T is defined over ℚ we have σ(u(f)) = u(σ(f)) for all σ ∈ 𝔾. From the definition (2.3) of v we infer that σ(v) = v for all σ ∈ 𝔾. It follows that v^{−1}T v is defined over ℚ. This finishes the proof.

Lemma 2.2 is somewhat surprising. The result is not true for linear algebraic groups G which are not solvable-by-finite. There are for example ℚ-defined finite solvable subgroups of SL(n, ℂ) which are maximal amongst solvable subgroups. We finish now the proof of Proposition 2.1.

Proof of Proposition 2.1 (Conjugacy): We shall proof first that any two maximal ℚ-defined d-subgroups of a ℚ-defined solvable-by-finite linear algebraic group G are conjugate by an element of U(G)(ℚ). Let S be the maximal ℚ-defined d-subgroup of G constructed in the existence proof above. Let S_{1} be another maximal ℚ-defined dsubgroup of G. The connected component S_{1}^{∘} is a ℚ-defined torus in G^{∘}. As such it is contained in a maximal ℚ-defined torus which is conjugate by an element of U(G)(ℚ) to S^{∘} (see [P1]). Hence we can from now on assume that S_{1}^{∘} ≤ S^{∘}. We distinguish two cases: Case 1: The dimension of S is 0. In this case S is a maximal ℚ-defined finite subgroup of G with G = U(G) · S. A simple argument shows that S_{1} is also finite. Consider first the case that U(G) is abelian. Make U(G)(\overline{ℚ}) into an S_{1}-module through conjugation. Note that for every f ∈ S_{1} there is a unique v(f) ∈ U(G)(\overline{ℚ}) such that v(f)f ∈ S. The function v : S_{1} → U(G)(\overline{ℚ}) satisfies v(f_{1}f_{2}) = v(f_{1})f_{1}v(f_{2})f_{1}^{−1} for all f_{1}, f_{2} ∈ S_{1}. Hence it is a derivation of the finite group S_{1} into the S_{1}-module U(G)(\overline{ℚ}). We also have σ(v(f)) = v(σ(f)) for every σ in the Galois group 𝔾 aand every f ∈ S_{1}. We find an element v ∈ U(G)(ℚ) such that v(f) = v^{−1}fvf^{−1} for all f ∈ S_{1}, see (2.3). Clearly then v^{−1}S_{1}v ≤ S and the result follows for abelian unipotent radical U(G). The general case is proved by induction on the nilpotency class of U(G). As done twice before we just compare G to G/Z where Z is the center of U(G). We skip the details but remark that it is relevant here that the map U(G)(ℚ) → U(G)/Z(ℚ) is surjective. Case 2: The dimension of S is > 0. We shall first argue that S_{1}^{∘} = S^{∘}. To do this it suffices to show that the dimensions of S_{1}^{∘} and of S^{∘} are the same. Note that we already have arranged that S_{1}^{∘} ≤ S^{∘}. So assume that the dimension of S_{1}^{∘} is strictly smaller than that of S^{∘}. Let N_{G}(S_{1}^{∘}) be the normalizer of S_{1}^{∘} in G. We have S_{1} ≤ N_{G}(S_{1}^{∘}) and also S^{∘} ≤ N_{G}(S_{1}) (note that S_{1}^{∘} ≤ S^{∘} and S^{∘} is abelian). Let H := N_{G}(S_{1}^{∘})/S_{1}^{∘} and π : N_{G}(S_{1}^{∘}) → H be the quotient homomorphism. Then π(S_{1}) is a ℚ-define finite subgroup of H and π(S) is a ℚ-defined torus of positive dimension in H. By Lemma 2.2 there is a ℚ-defined torus T of positive dimension in H which is normalized by π(S_{1}). Define S_{2} := π^{−1}(T) · S_{1}. It is easy to show that S_{2} is a ℚ-defined d-subgroup of G which is strictly bigger than S_{1}. This being a contradiction we can from now on assume that S_{1}^{∘} = S^{∘}. Let N := N_{G}(S^{∘}) be the normalizer of S^{∘} in G. We have S_{1}, S ≤ N. We may also consider the quotient N_{0} := N/S^{∘}. This is a ℚ-defined unipotent-by-finite linear algebraic group. We write π : N → N_{0} for the corresponding quotient homomorphism. Notice that both π(S_{1}) and π(S) are ℚ-defined maximal finite subgroups of N_{0}. By the first case of this prove there is a v ∈ U(N_{0})(ℚ) with vπ(S_{1})v^{−1} = π(S). Since S^{∘} consists of semi-simple elements the map π induces by restriction a ℚ-defined rational isomorphism π : U(N) → U(N_{0}). Because U(N) is unipotent the induced map π : U(N)(ℚ) → U(N_{0})(ℚ) is also surjective. Hence we find a u ∈ U(N)(ℚ) with π(u) = v. It is easy to prove that uS_{1}u^{−1} = S. It remains to be shown that u ∈ U(G)(ℚ). By (2.1) we find u_{1} ∈ U(G) and s ∈ S such that u = u_{1} · s. Applying the quotient homomorphism G → G/U(G) we find that s = 1. This finishes the proof of the first part of the conjugacy statement.

To prove the second we go exactly through the same steps as above but with all assumptions about groups or maps being defined over ℚ removed. Note that the corresponding Lemma 2.2 can in fact be proved by the method given above. We skip the details.

Let G be a solvable-by-finite ℚ-defined linear algebraic group decomposed as G = U(G) · S as in (2.1). It can be proved that there is a maximal ℚ-defined torus T ≤ S and a finite subgroup F ≤ S(\overline{ℚ}) such that S = T ·F. The finite group F can in general not be chosen as being defined over ℚ.

In the proofs to come we need following technical result concerning algebraic groups.

Lemma 2.3: Let G_{1}, G_{2} be two ℚ-defined d-groups and β : G_{1} → G_{2} a ℚ-defined rational map which is a group isomorphism. Then β^{−1} is rational and ℚ-defined.

Proof: Let us first remark that this lemma is well known (over any field of characteristic 0) if G_{1} and G_{2} are connected. To treat the more general case we think of G_{1}, G_{2} as closed subsets of some affine space. The map β is then the restriction of a polynomial map to G_{1}. We define \overline{ℚ}[G_{1}], \overline{ℚ}[G_{2}] to be the rings of regular functions on G_{1}, G_{2} over the algebraic closure \overline{ℚ} of ℚ in ℂ. We write β^{∗} : \overline{ℚ}[G_{2}] → \overline{ℚ}[G_{1}] for the comorphism induced by β. Clearly β^{∗} is injective. We shall prove that β is also surjective. Let G_{1}^{∘}, G_{2}^{∘} be the connected components of the identity in G_{1} respectively G_{2}. Then β(G_{1}^{∘}) = G_{2}^{∘} because β is regular and a group isomorphism. Let μ : G_{1}^{∘} → G_{2}^{∘} be the restriction of β to G_{1}^{∘}. Then β^{−1} is regular and ℚ-defined. Let g_{1}, …, g_{n} ∈ G_{1} be a system of representatives for the left cosets of G_{1}^{∘} in G_{1}. Define h_{i} := β(g_{i}) for i = 1, …, n. Then h_{1}, …, h_{n} is a system of representatives for the left cosets of G_{2}^{∘} in G_{2}. Let i be any element of \{1, …, n\}. We get \overline{ℚ}-defined regular isomorphisms of the cosets μ_{i} : C_{1}^{[i]} := G_{1}^{∘}g_{i} → C_{2}^{[i]} := G_{2}^{∘}h_{i} by defining μ_{i}(x) := μ(xg_{i}^{−1})h_{i}. The (regular) inverse of μ_{i} is given by μ_{i}^{−1}(x) := μ^{−1}(xh_{i}^{−1})g_{i}. Let Θ : \overline{ℚ}[C_{2}^{[1]}] × … × \overline{ℚ}[C_{2}^{[n]}] → \overline{ℚ}[C_{1}^{[1]}] × … × \overline{ℚ}[C_{1}^{[n]}] be the direct product of the comorphisms corresponding to the μ_{i}. Let k now be 1 or 2 and ρ_{k}^{[i]} : \overline{ℚ}[G_{k}] → \overline{ℚ}[C_{k}^{[i]}] be the restriction map. Define ρ_{k} to be the \overline{ℚ}-algebra homomorphism from \overline{ℚ}[G_{k}] to \overline{ℚ}[C_{k}^{[1]}] × … × \overline{ℚ}[C_{k}^{[n]}] defined by the ρ_{k}^{[1]}, …, ρ_{k}^{[n]}. We get an obviously commutative diagram of \overline{ℚ}-algebra homomorphisms:

(2.4)  \begin{matrix}\overline{ℚ}[G_{2}] & ⟶^{β^{∗}} & \overline{ℚ}[G_{1}] \\ ↓^{ρ_{2}} & & ↓^{ρ_{1}} \\ \overline{ℚ}[C_{2}^{[1]}] × … × \overline{ℚ}[C_{2}^{[n]}] & ⟶^{Θ} & \overline{ℚ}[C_{1}^{[1]}] × … × \overline{ℚ}[C_{1}^{[n]}] \end{matrix} .

The map Θ is bijective and ρ_{1}, ρ_{2} are obviously injective. Let ℑ_{i} be the kernel of ρ_{1}^{[i]}. Let i, j ∈ {1, …, n} and i ≠ j then the ideal 𝔞 := ℑ_{i} +ℑ_{j} is the kernel of the restriction map from \overline{ℚ}[G_{1}] to \overline{ℚ}[C_{1}^{[i]} ∩ C_{1}^{[j]}]. Since C_{1}^{[i]} ∩ C_{1}^{[j]} = ∅ Hilbert’s Nullstellensatz shows 𝔞 = \overline{ℚ}[G_{1}]. The Chinese remainder theorem implies that ρ_{1} and also ρ_{2} are isomorphisms. The diagram (2.4) shows that β^{∗} is an isomorphism. This means that β^{−1} is regular and \overline{ℚ}-defined. The Galois criterion shows that the regular map β^{−1} is ℚ-defined.

We shall discuss some facts concerning the class of ℚ-defined solvable-by-finite reduced linear algebraic groups. Let G be such a group. We always write U(G) for its unipotent radical and 𝔘(G) for the Lie algebra of U(G). Since U(G) is a ℚ-defined unipotent algebraic group the Lie algebra 𝔘(G) is also defined over ℚ. The exponential map exp : 𝔘(G) → U(G) gives us a ℚ-defined rational identification of 𝔘(G) with U(G). The automorphism group Aut(𝔘(G)) of the Lie algebra 𝔘(G) is then naturally a ℚdefined linear algebraic group. We have a ℚ-defined rational representation:

(2.5) α_{G} : G → Aut(𝔘(G)), α_{G}(g)(x) := exp^{−1}(g exp(x)g^{−1}) (g ∈ G, x ∈ 𝔘(G)).

The kernel of α_{G} coincides with the centralizer of U(G) in G and contains the center Z(U(G)) of U(G).

Definition 2.4: Let G be a solvable-by-finite ℚ-defined algebraic group. We call G reduced if ker(α_{G}) ≤ U(G).

We shall often need the following elementary results about reduced groups.

Lemma 2.5: Let G, G_{1} be solvable-by-finite ℚ-defined linear algebraic groups. Let S be a maximal ℚ-defined d-subgroup of G. If H is a subgroup of G we define Z_{U}(H) to be its subgroup consisting of those elements which centralize U(G). We have:

(i) G is reduced if and only if ker(α_{G}) = Z(U(G)). (ii) If G is reduced then we have Z(G) ≤ Z(U(G)). (iii) G is reduced if and only if Z_{U}(S) = {1}. (iv) Z_{U}(S) is a ℚ-defined normal subgroup of G and the quotient G/Z_{U}(S) is a

reduced solvable ℚ-defined linear algebraic group.

(v) If G_{1} ≤ G is a ℚ-defined subgroup of finite index in G and if G is reduced then G_{1} is reduced.

(vi) If G_{1} ≤ G is a ℚ-defined subgroup of finite index in G and if G_{1} is reduced then Z_{U}(S) is a finite ℚ-defined normal subgroup of G.

Proof: (i) and (ii) are evident. (iii): Assume that Z_{U}(S) = {1}. We have G = U(G) · S as stated in (2.1). Let u · s ∈ ker(α_{G}) with u ∈ U(G) and s ∈ S. We have α_{G}(u) = α_{G}(s)^{−1}. Since the left-hand side of this equation is unipotent and the right-hand side is semi-simple both sides are equal to 1. By our assumption we have s = 1. This proves one direction of our statement. The other direction is evident. (iv): The group Z_{U}(S) is evidently ℚ-defined, we prove that it is normal in G. Let g ∈ G be decomposed as g = us with u ∈ U(G) and s ∈ S. Then g^{−1}tg = s^{−1}ts for all t ∈ Z_{U}(S) which then shows that Z_{U}(S) is normal in G. We write Ĝ for the quotient group Ĝ = G/Z_{U}(S). Let Ŝ be the image of S in Ĝ. It is a maximal ℚdefined d-subgroup of Ĝ. Let t ∈ S represent an element of Z_{U}(Ŝ). This means that tut^{−1}u^{−1} ∈ Z_{U}(S) for all u ∈ U(G). The element tut^{−1}u^{−1} is also in U(G) and hence it is equal to 1. This implies that t ∈ Z_{U}(S). The result (iv) is then implied by (iii). (v): Since U(G) is connected we find U(G_{1}) = U(G) and the result follows. (vi): We again have U(G_{1}) = U(G). Let n be the index be the index of G_{1} in G and m := n!. Then g^{m} ∈ G_{1} for all g ∈ G. There is a maximal ℚ-defined d-subgroup S_{1} of G_{1} which is contained in S. The decompositions (2.1) for G_{1} and G show that s^{m} ∈ S_{1} for all s ∈ S. This implies using (iii) that s^{m} = 1 for all s ∈ Z_{U}(S). Since every linear group of finite exponent is finite we can infer that Z_{U}(S) is finite. Everything else follows from (iv).

In addition to the notation already introduced some more is required in the following. Let G be a solvable-by-finite ℚ-defined linear algebraic group which is also assumed to be reduced. We define

(2.6) ℨ𝔘(G) := exp^{−1}(Z(U(G))), ℨ(G^{∘}) := exp^{−1}(Z(G^{∘})), ℨ(G) := exp^{−1}(Z(G)).

The subspaces ℨ𝔘(G), ℨ(G^{∘}) and ℨ(G) of the Lie algebra 𝔘(G) are defined over ℚ and invariant under the action (2.5) of G. The ℚ-subspace ℨ𝔘(G)(ℚ) underlying ℨ𝔘(G) is defined to be ℨ𝔘(G)(ℚ) := exp^{−1}(Z(U(G))(ℚ)). We define analogously ℨ(G^{∘})(ℚ) = exp^{−1}(Z(G^{∘})(ℚ)) and ℨ(G)(ℚ) = exp^{−1}(Z(G)(ℚ)). All of the spaces ℨ𝔘(G)(ℚ), ℨ(G^{∘})(ℚ), ℨ(G)(ℚ) are invariant under G(ℚ). The group U(G) acts trivially on ℨ𝔘(G). Since G/U(G) is reductive we get a decompositions

(2.7)  ℨ𝔘(G) = 𝔜_{G} ⊕ ℨ(G^{∘}),  ℨ(G^{∘}) = 𝔜_{G}^{∘} ⊕ ℨ(G)

where 𝔜_{G} and 𝔜_{G}^{∘} are ℚ-defined G-invariant subspace. We again write 𝔜_{G}(ℚ), 𝔜_{G}^{∘}(ℚ) for the underlying (G(ℚ)-invariant) ℚ-vector spaces. Note that the centralizer of G in 𝔜_{G} ⊕ 𝔜_{G}^{∘} (that is the space of all elements of 𝔜_{G} ⊕ 𝔜_{G}^{∘} which are fixed by all g ∈ G) is trivial. The same result holds for the centralizer of G^{∘} in 𝔜_{G}. Note that the exponential map followed by the quotient homomorphism induces G/U(G)-module isomorphisms

(2.8) exp : 𝔜_{G}(ℚ) → Z(U(G))/Z(G^{∘})(ℚ),  exp : 𝔜_{G}^{∘}(ℚ) → Z(G^{∘})/Z(G)(ℚ)

where the action of G/U(G)(ℚ) on the right-hand groups is by conjugation.

We come now to the discussion of arithmetic subgroups of solvable-by-finite linear algebraic groups. We shall often use the following facts from the theory of arithmetic groups. Suppose that G is a ℚ-defined linear algebraic group and Γ ≤ G an arithmetic subgroup. If H is a ℚ-defined subgroup then H ∩ Γ is arithmetic in H. If H is also normal in G, then Γ projects to an arithmetic subgroup of G/H. We shall also use that if Γ is a Zariski-dense subgroup of G then Γ ∩ G^{∘} is Zariski-dense in G^{∘}. See [B2] and [P-R] for these results. Let G be a solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G be an arithmetic subgroup. It is known that Γ is a polycyclic group, in fact every solvable subgroup of GL(n, ℤ) is polycyclic (see [Se], Chapter 2). We define

(2.9)  U(G, Γ) := Γ ∩ U(G).

The finitely generated and torsion-free group U(G, Γ) is an arithmetic subgroup of U(G). It is always contained in U(G)(ℚ) because U(G) is uniquely divisible. Also U(G, Γ) is always Zariski-dense in U(G). It is further known that the ℚ-span of exp^{−1}(U(G, Γ)) is equal to 𝔘(G)(ℚ). See [Se], chapter 6 for these properties of U(G, Γ). Let S ≤ G be a maximal ℚ-defined d-subgroup. Then Γ ∩ S is an arithmetic subgroup of S and U(G, Γ) · (Γ ∩ S) has finite index in Γ. If Γ is a linear group we write F(Γ) for its Fitting subgroup, that is its biggest normal nilpotent subgroup. The following lemma is important in the proofs of Theorems A, A’ and B. Its proof uses an argument from [P2]. We write ⟨ M ⟩ for the subgroup generated by a substset M ⊆ G of a group G.

Lemma 2.6: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G an arithmetic subgroup. Then F(Γ) = U(G, Γ).

Proof: Clearly U(G, Γ) is contained in F(Γ). Let γ be an element from F(Γ), put A := U(G, Γ) · ⟨γ⟩ and let \overline{A} be the Zariski-closure of A. Then \overline{A} is a nilpotent linear algebraic group. Let γ = γ_{u} · γ_{s} be the Jordan decomposition of γ, that is γ_{u} is unipotent, γ_{s} is semi-simple and the two commute. We also have γ_{u}, γ_{s} ∈ \overline{A}. From U(G, Γ) ≤ A we infere that U(G) ≤ \overline{A}. Hence the group U(G) · ⟨γ_{s}⟩ is nilpotent. This implies that 𝔘(G)⋊⟨α_{G}(γ_{s})⟩ is nilpotent. Since α_{G}(γ_{s}) is a semi-simple transformation of 𝔘(G) this can only happen if α_{G}(γ_{s}) = 1. We infer that γ_{s} = 1 because G is reduced. Hence γ is unipotent. Let now G = U(G)·S be decomposed as in (2.1). Let γ = γ_{1} ·γ_{2} with γ_{1} ∈ U(G) and γ_{2} ∈ S. Since γ and γ_{1} are unipotent we have γ_{2} = 1 and the result follows.

Let G be a solvable-by-finite linear algebraic group defined over ℚ and S ≤ G be a maximal d-subgroup of G which is also defined over ℚ. We then have G = U(G) · S as in (2.1). Then G/U(G) is a ℚ-defined d-group (in fact isomorphic to S). The connected component (G/U(G))^{∘} is a ℚ-defined maximal torus in G/U(G). We write π_{U} : G → G/U(G) for the quotient homomorphism. Let Γ ≤ G be a Zariski-dense arithmetic subgroup. Then π_{U}(Γ) is a finitely generated abelian-by-finite group which is Zariski-dense in G/U(G). Its abelian subgroup π_{U}(Γ) ∩ (G/U(G))^{∘} is Zariski-dense in (G/U(G))^{∘}. We define

(2.10)  M(Γ) := π_{U}(Γ) ∩ (G/U(G))^{∘},  F (Γ) := π_{U}(Γ)/M(Γ).

It is easy to see that M(Γ) is a finitely generated abelian group and F (Γ) is a finite group. We obtain an exact sequence

(2.11)  {1} → M(Γ) → π_{U}(Γ) → F (Γ) → {1}.

We also think of M(Γ) as a F (Γ)-module the action being induced by conjugation by representatives taken from π_{U}(Γ). Suppose now further that G is reduced. The action (by conjugation) of Γ on U(G, Γ) leads to an action of π_{U}(Γ) on the spaces 𝔜_{G}(ℚ), 𝔜_{G}^{∘}(ℚ) introduced in (2.6), (2.7), (2.8). The action of M(Γ) on 𝔜_{G}^{∘}(ℚ) is trivial. If A is a group and B, C are A-modules we write Hom_{A}(B, C) for the group of Aequivariant homomorphisms. For i ∈ ℕ we denote the i-th cohomology group of A with coefficients in B by H^{i}(A, B) (see [W] for the definition). By definition H^{0}(A, B) is the subgroup of those elements in B which are left fixed by all elements of A. The first cohomology group H^{1}(A, B) is the group of derivations modulo its sugroup of inner derivations. We need the following computation of cohomology groups.

Lemma 2.7: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup. Then the following hold.

(i) H^{0}(M(Γ), 𝔜_{G}(ℚ)) = {0}. (ii) H^{1}(π_{U}(Γ), 𝔜_{G}(ℚ)) = {0}. (iii) H^{1}(π_{U}(Γ), 𝔜_{G}^{∘}(ℚ)) ≅ Hom_{F (Γ)}(M(Γ), 𝔜_{G}^{∘}(ℚ)), where the isomorphism is given

by restricting derivations from π_{U}(Γ) to M(Γ). (iv) If π_{U}(Γ) is nilpotent then H^{1}(π_{U}(Γ), 𝔜_{G}^{∘}(ℚ)) = {0}.

Proof: (i): Let v ∈ H^{0}(M(Γ), 𝔜_{G}(ℚ)) then v is an element in 𝔜_{G}(ℚ)) which is left fixed by all elements of M(Γ). Consider u = exp(v) ∈ Z(U(G)). Then u is centralized by M(Γ). Since M(Γ) is Zariki-dense in (G/U(G))^{∘} it is even centralized by (G/U(G))^{∘}. Since π_{U}(G^{∘}) = (G/U(G))^{∘} the element u is centralized by G^{∘}. Since Z(U(G)) ≤ G^{∘} we have u ∈ Z(G^{∘}) and v = 0. (ii): From (i) we can infer that H^{1}(M(Γ), 𝔜_{G}(ℚ)) = {0}. This result can be found in [Se], chapter 3, theorem 2^{∗∗}. Since 𝔜_{G}(ℚ) is a ℚ-vector space the restriction homomorphism H^{1}(π_{U}(Γ), 𝔜_{G}(ℚ)) → H^{1}(M(Γ), 𝔜_{G}(ℚ)) is injective.

(iii): Since M(Γ) acts trivially on 𝔜_{G}^{∘}(ℚ)) the exact sequence (2.11) leads to the 5-term exact sequence of cohomology

(2.12) {0} → H^{1}(F (Γ), 𝔜_{G}^{∘}(ℚ)) → H^{1}(π_{U}(Γ), 𝔜_{G}^{∘}(ℚ)) → Hom_{F (Γ)}(M(Γ), 𝔜_{G}^{∘}(ℚ)) → H^{2}(F (Γ), 𝔜_{G}^{∘}(ℚ)).

We have H^{1}(F (Γ), 𝔜_{G}^{∘}(ℚ)) = {0} and H^{2}(F (Γ), 𝔜_{G}^{∘}(ℚ)) = {0} because F (Γ) is a finite group. We remark that the isomorphism between H^{1}(π_{U}(Γ), 𝔜_{G}^{∘}(ℚ)) and Hom_{F (Γ)}(M(Γ), 𝔜_{G}^{∘}(ℚ)) which then follows from (2.12) is given by the restriction of derivations from π_{U}(Γ) to M(Γ). (iv): If π_{U}(Γ) is nilpotent then F (Γ) acts unipotently on M(Γ) (that is we have a composition series of F (Γ)-submodules in M(Γ) such that F (Γ) acts trivially on its factors) and trivially on ℚ ⊗_{ℤ} M(Γ). Let φ ∈ Hom_{F (Γ)}(M(Γ), 𝔜_{G}^{∘}(ℚ)) and Im(φ) ≤ 𝔜_{G}^{∘}(ℚ) its image. It follows from our definitions that exp(Im(φ)) is contained in Z(G) hence Im(φ) = {0}.

The following definition is motivated by Lemma 2.7. It is relevant in the formulation of Theorem A’.

Definition 2.8: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G a Zariski-dense arithmetic subgroup. We call Γ separated in G if

(2.13)  Hom_{F (Γ)}(M(Γ), Z(G^{∘})/Z(G)(ℚ)) = {0}.

Clearly if G is connected then every arithmetic subgroup is separated. Note that H^{1}(π_{U}(Γ), 𝔜_{G}^{∘}(ℚ)) = {0} if Γ is separated in G. It follows from Lemma 2.7 (iv) that Γ is separated in G if Γ/U(G, Γ) is nilpotent.

We are now ready to give the Proof of Theorems A, A’: By Lemma 2.6 φ induces an isomorphism φ : U(G_{1}, Γ_{1}) → U(G_{2}, Γ_{2}). By results of Malcev (see [Mc]) we find a ℚ-defined rational isomorphism Φ_{u} : U(G_{1}) → U(G_{2}) such that Φ_{u}(γ) = φ(γ) for all γ ∈ U(G_{1}, Γ_{1}). This rational isomorphism induces a ℚ-defined Lie algebra isomorphism Φ̃_{u} : 𝔘(G_{1}) → 𝔘(G_{2}) by Φ̃_{u} := exp^{−1} ∘Φ_{u}∘exp. The isomorphism Φ̃_{u} gives us a ℚ-defined rational isomorphism Ψ_{u} : Aut(𝔘(G_{1})) → Aut(𝔘(G_{2})) of the automorphism groups of the corresponding Lie algebras defined by Ψ_{u}(Θ) := Φ̃_{u} ∘ Θ ∘ Φ̃_{u}^{−1} where Θ ∈ Aut(𝔘[G_{1}]). In the following k is always supposed to be 1 or 2. Let α_{k} = α_{G_{k}} : G_{k} → Aut(𝔘(G_{k})) be the homomorphism defined in (2.5). The map α_{k} is a ℚ-defined rational homomorphism of G_{k} into the automorphism group Aut(𝔘(G_{k})). We have ker(α_{k}) = Z(U(G_{k})). Notice also that Ψ_{u}(α_{1}(u)) = α_{2}(Φ_{u}(u)) holds for all u ∈ U(G_{1}).

By a straightforward computation using our definitions we find for γ ∈ Γ_{1}:

(2.14)  Ψ_{u}(α_{1}(γ)) = α_{2}(φ(γ)).

The groups α_{k}(Γ_{k}) are Zariski-dense arithmetic subgroups in α_{k}(G_{k}). We infer from (2.14) that Ψ_{u}(α_{1}(G_{1})) = α_{2}(G_{2}). Hence Ψ_{u} gives us a ℚ-defined rational isomorphism between α_{1}(G_{1}) and α_{2}(G_{2}). We now choose a maximal ℚ-defined d-subgroup S_{1} in G_{1}. Then Ŝ_{2} := Ψ_{u}(α_{1}(S_{1})) is a maximal ℚ-defined d-subgroup of α_{2}(G_{2}). Consider now the ℚ-defined algebraic subgroup H := α_{2}^{−1}(Ŝ_{2}) of G_{2} and let S_{2} be a maximal ℚ-defined subgroup in H. Clearly S_{2} is also a maximal ℚ-defined d-subgroup in G_{2} and α_{2} induces a ℚ-defined rational homomorphism β := α_{2}|_{S_{2}} : S_{2} → Ŝ_{2} which is also a group isomorphism. The latter is implied by the fact that the kernel of α_{2} is unipotent. Hence β^{−1} is rational and ℚ-defined by Lemma 2.3. We shall now define a map Φ_{0} which after some slight modification will serve as our Φ. Given an element g ∈ G_{1} we may uniquely decompose it as g = t_{g} · u_{g} with t_{g} ∈ S_{1} and u_{g} ∈ U(G_{1}). We put

(2.15)  Φ_{0}(g) := β^{−1} ∘ Ψ_{u} ∘ α_{1}(t_{g}) · Φ_{u}(u_{g}).

Some straightforward considerations show that Φ_{0} : G_{1} → G_{2} is a ℚ-defined rational isomorphism. It is for example necessary to show

(2.16)  Φ_{u}(t · u · t^{−1}) = β^{−1}(Ψ_{u}(α_{1}(t))) · Φ_{u}(u) · (β^{−1}(Ψ_{u}(α_{1}(t))))^{−1}

for t ∈ S_{1} and u ∈ U(G_{1}). This can easily be done on the level of the Lie algebras 𝔘(G_{1}), 𝔘(G_{2}). From what was said above it follows that

(2.17)  α_{2}(Φ_{0}(g)) = Ψ_{u}(α_{1}(g))  for all g ∈ G_{1}.

For γ ∈ Γ_{1} we get from (2.14), (2.17) that α_{2}(Φ_{0}(γ)φ(γ)^{−1}) = 1, hence

(2.18)  v(γ) := Φ_{0}(γ) · φ(γ)^{−1} ∈ Z(U(G_{2}))(ℚ).

Note that if γ ∈ U(G_{1}, Γ_{1}) then v(γ) = 1. We have now constructed a ℚ-defined rational isomorphism Φ_{0} : G_{1} → G_{2} and a map v : Γ_{1} → Z(U(G_{2}))(ℚ) so that the analogue of (1.1) is satisfied. Our construction depends basically on the choice of the d-subgroup S_{2} in H. We shall now show that by possibly altering this choice we can make the new v have its values in Z(G_{2}^{∘})(ℚ). Consider the ℚ-vector spaces ℨ𝔘(G_{2})(ℚ), ℨ(G_{2}^{∘})(ℚ) and 𝔜_{G_{2}}(ℚ) defined in (2.6), (2.7). We shall use now the notation introduced just before Lemma 2.7. Make ℨ𝔘(G_{2})(ℚ) into a Γ_{1}-module by defining γ · x := exp^{−1}(φ(γ) exp(x)φ(γ)^{−1}) for γ ∈ Γ_{1} and x ∈ ℨ𝔘(G_{2})(ℚ). In fact U(G_{1}, Γ_{1}) acts trivially and we may consider ℨ𝔘(G_{2})(ℚ) as a π_{U}(Γ_{1}) module. Both ℨ(G_{2}^{∘})(ℚ) and 𝔜_{G_{2}}(ℚ) are submodules. Define ṽ : Γ_{1} → ℨ𝔘(G_{2})(ℚ) by ṽ(γ) := exp^{−1}(v(γ)) (γ ∈ Γ_{1}). An obvious computation shows that the map ṽ is a derivation of Γ_{1} into ℨ𝔘(G_{2})(ℚ). That is ṽ(γ_{1}γ_{2}) = ṽ(γ_{1}) + γ_{1} · ṽ(γ_{2}) holds for γ_{1}, γ_{2} ∈ Γ_{1}. Hence the component of ṽ in 𝔜_{G_{2}}(ℚ)) represents an element of H^{1}(π_{U}(Γ_{1}), 𝔜_{G_{2}}(ℚ)). By Lemma 2.6 φ induces an isomorphism between π_{U}(Γ_{1}) and π_{U}(Γ_{1}). We infer from Lemma 2.7 (ii) that H^{1}(π_{U}(Γ_{1}), 𝔜_{G_{2}}(ℚ)) = {0}. We have proved now that we may alter the derivation ṽ by an inner derivation so that the result has values in ℨ(G_{2}^{∘})(ℚ). This means that there is a x ∈ Z(U(G_{2}))(ℚ) so that

(2.19)  v(γ) · x φ(γ) x^{−1} φ(γ)^{−1} ∈ Z(G_{2}^{∘})(ℚ)  for all γ ∈ Γ_{1}.

Define Φ(g) = x Φ_{0}(g) x^{−1} for all g ∈ G_{1} and

(2.20)  u(γ) := Φ(γ) φ(γ)^{−1} = x Φ_{0}(γ) x^{−1}φ(γ)^{−1} = v(γ) · x φ(γ) x^{−1} φ(γ)^{−1}.

Theorem A is implied by (2.19), (2.20). Theorem A’ is proved by the same argument but using 𝔜_{G_{2}}^{∘}(ℚ) instead of 𝔜_{G_{2}}(ℚ). The relevant vanishing result for the cohomology follows from Lemma 2.7 (iii).

The choices made for the construction of the homomorphism Φ in Theorem A’ imply certain compatibility properties. These are formulated in Theorem C and Proposition 2.10. In the proofs to come we shall several times use an argument which we formulate in the following simple lemma.

Lemma 2.9: Let G be a reduced solvable ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup. Let Φ, Ψ be two ℚ-defined rational automorphisms of G. Assume that there is a homomorphism w : Γ → Z(G) which is trivial on F(Γ) such that Φ(γ) = w(γ) · Ψ(γ) for all γ ∈ Γ. Then Φ = Ψ and w ≡ 1.

Proof: Define Θ(g) := Φ(g) · Ψ(g)^{−1} for g ∈ G. This is a ℚ-defined rational map from G to G. We have Θ(Γ) ⊆ Z(G). We conclude Θ(G) ⊆ Z(G) because Γ is Zariski-dense in G. The computation

(2.21)

Θ(g_{1}g_{2}) = Φ(g_{1}g_{2}) Ψ(g_{1}g_{2})^{−1}
= Φ(g_{1}) Φ(g_{2}) · Ψ(g_{2})^{−1} Φ(g_{1})^{−1} Φ(g_{1}) · Ψ(g_{1})^{−1} = Θ(g_{1}) Θ(g_{2}).

then shows that then Θ is a homomorphism from G to G. If γ ∈ Γ ∩ U(G) = U(G, Γ) then Θ(γ) = 1 by Lemma 2.6 and our assumption. Since U(G, Γ) is Zariski-dense in U(G) we infer that Θ is trivial on U(G). Let G be decomposed as in (2.1). The restriction of Θ to S is a ℚ-defined rational homomorphism of the d-group S into the unipotent group Z(G) and hence has to be trivial. We conclude that Θ is trivial on G. This proves the lemma.

Proof of Theorem B: We shall use here the notation introduced just before Lemma 2.7. Suppose that Γ is not separated in G. Then by Lemma 2.7 (iii) there is a derivation w̃ : π_{U}(Γ) → 𝔜_{G}(ℚ) which is not an inner derivation. Multiplying w by a suitable integer we may assume that w̃(γ) ∈ exp^{−1}(Z(U(G, Γ)) ∩ G^{∘}) for all γ ∈ π_{U}(Γ). The restriction w̃_{0} of w̃ to M(Γ) is a homomorphism w̃_{0} : M(Γ) → ℨ(G^{∘}). Moreover the image of w̃_{0} is not contained in ℨ(G). This follows because the isomorphism described in Lemma 2.7 (iii) is given by restriction from π_{U}(Γ) to M(Γ) (see the proof of Lemma 2.7 (iii)). Define w(γ) := exp(w̃(π_{U}(γ))) for γ ∈ Γ and w_{0}(γ) := exp(w̃_{0}(π_{U}(γ))) for γ ∈ Γ ∩ G^{∘}. Notice w_{0} is the restriction of w to Γ∩G^{∘} and that the image of w_{0} is not contained in Z(G). Define further φ(γ) := w(γ)·γ for γ ∈ Γ. By our assumptions we have φ(γ) ∈ Γ for all γ ∈ Γ. It is straightforward to see that φ is an automorphism of Γ. Suppose now that there are Φ and u as described in the statement of the theorem. In particular we have Φ(γ) = u(γ)w_{0}(γ)γ for all γ ∈ Γ ∩ G^{∘}. Since G^{∘} is reduced by Lemma 2.5 (v) and since Γ ∩ G^{∘} is Zariski-dense in G^{∘} we are in the position to apply Lemma 2.9 and obtain that w_{0}(γ) = u(γ)^{−1} for all γ ∈ Γ ∩ G^{∘}. This contradicts the fact that the image of w_{0} is not contained in Z(G) and the Theorem is proved.

Proof of Theorem C: Let φ_{1}, φ_{2} ∈ Aut(Γ) and put φ_{3} := φ_{1}∘φ_{2}. Let Φ_{1}, Φ_{2}, Φ_{3} : G → G be the algebraic isomorphisms corresponding to φ_{1}, φ_{2}, φ_{3} constructed in Theorem A’. Let u_{1}, u_{2}, u_{3} : Γ → Z(G)(ℚ) be the corresponding homomorphisms. We shall show that Φ_{3} = Φ_{1} ∘ Φ_{2}. For γ ∈ Γ we have from (1.1):

(2.22) Φ_{1} ∘ Φ_{2}(γ) = Φ_{1}(u_{2}(γ)) · u_{2}(φ_{2}(γ)) · φ_{1} ∘ φ_{2}(γ),  Φ_{3}(γ) = u_{3}(γ) · φ_{1} ∘ φ_{2}(γ).

The formulas (2.22) and Theorem A’ show that Φ_{3} and Φ_{1} ∘ Φ_{2} satisfy the hypotheses of Lemma 2.9. We conclude that Φ_{3} = Φ_{1} ∘ Φ_{2}. This shows the existence of the homomorphism D. The properties which relate D and u are simple consequences of Theorem A’. The uniqueness of D, u follows from Lemma 2.9 by a straightforward argument.

Before we proceed we need some terminology. Let Δ be a group and M an abelian group. A Δ-bimodule structure on M is an action (δ, m) ↦ δ · m together with an action from the right (m, δ) ↦ m · δ (δ ∈ Δ, m ∈ M) of Δ on M such that (δ · m) · ρ = δ · (m · ρ) for all δ, ρ ∈ Δ and m ∈ M. A bimodule derivation of Δ into the bimodule M is a map λ : Δ → M such that

(2.23)  λ(δ_{1}δ_{2}) = λ(δ_{1}) · δ_{2} + δ_{1} · λ(δ_{2})  for all δ_{1}, δ_{2} ∈ Δ.

A bimodule derivation λ is called inner if there is an m ∈ M so that λ(δ) = δ·m−m·δ. The first cohomology group H_{bi}^{1}(Δ, M) of the group Δ in its bimodule M is defined to be the group of bimodule derivations modulo its subgroup of inner derivations. In the following we formulate some further properties of the homomorphism D and the map u from Theorem C.

Proposition 2.10: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is separated in G. Let D : Aut(Γ) → Aut_{ℚ}(G) and u : Aut(Γ) → Hom(Γ, Z(G)(ℚ)) be defined as in Theorem C. Then the following hold:

(i) If φ is an inner automorphism of Γ, then D_{φ} is the corresponding inner automorphism of G and u_{φ} ≡ 1.

(ii) Take on Hom(Γ, Z(G)(ℚ)) the Aut(Γ)-bimodule structure given by φ_{1} · v · φ_{2} := D_{φ_{1}} ∘ v ∘ φ_{2}. Then the map φ → u_{φ} is a bi-module derivation of Aut(Γ) into Hom(Γ, Z(G)(ℚ)).

Proof: (i): Let δ ∈ Γ such that φ(γ) = δγδ^{−1} for all γ ∈ Γ. Define a map Θ : G → G by Θ(g) := D_{φ}(g)γg^{−1}γ^{−1}. By the same arguments as in the proof of Theorem C we find that Θ is a ℚ-defined rational homomorphism from G into Z(G) which is trivial on U(G). This implies that Θ ≡ 1. (ii) follows by a straightforward computation.

3. Embedding Results

This section contains various embedding results. The first is a restricted but also more specific version of Theorem D.

Proposition 3.1: Let G be a reduced connected solvable ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup. Let Δ ≥ Γ be a finite extension group of Γ such that Γ is normal in Δ. Then there is a ℚ-defined linear algebraic group H which contains a ℚ-defined algebraic subgroup of finite index rationally isomorphic over ℚ to G and an arithmetic subgroup Δ_{0} ≤ H(ℚ) which is isomorphic to Δ. If Δ/Γ is solvable then H also has this property.

In the proof of Proposition 3.1 two technical ingredients are required. The first is the following simple fact about bimodule cohomology. For the definitions see the end of Section 2.

Lemma 3.2: Let A be a finite group and V a finite dimensional ℚ-vector space with an A-bimodule structure. Then H_{bi}^{1}(A, V) = 0.

Proof: Define an action from the left of A on V by a ∗ v := a · v · a^{−1} (a ∈ A, v ∈ V). Given a bimodule derivation f : A → V define f̃ : A → V by f̃(a) := f(a) · a^{−1} (a ∈ A). Then f̃ is a derivation in the ordinary sense of A into the left A-module defined above. It is easy to see that f is inner (in the bimodule sense) if and only if f̃ is inner (in the ordinary sense). Our result then follows from the analogous result in the standard cohomology theory of finite groups, see [W].

The second ingredient in the proof of Proposition 3.1 is an induction procedure which we describe now. This will also be of importance in the arguments to come. Let G ≤ GL(n, ℂ) be a linear algebraic group and Γ ≤ G a subgroup. Let Γ ≤ Δ be a finite extension group so that Γ is normal in Δ. Write H = \overline{Γ} for the Zariski-closure of Γ. Let Δ = Γr_{1} ∪ … ∪ Γr_{m} be the decomposition of Δ into left cosets. Assume that there are rational isomorphisms f_{1}, …, f_{m} : G → G so that

(3.1)  f_{i}(γ) = r_{i}γr_{i}^{−1}  (i = 1, …, m)

for all γ ∈ Γ. We then have f_{i}(H) = H for i = 1, …, m. Let ψ : Δ → GL(mn, ℂ) be the homomorphism inducing the injection of Γ into GL(n, ℂ) with respect to the chosen coset decomposition. The map ψ has the property

(3.2)  ψ(γ) = ( \begin{matrix}r_{1}γr_{1}^{−1} & 0 & … & 0 \\ 0 & r_{2}γr_{2}^{−1} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & r_{m}γr_{m}^{−1} \end{matrix} )  for all γ ∈ Γ.

If r_{i}r_{j} = γ_{jk}r_{k} with γ_{jk} ∈ Γ then the matrix ψ(r_{i}) has γ_{jk} in the (j, k)-position. The entries of ψ(r_{i}) not specified by this are equal to 0. By our assumptions we may now extend the injective homomorphism ψ : Γ → GL(mn, ℂ) to a injective rational homomorphism Ψ : H → GL(nm, ℂ) by the definition

(3.3)  Ψ(h) = ( \begin{matrix}f_{1}(h) & 0 & … & 0 \\ 0 & f_{2}(h) & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & f_{m}(h) \end{matrix} )  (h ∈ H).

We define now

(3.4)  I(G, Γ, Δ) := Ψ(H) · ψ(Δ).

It is clear that ψ(Δ) normalizes Ψ(H), hence I(G, Γ, Δ) is a subgroup of GL(mn, ℂ). It is furthermore easy to see that Ψ(H) has index m in I(G, Γ, Δ). Hence I(G, Γ, Δ) is an algebraic subgroup of GL(mn, ℂ). We have now:

Lemma 3.3: Let G ≤ GL(n, ℂ) be a linear algebraic group defined over ℚ and Γ ≤ G(ℤ) be a subgroup of finite index. Define H to be the Zariski-closure of Γ. Let Γ ≤ Δ be a finite extension group so that Γ is normal in Δ. Let Δ = Γr_{1} ∪ … ∪ Γr_{m} be the decomposition of Δ into left cosets. Assume that there are rational isomorphisms f_{1}, …, f_{m} satisfying (3.1) which are ℚ-defined when restricted to H. Then I(G, Γ, Δ) is a ℚ-defined linear algebraic group and Ψ(H) is a ℚ-defined algebraic subgroup of I(G, Γ, Δ) which rationally isomorphic over ℚ to H. The subgroup ψ(Δ) of I(G, Γ, Δ) is contained in I(G, Γ, Δ)(ℚ) and is arithmetic. If G and Δ/Γ are solvable then so is I(G, Γ, Δ).

The proof uses only elementary considerations, see [G-P1] and [G-P2] for more comments.

Proof of Proposition 3.1: We have to start with some preliminary considerations. Define

(3.5) V := { w : Γ → Z(G)(ℚ)  w is a homomorphism trivial on U(G, Γ) }.

We shall sometimes think of V as a ℚ-vector space and write the composition additively. For w ∈ V define

(3.6)  Λ_{w} : Γ → G,  Λ_{w}(γ) := w(γ) · γ (γ ∈ Γ).

It is easy to see that Λ_{w} is an injective group homomorphism. The image Γ_{w} := Λ_{w}(Γ) is commensurable with Γ and hence is an arithmetic subgroup of G(ℚ), see [G-P2] Lemma 3.8. Let Δ_{w} > Γ_{w} be an extension group of Γ_{w} so that there is an isomorphism Λ_{w} : Δ → Δ_{w} which restricts to the original Λ_{w} on Γ. The existence of Δ_{w} can be seen by constructing its multiplication table. Define φ_{δ} : Γ → Γ for δ ∈ Δ by φ_{δ}(γ) := δ γ δ^{−1}. Let D_{φ_{δ}} : G → G be the corresponding ℚ-defined rational isomorphism constructed in Theorem C. Note that Γ is separated in G since G is connected. We now define a Δ-bimodule structure on V by

(3.7)  δ · w · μ := D_{φ_{δ}} ∘ w ∘ φ_{μ}  (δ, μ ∈ Δ, w ∈ V).

From the definition (3.7) and Proposition 2.10 (i) it is obvious that γ · w = w and w · γ = w for all γ ∈ Γ and w ∈ V. Hence V is a bi-module for the finite group A := Δ/Γ. The map δ ↦ u_{φ_{δ}} is a derivation of Δ into V. Since this derivation is trivial on Γ by Proposition 2.10 (i) we may use Lemma 3.2 and obtain that it is inner. This means that there is a w ∈ V such that

(3.8)  u_{φ_{δ}}(γ) = w(φ_{δ}(γ)) − D_{φ_{δ}}(w(γ))  for all δ ∈ Δ, γ ∈ Γ.

Consider now μ ∈ Δ_{w} and ρ ∈ Γ_{w}. Find δ ∈ Δ and γ ∈ Γ such that Λ_{w}(δ) = μ and Λ_{w}(γ) = ρ = w(γ) · γ. Do the following two obvious computations:

(3.9) \begin{matrix}μ ρ μ^{−1} & = Λ_{w}(δ γ δ^{−1}) = w(δ γ δ^{−1}) · δ γ δ^{−1} = w(φ_{δ}(γ)) · φ_{δ}(γ), \\ D_{φ_{δ}}(ρ) = & D_{φ_{δ}}(w(γ) · γ) = D_{φ_{δ}}(w(γ)) · D_{φ_{δ}}(γ) = u_{φ_{δ}}(γ) · D_{φ_{δ}}(w(γ)) · φ_{δ}(γ). \end{matrix}

From (3.8) and (3.9) we conclude that μ ρ μ^{−1} = D_{φ_{δ}}(ρ). The proposition is then proved by applying Lemma 3.3.

The next result characterizes arithmetic subgroups of reduced solvable linear algebraic groups.

Theorem 3.4: Let Γ be an arithmetic subgroup in a solvable ℚ-defined linear algebraic group G. Then Γ is isomorphic to a Zariski-dense arithmetic subgroup contained in H(ℚ) for a reduced solvable ℚ-defined linear algebraic group H if and only if the Fitting subgroup of Γ is torsion-free.

The following is an immediate corollary of Theorem 3.4.

Corollary 3.5: Let Γ be an arithmetic subgroup of a solvable-by-finite ℚ-defined linear algebraic group. Then there is a subgroup Γ_{0} of finite index in Γ and a reduced solvable ℚ-defined linear algebraic group H so that Γ_{0} is isomorphic to an arithmetic Zariskidense subgroup of H(ℚ).

Proof of Theorem 3.4: The only if part of our proposition follows from Lemma 2.6 because a unipotent linear algebraic group is torsion-free. We proceed by proving the if statement. Let Γ̃ ≤ G(ℤ) ∩ Γ be a subgroup of finite index which is normal in Γ. Let γ ∈ Γ be given. We have γΓ̃γ^{−1} = Γ̃. This implies that there is a ℚ-defined rational isomorphism f : G → G with f(h) = γhγ^{−1} for all h ∈ G (see [G-P2] Lemma 2.2). We now use Lemma 3.3 to replace G by one of its solvable finite extensions so that Γ is a subgroup of G(ℚ). Let G be decomposed as in (2.1) and let T ≤ S be a ℚ-defined torus which is of finite index in S. Then Γ ∩ (U(G) · T) is of finite index in Γ. Then there is a torsion-free subgroup Γ_{0} ≤ Γ∩(U(G)·T) which is normal and of finite index in Γ. Let G_{0} := \overline{Γ}_{0} be the Zariski-closure of Γ_{0} in G. Then G_{0} is a solvable ℚ-defined linear algebraic group and if G_{0} = U(G_{0}) · S_{0} is its decomposition (2.1) then S_{0} is a commutative ℚ-defined d-group. We shall now prove that Γ_{0} is isomorphic to a Zariski-dense arithmetic subgroup of a reduced solvable ℚ-defined linear algebraic group. Let S_{1} be the kernel of α_{G_{0}} in S_{0}. Then S_{1} can also be described as the centralizer of U(G_{0}) in S_{0}. Choose a ℚ-defined d-subgroup S_{2} ≤ S_{0} such that S = S_{1} · S_{2} and such that S_{1} ∩ S_{2} is finite. The possibility to choose such a S_{2} follows from the well known (contravariant) equivalence between the category of ℚ-defined commutative d-groups and the category of continuous finitely generated ℤ-modules for the absolute Galois group of ℚ. See Section 5 for more details on this equivalence of categories. The ℚ-defined finite group S_{1} ∩ S_{2} is normal in G_{0} and hence Γ_{0} ∩ S_{1} ∩ S_{2} = {1} since Γ_{0} is torsion-free. Define

(3.10) Ĝ_{0} := G_{0}/(S_{1}∩S_{2}), Ŝ_{0} := S/(S_{1}∩S_{2}), Ŝ_{1} := S_{1}/(S_{1}∩S_{2}), Ŝ_{2} := S_{2}/(S_{1}∩S_{2}).

Then Ĝ_{0} is a solvable ℚ-defined linear algebraic group which decomposes as Ĝ_{0} = U(Ĝ_{0}) · Ŝ_{0}. Let Γ̂_{0} be the image of Γ_{0} in Ĝ_{0}. Note that Γ̂_{0} a Zariski-dense arithmetic subgroup of Ĝ_{0}(ℚ) isomorphic to Γ_{0}. Let s ∈ S_{0} represent an element of Ŝ_{0} which centralizes U(Ĝ_{0}). Then sus^{−1}u^{−1} ∈ S_{1} ∩ S_{2} for all u ∈ U(G_{0}). This implies sus^{−1}u^{−1} = 1 for all u ∈ U(G_{0}) and hence s ∈ S_{1}. We infer that Ŝ_{1} is the centralizer of U(Ĝ_{0}) in Ŝ. We also have Ŝ = Ŝ_{1} · Ŝ_{2} and Ŝ_{1} ∩ S_{2} = {1}. This shows that U(Ĝ_{0}) · Ŝ_{2} is a reduced solvable ℚ-defined linear algebraic group. Consider now the residue map π : Ĝ_{0} → Ĝ_{0}/(U(Ĝ_{0})·Ŝ_{2}) ≅ Ŝ_{1}. The image π(Γ̂_{0}) of Γ̂_{0} is an arithmetic subgroup of Ĝ_{0}/(U(Ĝ_{0})·Ŝ_{2})(ℚ) and hence a finitely generated abelian group. Let π(Γ̂_{0})_{tor} be its torsion subgroup and C := \overline{π(Γ̂_{0})_{tor}} its Zariski-closure in S_{1}. The group C is finite and ℚ-defined and is contained in the center of Ĝ_{0}. We define

(3.11)  G̃_{0} := Ĝ_{0}/C, S̃ := Ŝ/C, S̃_{1} := Ŝ_{1}/C, S̃_{2} := Ŝ_{2} · C/C ≅ Ŝ_{2}.

The solvable ℚ-defined linear algebraic group G̃_{0} decomposes as G̃_{0} = U(G̃_{0}) · S̃ and S̃ = S̃_{1} · S̃_{2} with S̃_{1} ∩ S̃_{2} = {1}. The ℚ-defined torus S̃_{1} is contained in the center of G̃_{0} and its complement U(G̃_{0}) · S̃_{2} is a reduced solvable ℚ-defined linear algebraic group because it is isomorphic to U(Ĝ_{0}) · Ŝ_{2} by a ℚ-defined rational isomorphism. Let Γ̃_{0} be the image of Γ̂_{0} in G̃_{0}. This group is isomorphic to Γ_{0} because Γ̂_{0} ∩ C = {1}. Consider now the residue maps

(3.12)  π_{1} : G̃_{0} → G̃_{0}/S̃_{1} ≅ U(G̃_{0}) · S̃_{2},  π_{2} : G̃_{0} → G̃_{0}/(U(G̃_{0}) · S̃_{2}) ≅ S̃_{1}.

The groups π_{1}(Γ̃_{0}), π_{2}(Γ̃_{0}) are Zariski-dense arithmetic subgroups of the ℚ-defined linear algebraic groups G̃_{0}/S̃_{1} ≅ U(G̃_{0}) · S̃_{2}, G̃_{0}/(U(G̃_{0}) · S̃_{2}) ≅ S̃_{1}. Since Γ̃_{0} is an arithmetic subgroup of the two latter groups it gets mapped injectively by the direct product of the residue maps π_{1} × π_{2} to a subgroup of finite index in π_{1}(Γ̃_{0}) × π_{2}(Γ̃_{0}). It is easy to see that π_{2}(Γ̃_{0}) is now finitely generated abelian and torsion-free. Choose an r ∈ ℕ ∪ {0} and an isomorphism π_{2}(Γ̃_{0}) ≅ ℤ^{r}. Put U_{1} := G_{a}^{r} and

(3.13)  H_{0} := (U(G̃_{0}) · S̃_{2}) × U_{1}.

From what was said above we infer that H_{0} is a reduced solvable ℚ-defined linear algebraic group. Embed now ℤ^{r} as a Zariski-dense arithmetic subgroup into U_{1}(ℚ). This shows that π_{1}(Γ̃_{0}) × π_{2}(Γ̃_{0}) is isomorphic to a Zariski-dense arithmetic subgroup of H_{0}(ℚ). Hence Γ_{0} is isomorphic to an arithmetic subgroup Γ_{1} contained in H_{0}(ℚ). The Zariski-closure \overline{Γ_{1}} of Γ_{1} in H_{0} is of of finite index in H_{0}. By Lemma 2.5 this implies that \overline{Γ_{1}} is reduced. We replace H_{0} by \overline{Γ_{1}} and have thus shown that Γ_{0} is isomorphic to the Zariski-dense arithmetic subgroup Γ_{1} contained in the ℚ-points of the reduced solvable linear algebraic group H_{0}. Let Δ be an extension group of Γ_{1} such that Δ is isomorphic to the Γ we startet with and such that Γ_{1} is normal and of finite index in Δ. We use now Lemma 3.3 to construct a solvable ℚ-defined linear algebraic group H such that Δ is isomorphic to a Zariski-dense subgroup of H(ℚ). We may further assume that H_{0} is a ℚ-defined subgroup of finite index in H. Let S be a maximal ℚ-defined d-subgroup of H. From Lemma 2.5 we infer that Z_{U}(S) is a finite ℚ-defined normal subgroup of H. The intersection Δ ∩ Z_{U}(S) is then a finite normal subgroup of Δ. If it is non-trivial it contains a non-trivial abelian characteristic subgroup. This subgroup is contained in the Fitting subgroup of Δ. Our assumption implies that Δ∩Z_{U}(S) = {1}. We replace H by its quotient H/Z_{U}(S) and have proved the result using Lemma 2.5.

Finally for this Section we give the proof of Theorem D.

Proof of Theorem D: Let Δ be a finite extension of the solvable arithmetic group Γ. It is easily shown that there is a subgroup Γ_{0} ≤ Γ which is normal and of finite index in Δ and which has torsion-free Fitting subgroup. We now apply Theorem 3.4 (to Γ_{0}) and find a reduced ℚ-defined linear algebraic group H and a Zariski-dense arithmetic subgroup Γ_{1} ≤ H(ℚ) which is isomorphic to Γ_{0}. Let Δ_{1} the finite extension group of Γ_{1} corresponding to the finite extension Δ of Γ_{0}. Put Γ_{2} := Γ_{1} ∩ H^{∘} and let Γ_{3} be a subgroup of finite index in Γ_{2} which is normal in Δ_{1}. Theorem D follows by using Proposition 3.1 for the extension Δ_{1} of Γ_{3} ≤ H^{∘}(ℚ).

4. Automorphism Groups

This section describes the automorphism group of certain arithmetic polycyclic groups. The main results are contained in Theorems 4.7 and 4.9. We first prove a structure theorem for the group of rational automorphisms of a reduced solvable-by-finite linear algebraic group. The group of rational automorphisms 𝔄𝔲𝔱(G) of a ℚ-defined linear algebraic group G is in general a locally algebraic group of type ALA (see [B-S], chapter 5) that is an extension of a ℚ-defined linear algebraic group by an arithmetic group. In case G is solvable-by-finite and reduced we can be much more specific. Before we can describe our result some preliminary considerations in which we identify 𝔄𝔲𝔱(G) with a ℚ-defined linear algebraic group are required. Our approach is similar to that of [B-S]. Let G be a reduced ℚ-defined solvable-by-finite linear algebraic group which is decomposed as in (2.1). Given Φ ∈ 𝔄𝔲𝔱(G) we have Φ(U(G)) = U(G) and let ρ_{U}(Φ) : U(G) → U(G) be the restriction of Φ to U(G). We also obtain an isomorphism Φ_{S} : U(G)\G → U(G)\G induced by Φ. Both ρ_{U}(Φ) and Φ_{S} are rational maps. We define

(4.1)

𝔄𝔲𝔱^{1}(G) := { Φ ∈ 𝔄𝔲𝔱(G)  Φ_{S} = id_{U(G)\G} },
𝔄_{S}(G) := { Φ ∈ 𝔄𝔲𝔱(G)  Φ(S) = S },
𝔄_{S}^{1}(G) := { Φ ∈ 𝔄_{S}(G)  Φ|_{S} = id_{S} }.

Consider the restriction homomorphism ρ_{U} : 𝔄𝔲𝔱(G) → 𝔄𝔲𝔱(U(G)). The automorphism group 𝔄𝔲𝔱(U(G)) is a ℚ-defined linear algebraic group since it can be identified via the exponential map with the automorphism group of the Lie algebra 𝔘(G). Let Φ ∈ 𝔄_{S} be in the kernel of ρ_{U}. Then Φ satisfies Φ(sus^{−1}) = sus^{−1} = Φ(s)uΦ(s)^{−1} for all s ∈ S and u ∈ U(G). Hence ρ_{U} is injective on 𝔄_{S}(G) because G is reduced. We shall show now that ρ_{U}(𝔄_{S}(G)) is a ℚ-defined algebraic subgroup of 𝔄𝔲𝔱(U(G)). Let ρ_{S} : 𝔄_{S}(G) → 𝔄𝔲𝔱(S) be the restriction homomorphism and let σ ∈ 𝔄𝔲𝔱(S) be given. Then ρ_{U}(ρ_{S}^{−1}(σ)) consists of those Ψ ∈ 𝔄𝔲𝔱(U(G)) which satisfy Ψ(sus^{−1}) = σ(s)Ψ(u)σ(s)^{−1} for all s ∈ S and u ∈ U(G). This shows that ρ_{U}(ρ_{S}^{−1}(σ)) is a ℚ-defined Zariski-closed subset of 𝔄𝔲𝔱(U(G)). We shall now show that ρ_{S}(𝔄_{S}(G)) is finite. Let Ŝ := α_{G}(S) be the image of S in Aut(𝔘(G)) and β : S → Ŝ the restriction of α_{G} to S. By Lemma 2.3 β is a ℚdefined rational isomorphism because G is reduced. An element σ ∈ 𝔄𝔲𝔱(S) gives now rise to the automorphism σ̂ := β^{−1} ∘ σ ∘ β ∈ 𝔄𝔲𝔱(Ŝ). To an element Ψ ∈ 𝔄𝔲𝔱(U(G)) corresponds via the exponential map to an automorphism Ψ̂ ∈ Aut(𝔘(G)). An element σ ∈ 𝔄𝔲𝔱(S) is in the image ρ_{S}(𝔄_{S}) if there is a Ψ ∈ 𝔄𝔲𝔱(U(G)) such that Ψ(sus^{−1}) = σ(s)Ψ(u)σ(s)^{−1} for all s ∈ S and u ∈ U(G). This implies that Ψ̂ is in the normalizer of Ŝ in Aut(𝔘(G)) and that σ̂(x) = Ψ̂xΨ̂^{−1} for all x ∈ Ŝ. Since the normalizer of Ŝ in Aut(𝔘(G)) induces (by conjugation) only a finite group of automorphisms of Ŝ we have proved that ρ_{S}(𝔄_{S}(G)) is finite. Since

(4.2)  ρ_{U}(𝔄_{S}(G)) = ⋃_{σ∈𝔄𝔲𝔱(S)} ρ_{U}(ρ_{S}^{−1}(σ))

we have shown that ρ_{U}(𝔄_{S}) is a ℚ-defined algebraic subgroup of 𝔄𝔲𝔱(U(G)). We further infer from the above that 𝔄_{S}^{1}(G) has finite index in 𝔄_{S}(G) and that ρ_{U}(𝔄_{S}^{1}(G)) is a ℚ-defined subgroup of finite index in ρ_{U}(𝔄_{S}(G)). By results of Malcev [Mc] the split extension U(G)⋊𝔄𝔲𝔱(U(G)) is a ℚ-defined algebraic group. We conclude that U(G) ⋊ ρ_{U}(𝔄_{S}(G)) also has this property. Consider now the homomorphism

(4.3)  Θ : U(G) ⋊ ρ_{U}(𝔄_{S}(G)) → 𝔄𝔲𝔱(G),  Θ : (u, ρ_{U}(Φ)) ↦ In_{u} ∘ Φ

where In_{u} is the inner automorphism corresponding to u ∈ U(G), that is In_{u}(g) = ugu^{−1} for all g ∈ G. The homomorphism Θ is surjective because (Proposition 2.1) all maximal d-subgroups of G are conjugate by elements of U(G). The kernel ker(Θ) of Θ is easily seen to be a ℚ-defined algebraic subgroup of U(G) ⋊ ρ_{U}(𝔄_{S}(G)). We have now identified 𝔄𝔲𝔱(G) with the ℂ-points of a ℚ-defined algebraic group. Note that Θ gives by restriction rise to a surjective homomorphism

(4.4)  Θ : U(G) ⋊ ρ_{U}(𝔄_{S}^{1}(G)) → 𝔄𝔲𝔱^{1}(G),  Θ : (u, ρ_{U}(Φ)) ↦ In_{u} ∘ Φ

This shows that 𝔄𝔲𝔱^{1}(G) is of finite index in 𝔄𝔲𝔱(G). By the same arguments as above we find that Θ also induces a group isomorphism

(4.5)  Θ : (U(G) ⋊ ρ_{U}(𝔄_{S}(G))/ ker(Θ))(ℚ) → 𝔄𝔲𝔱ℚ(G).

We have to introduce some notation concerning automorphisms of abstract groups. Let Γ be a group. We call an element φ ∈ Aut(Γ) F-central if φ(γ)γ^{−1} ∈ Z(Γ) for all γ ∈ Γ and if φ is the identity when restricted to the Fitting subgroup F(Γ). We define

(4.6)  Aut_{0}(Γ) := { φ ∈ Aut(Γ)  φ is F-central }.

As already mentioned Aut_{0}(Γ) is abelian and normal in Aut(Γ). We can now formulate the first main result of this section.

Proposition 4.1: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is separated in G. Then the following hold.

(i) 𝔄𝔲𝔱(G) is a ℚ-defined linear algebraic group. (ii) Let D : Aut(Γ) → 𝔄𝔲𝔱(G) be the homomorphism defined in Theorem C. Then the

image of D is an arithmetic subgroup of 𝔄𝔲𝔱(G) and the kernel of D is Aut_{0}(Γ).

While (i) is already established by what was said in the begining of this section we need some further considerations concerning arithmetic subgroups of reduced solvableby-finite algebraic groups before we can prove (ii).

Definition 4.2: Let G be a solvable-by-finite ℚ-defined linear algebraic group and S ≤ G a maximal ℚ-defined d-subgroup. An arithmetic subgroup Γ ≤ G(ℚ) is called S-split if Γ = U(G, Γ) · (Γ ∩ S). Two arithmetic subgroups Γ_{2} ≤ Γ_{1} are called S-related if pr_{S}(Γ_{2}) = pr_{S}(Γ_{1}) where pr_{S} : G → G/U(G) ≅ S is the projection homomorphism.

Suppose we are in the situation of the definition and Γ ≤ G is a Zariski-dense and S-split arithmetic subgroup. Then Γ ∩ S is Zariski-dense in S beause Γ ∩ S = pr_{S}(Γ). The proof of (ii) in Proposition 4.1 will first be carried through for S-split arithmetic groups. The following lemma will then allow us the reduction to this case.

Lemma 4.3: Let G be a solvable-by-finite ℚ-defined linear algebraic group and S ≤ G a maximal ℚ-defined d-subgroup. Let further Γ ≤ G(ℚ) be an arithmetic subgroup. Then there is an S-split arithmetic subgroup Γ̃ < G(ℚ) which contains Γ such that Γ and Γ̃ are S-related.

If Γ_{2} ≤ Γ_{1} are two groups we define

(4.7)  Aut(Γ_{1})_{Γ_{2}} := { φ ∈ Aut(Γ_{1})  φ(Γ_{2}) = Γ_{2} }

to be the stabilizer of Γ_{2} in Aut(Γ_{1}) and ρ_{Γ_{1}/Γ_{2}} : Aut(Γ_{1})_{Γ_{2}} → Aut(Γ_{2}) to be the homomorphism given by reduction. The following lemma is needed in the proof of Lemma 4.3 and also in a later argument.

Lemma 4.4: Let U be a unipotent ℚ-defined linear algebraic group and 𝔘 its Lie algebra. Let Ψ̂ ∈ Aut(𝔘) be the element which is associated to Ψ ∈ 𝔄𝔲𝔱(U) via the exponential map and

(4.8)  Δ̂ := { Ψ̂  Ψ ∈ Δ } ≤ Aut(𝔘)

the subgroup corresponding to a subgroup Δ ≤ 𝔄𝔲𝔱(U). Let Λ ≤ U(ℚ) be an arithmetic subgroup. We have:

(i) Let x_{1}, …, x_{n} be finitely many elements of U(G)(ℚ) then ⟨ Λ, x_{1}, …, x_{n} ⟩ contains Λ as a subgroup of finite index.

(ii) Let Δ ≤ 𝔄𝔲𝔱(U) be a subgroup so that Δ̂ ≤ Aut(𝔘) is an arithmetic subgroup contained in the group of ℚ-points of a ℚ-defined algebraic subgroup of Aut(𝔘). Then there is a subgroup Λ̃ ≤ U(ℚ) which contains Λ as a subgroup of finite index and which is left invariant by Δ.

(iii) Let Λ_{2} ≤ Λ_{1} ≤ U(ℚ) be two arithmetic subgroups. Then ρ_{Λ_{1}/Λ_{2}}(Aut(Λ_{1})_{Λ_{2}}) has

finite index in Aut(Λ_{2}). (iv) Let H ≤ 𝔄𝔲𝔱(U) be such that Ĥ is a ℚ-defined algebraic subgroup of Aut(𝔘). Put

(4.9)  Δ_{Λ} := { Φ ∈ H  Φ(Λ) = Λ }.

Then Δ̂_{Λ} is an arithmetic subgroup of Ĥ.

Lemma 4.4 is a straightforward exercise using the theory of arithmetic subgroups of unipotent algebraic groups, see [Se], Chapter 6. we skip the details.

Proof of Lemma 4.3: Let

(4.10)  pr_{U} : G → U(G),  pr_{S} : G → S

be the projection maps associated to the decomposition (2.1) of G. Then pr_{S}(Γ) is an arithmetic subgroup of S and U(G, Γ) · (Γ ∩ S) has finite index in Γ. Let γ_{1}, …, γ_{n} be coset representatives for U(G, Γ) · (Γ ∩ S) in Γ and x_{1} := pr_{U}(γ_{1}), …, x_{n} := pr_{U}(γ_{n}). By a combination of Lemma 4.4 (i), (ii) we find a subgroup Λ̃ ≤ U(ℚ) which contains U(G, Γ) as a subgroup of finite index and also the x_{1}, …, x_{n} and which is normalized by pr_{S}(Γ). Then Γ̃ := Λ̃ · pr_{S}(Γ) satisfies the requirements of the lemma.

Lemma 4.5: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ_{2} ≤ Γ_{1} ≤ G(ℚ) two S-related arithmetic subgroups. Then ρ_{Γ_{1}/Γ_{2}} is injective and its image has finite index in Aut(Γ_{2}).

Proof: We first prove that ρ_{Γ_{1}/Γ_{2}} is injective. Suppose that φ ∈ Aut(Γ_{1})_{Γ_{2}} is the identitity when restricted to Γ_{2}. Let γ ∈ U(G, Γ_{1}) and choose n ∈ ℕ so that γ^{n} ∈ U(G, Γ_{2}). Then φ(γ^{n}) = φ(γ)^{n} = γ^{n}. Since U(G) is uniquely divisible we get φ(γ) = γ. This shows that the restriction of φ to U(G, Γ_{1}) is the identity. Let now γ = us ∈ Γ_{1} with u ∈ U(G)(ℚ) and s ∈ S. Since Γ_{2} and Γ_{1} are S-related there is a v ∈ U(G)(ℚ) such that vs ∈ Γ_{2}. Note that uv^{−1} ∈ U(G, Γ_{1}). We then have

(4.11)  φ(us) = φ(uv^{−1} · vs) = φ(uv^{−1}) · φ(vs) = us.

This shows that ρ_{Γ_{1}/Γ_{2}} is injective. Next we prove that ρ_{Γ_{1}/Γ_{2}}(Aut(Γ_{1})_{Γ_{2}}) has finite index in Aut(Γ_{2}). By Lemma 2.6 we have a reduction homomorphism ρ_{U} : Aut(Γ_{1}) → Aut(U(G, Γ_{1})) which maps Aut(Γ_{1})_{Γ_{2}} into Aut(U(G, Γ_{1}))_{U(G,Γ_{2})}. Put

(4.12)  B := { φ ∈ Aut(Γ_{2})  ρ_{U}(φ) ∈ ρ_{U(G,Γ_{2})}(Aut(U(G, Γ_{1})_{U(G,Γ_{2})})) }.

By Lemma 4.4 (iii) ρ_{U(G,Γ_{1})/U(G,Γ_{2})}(Aut(U(G, Γ_{1}))_{U(G,Γ_{2})}) has finite index in the group Aut(U(G, Γ_{2})) and hence B has finite index in Aut(Γ_{2}). We prove that B = ρ_{Γ_{1}/Γ_{2}}(Aut(Γ_{1})_{Γ_{2}}). It is evident that B ≥ ρ_{Γ_{1}/Γ_{2}}(Aut(Γ_{1})_{Γ_{2}}), so let φ ∈ B. Take ψ ∈ Aut(U(G, Γ_{1}))_{U(G,Γ_{2})} such that ρ_{U(G,Γ_{1})/U(G,Γ_{2})}(ψ) = ρ_{U}(φ). As above let γ = us ∈ Γ_{1} with u ∈ U(G)(ℚ) and s ∈ S(ℚ). Since Γ_{2} and Γ_{1} are S-related there is a v ∈ U(G)(ℚ) such that vs ∈ Γ_{2}. Define a map φ_{1} : Γ_{1} → Γ_{1} by putting

(4.13)  φ_{1}(γ) = φ_{1}(us) = φ_{1}(uv^{−1} · vs) := ψ(uv^{−1}) · φ(vs).

We shall show that φ_{1} is in Aut(Γ_{1})_{Γ_{2}} and satisfies ρ_{Γ_{1}/Γ_{2}}(φ_{1}) = φ. Notice first of all that the value of φ_{1} does not depend on the choice of v. This follows by a straightforward computation. Next we show that φ_{1} is a homomorphism. Let γ_{1}, γ_{2} be two elements of Γ_{1} and γ_{1} = u_{1}s_{1}, γ_{2} = u_{2}s_{2} where u_{1}, u_{2} ∈ U(G)(ℚ) and s_{1}, s_{2} ∈ S. Choose v_{1}, v_{2} ∈ U(G)(ℚ) such that v_{1}s_{1}, v_{2}s_{2} ∈ Γ_{2}. Then γ_{1}γ_{2} = u_{1}s_{1}u_{2}s_{1}^{−1} · s_{1}s_{2} and v_{1}s_{1}v_{2}s_{1}^{−1} · s_{1}s_{2} ∈ Γ_{2}. By definition we find

(4.14)  φ_{1}(γ_{1}γ_{2}) = ψ(u_{1}s_{1}u_{2}s_{1}^{−1} · (v_{1}s_{1}v_{2}s_{1}^{−1})^{−1}) · φ(v_{1}s_{1}v_{2}s_{1}^{−1} · s_{1}s_{2}).

On the other hand we have φ_{1}(γ_{1})φ_{1}(γ_{2}) = ψ(u_{1}v_{1}^{−1})φ(v_{1}s_{1}) · ψ(u_{2}v_{2}^{−1})φ(v_{2}s_{2}). A small computation shows that φ_{1}(γ_{1})φ_{1}(γ_{2}) = φ_{1}(γ_{1}γ_{2}) is implied by

(4.15)  φ(v_{1}s_{1}) ψ(u_{2}v_{2}^{−1}) φ(v_{1}s_{1})^{−1} = ψ(v_{1}s_{1} u_{2}v_{2}^{−1} (v_{1}s_{1})^{−1}).

Let x be the left-hand side and y be the right-hand side of (4.15). Choose a ℓ ∈ ℕ such that (u_{2}v_{2}^{−1})^{ℓ} ∈ U(G, Γ_{2}). We infer from (4.15) that x^{ℓ} = y^{ℓ}. Since U(G)(ℚ) is uniquely divisible we have x = y and φ_{1} is a homomorphism. Next we show that φ_{1} is injective. Let γ = us ∈ Γ_{1} and vs ∈ Γ_{2} (u, v ∈ U(G)(ℚ), s ∈ S) with φ_{1}(γ) = 1. We then have 1 = φ_{1}(γ) = ψ(uv^{−1})φ(vs). It follows that φ(vs) ∈ U(G, Γ_{2}) which implies vs ∈ U(G, Γ_{2}) and also s = 1. We infer that γ ∈ U(G, Γ_{1}) and φ_{1}(γ) = ψ(γ). This implies γ = 1 and hence the injectivity of φ_{1}. follows. Next we show that φ_{1} is surjective. Let γ = us ∈ Γ_{1} and vs ∈ Γ_{2} (u, v ∈ U(G)(ℚ), s ∈ S). Define then γ′ := ψ^{−1}(uv^{−1})φ^{−1}(vs). We show that φ_{1}(γ′) = γ. Let v′ ∈ U(G)(ℚ) and s′ ∈ S be chosen such that φ^{−1}(vs) = v′s′. Define u′ := ψ^{−1}(uv^{−1})v′ ∈ U(G)(ℚ), then γ′ = u′s′ and φ_{1}(γ′) = ψ(u′v^{′−1})φ(v′s′) = γ follows. From our definition of φ_{1} it is evident that φ_{1} is in Aut(Γ_{1})_{Γ_{2}} and that ρ_{Γ_{1}/Γ_{2}}(φ_{1}) = φ is satisfied.

Lemma 4.5 is rather surprising since it is hopelessly false in the general situation of two arithmetic groups Γ_{2} ≤ Γ_{1}.

Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group which is decomposed as in (2.1) and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup. We define the following subgroup of 𝔄_{S}^{1}(G):

(4.16)  𝔄_{S}^{1}(G, Γ) := { Φ ∈ 𝔄_{S}^{1}(G)  Φ(U(G, Γ)) = U(G, Γ) }.

We infer from Lemma 2.6 that F(Γ) = U(G, Γ). Given a φ ∈ Aut(Γ) we then have φ(U(G, Γ)) = U(G, Γ). Hence φ induces an isomorphism φ_{S} : U(G, Γ)\Γ → U(G, Γ)\Γ. We define the following subgroups of Aut(Γ):

(4.17)

Aut^{1}(Γ) : = { φ ∈ Aut(Γ)  φ_{S} = id_{U(G,Γ)\Γ} },
In_{U}(Γ) : = { In_{u} ∈ Aut(Γ)  u ∈ U(G, Γ) }
A_{S}^{1}(G, Γ) : = { Φ|_{Γ}  Φ ∈ 𝔄_{S}^{1}(G, Γ) } ≤ Aut^{1}(Γ)

The following is a simple consequence of our definitions.

Lemma 4.6: Let G be a ℚ-defined reduced solvable-by-finite linear algebraic group and Γ ≤ G a Zariski-dense arithmetic subgroup which is separated in G, then the following hold:

(i) Let D : Aut(Γ) → 𝔄𝔲𝔱(G) be the homomorphism of Theorem C. Then Aut^{1}(Γ) is the preimage of 𝔄𝔲𝔱^{1}(G) under D.

(ii) Aut^{1}(Γ) is of finite index in Aut(Γ). (iii) 𝔄_{S}^{1}(G, Γ) is an arithmetic subgroup of 𝔄_{S}^{1}(G). (iv) 𝔄_{S}^{1}(G, Γ) is by restriction isomorphic to A_{S}^{1}(G, Γ).

Proof: (i): A φ ∈ Aut(Γ) is in Aut^{1}(Γ) if and only if φ(γ)·γ^{−1} ∈ U(G, Γ) for all γ ∈ Γ. Correspondingly a Φ ∈ 𝔄𝔲𝔱(G) is in 𝔄𝔲𝔱^{1}(G) if and only if Φ(g) · g^{−1} ∈ U(G) for all g ∈ G. The statement then follows from the properties of D formulated in Theorem C. (ii): follows from (i) and the fact that 𝔄𝔲𝔱^{1}(G) has finite index in 𝔄𝔲𝔱(G). (iii): follows from Lemma 4.4 (iv). (iv): holds since Γ is Zariski-dense in G.

If G is a solvable-by-finite linear algebraic group decomposed as in (2.1) it follows from Proposition 2.1 that

(4.18)  𝔄𝔲𝔱(G) = In_{U}(G) · 𝔄_{S}(G).

The following theorem contains a corresponding result but for arithmetic groups.

Theorem 4.7: Let G be a ℚ-defined solvable-by-finite and reduced linear algebraic group and Γ ≤ G an arithmetic subgroup which is separated in G, then the subgroup A(Γ) := Aut_{0}(Γ) · In_{U}(Γ) · A_{S}^{1}(G, Γ) is of finite index in Aut(Γ).

Before we can prove Theorem 4.7 we have to establish the following finiteness result.

Proposition 4.8: Let G = U ⋊ S be a ℚ-defined linear algebraic group where U is unipotent and S is a finite extension of a torus. Let Γ ≤ G(ℚ) be an arithmetic subgroup which can be decomposed as Γ = Γ_{U} ⋊ Γ_{S} where Γ_{U} ≤ U(ℚ) and Γ_{S} ≤ S(ℚ) are arithmetic subgroups and where Γ_{S} is Zariski-dense in S. Assume further that Γ is separated in G. Then the set

(4.19)  C(G, Γ) := { Δ ≤ Γ  Δ = uΓ_{S}u^{−1} for some u ∈ U(ℚ) }

falls into finitely many Γ_{U}-conjugacy classes.

Proof: We first consider the case when U is abelian. Let U_{1} ≤ U be the centralizer of S in U. Since S acts by semi-simple transformations on U there is a ℚ-defined, S-invariant subspace U_{2} ≤ U complementary to U_{1} in U. We also fix a finite system γ_{1}, …, γ_{n} of Γ_{S}. A derivation of Γ_{S} in U_{2} is a map c : Γ_{S} → U_{2} satisfying c(γ_{1}γ_{2}) = c(γ_{1})γ_{1}c(γ_{2})γ_{1}^{−1}. A derivation c is called inner if there is a v ∈ U_{2} with c(γ) = vγv^{−1}γ^{−1} for all γ ∈ Γ_{S}. A derivation is determined by its values on the generators γ_{1}, …, γ_{n} of Γ_{S}. We call an n-tuple (u_{1}, …, u_{n}) ∈ U_{2}^{n} admissible if the is a derivation c : Γ_{S} → U_{2} with c(γ_{1}) = u_{1}, …, c(γ_{n}) = u_{n}. Taking a ℚ-basis of U(ℚ) it is easy to see that (u_{1}, …, u_{n}) ∈ U_{2}^{n} is admissible if and only if certain polynomial equations with rational coefficients between the coordinates of the (u_{1}, …, u_{n}) are satisfied. Hence

(4.20)  B := { (u_{1}, …, u_{n}) ∈ U_{2}^{n}  (u_{1}, …, u_{n}) is admissible }

is a ℚ-defined Zariski-closed subset of U_{2}^{n}. Consider the map

(4.21)  I : U_{2} → B,  I(v) := (vγ_{1}v^{−1}γ_{1}^{−1}, …, vγ_{1}v^{−1}γ_{1}^{−1})  (v ∈ U_{2}).

It is easy to see that I is a ℚ-defined, injective rational map (note that Γ_{S} is Zariskidense in S). Given an element v ∈ U(ℚ) we write v_{2} for its component in U_{2}. We shall be able to use the map I because of the following evident implications.

(4.22)  vΓ_{S}v^{−1} ≤ Γ ⟺ v_{2}Γ_{S}v_{2}^{−1} ≤ Γ ⟺ I(v_{2}) ∈ Γ_{U}^{n}

There is furthermore a ℚ-defined rational action of U_{2} on B given by

(4.23)  v · (u_{1}, …, u_{n}) := (u_{1}vγ_{1}v^{−1}γ_{1}^{−1}, …, u_{n}vγ_{n}v^{−1}γ_{n}^{−1})

for v ∈ U_{2} and (u_{1}, …, u_{n}) ∈ B. Adapting the proof of Lemma 2.7 appropriately we have H^{1}(Γ_{S}, U_{2}) = {0}. This means that every derivation of Γ_{S} in U_{2} is inner. This in turn implies that the map I is also bijective and that the action of U_{2} on B is transitive. We also infer that B is connected and smooth. Hence I is a ℚ-defined rational isomorphism. Let now 𝔸 be the ring of adeles of ℚ and 𝔸_{f} be the subring of finite adeles, that 𝔸_{f} is 𝔸 with the component consistng of the real numbers left off. Let 𝔸_{f}^{∞} be the ring of finite integral adeles which is a compact and open subring of 𝔸_{f}. See [P-R], Chapter 5 for more details. It is for example easy to see that there is an open compact subgroup V ≤ U_{2}(𝔸_{f}^{∞}) such that V ∩ U_{2}(ℚ) = Γ_{U}. By [P-R] Proposition 5.2 we infer that I : U_{2} → B(𝔸_{f}) is a homeomorphism. Hence I^{−1}(V^{n} ∩ B(𝔸_{f})) is compact and falls into finitely many orbits under translation by elements of V. Since we have V ∩ U_{2}(ℚ) = Γ_{U} the set I^{−1}(V^{n} ∩ B(ℚ)) = I^{−1}(Γ_{U}^{n} ∩ B) falls into finitely many orbits under translation by elements of Γ_{U}. This proves the result in case U is abelian.

In the general case we use an induction on the dimension of U. First of all we define Ū := Z(U)\U and Ḡ := Z(U)\G ≅ Ū ⋊S, We write π : G → Ḡ for the projection map. We shall call two subgrups Δ_{1}, Δ_{2} congruent (Δ_{1} ≡ Δ_{2}) if and only if π(Δ_{1}) = π(Δ_{2}). Then π(Γ) = π(Γ_{U}) ⋊ π(Γ_{S}) satisfies the hypotheses of the proposition. By induction this implies that there are finitely many w_{1}, …, w_{k} ∈ U(ℚ) so that w_{i}Γ_{S}w_{i}^{−1} ≤ Γ for i = 1, …, k and so that every subgroup uΓ_{S}u^{−1} ≤ Γ (u ∈ U(ℚ)) can be conjugated by an element of Γ_{U} to satisfy uΓ_{S}u^{−1} ≡ w_{i}Γ_{S}w_{i}^{−1} for some i ∈ {1, …, k}. Put

(4.24)  M_{i} := \{ Δ ≤ Γ  \begin{matrix}Δ = uΓ_{S}u^{−1} for some u ∈ U(ℚ), \\ Δ ≡ w_{i}Γ_{S}w_{i}^{−1} \end{matrix} \}

for i = 1, …, k. It suffices to show that each of the M_{i} intersects only finitely many Γ_{U}-conjugacy classes. Before we do this we have to add some facts about centralizers. If Σ ≤ S is a subgroup put

(4.25)

Z(U, Σ) : = { u ∈ U(ℚ)  σuσ^{−1} = u for all σ ∈ Σ },
Z̄(U, Σ) : = { u ∈ U(ℚ)  π(σuσ^{−1}) = π(u) for all σ ∈ Σ }.

Since Γ_{S} is Zariski-dense in S we have Z(U, Γ_{S}) = Z(U, S) and Z̄(U, Γ_{S}) = Z̄(U, S). We shall prove now that

(4.26)  Z̄(U, S) = Z(U)(ℚ) · Z(U, S).

This can be done by transfering the question to the Lie algebra 𝔘 of U. In fact, similar to (2.5) we use the exponential map to establish an action of S on 𝔘. Then there is an S-invariant complement to the center of 𝔘. Decomposing exp^{−1}(u) (u ∈ Z̄(U, S)) accordingly and applying exp again we find the decomposition of u in (4.26). The details are straightforward, we skip them. Finally we show that each of the M_{i} in (4.24) intersects only finitely many Γ_{U}- conjugacy classes. We replace Γ_{S} by w_{i}Γ_{S}w_{i}^{−1} and S by w_{i}Sw_{i}^{−1}. Notice that Γ = Γ_{U} · w_{i}Γ_{S}w_{i}^{−1}. If u ∈ M_{i} then uw_{i}^{−1} ∈ Z̄(U, w_{i}Sw_{i}^{−1}) and according to (4.26) (applied to w_{i}Sw_{i}^{−1}) there is a z_{u} ∈ Z(U)(ℚ) so that

(4.27)  z_{u} · w_{i}Γ_{S}w_{i}^{−1} · z_{u}^{−1} = uw_{i}^{−1} · w_{i}Γ_{S}w_{i}^{−1} · w_{i}u^{−1}.

To finish we apply the first part of the proof to the groups Γ_{U} ∩ Z(U) ≤ Z(U), w_{i}Γ_{S}w_{i}^{−1} ≤ w_{i}Sw_{i}^{−1} which are contained in Z(U) ⋊ w_{i}Sw_{i}^{−1}.

The proof just given is reminiscent of the proof of Theorem 5.10 from [P-R].

Proof of Theorem 4.7: Clearly A(Γ) ≤ Aut^{1}(Γ) and by Lemma 4.6 (ii) we need only to prove that A(Γ) is of finite index in Aut^{1}(Γ). We shall do this first for S-split groups Γ and afterwards consider the general case.

Let Γ be S-split. Let π : G(ℚ) → Z(G)(ℚ)\G(ℚ) be the projection map and call two subgroups Δ_{1}, Δ_{2} ≤ G(ℚ) congruent (Δ_{1} ∼ Δ_{2}) if π(Δ_{1}) = π(Δ_{2}). Consider the set

(4.28)  C_{Z} := \{ Δ ≤ Γ  \begin{matrix}Δ ∩ Z(G)(ℚ) = \{1\}, \\ Δ ∼ v(Γ ∩ S)v^{−1} for some v ∈ U(G)(ℚ) \end{matrix} \} / ∼ .

The subgroup Γ ∩ S represents a class in C_{Z} Using Theorem C it is straightforward to see that Aut^{1}(Γ) acts on C_{Z} where a φ ∈ Aut^{1}(Γ) sends a class of C_{Z} represented by Δ ≤ Γ to the class represented by φ(Δ). The group U(G, Γ) acts on C_{Z} by conjugation. The map

(4.29)  C_{Z} → C(Z(G)\G, Z(Γ)\Γ),  Δ ↦ π(Δ)

is well-defined, U(G, Γ)-equivariant and injective. Proposition 4.8 then implies that U(G, Γ)\C_{Z} is finite. Let B be the stabilizer of the class of Γ ∩ S in this set. Then B has finite index in Aut^{1}(Γ). Assume that φ ∈ B. Then there is a w ∈ U(G, Γ) such that π(φ(Γ ∩ S)) = π(w(Γ ∩ S)w^{−1}). Put φ_{1} := In_{w}^{−1} ∘ φ. Then π(φ_{1}(Γ ∩ S)) = π(Γ ∩ S). Since φ_{1} ∈ Aut^{1}(Γ) we find that for every s ∈ Γ ∩ S there is a z(s) ∈ Z(Γ) such that φ(s) = z(s) · s. It is easy to see that

(4.30)  ψ(γ) := u z(s) s  (γ = u · s ∈ Γ, u ∈ U(G, Γ), s ∈ Γ ∩ S)

defines an element of Aut_{0}(Γ). Put φ_{2} := ψ^{−1} · φ_{1}. We then have φ_{2}(s) = s for all s ∈ Γ ∩ S. By an argument similar to that of Lemma 2.9 we infer that D_{φ_{2}} satisfies D_{φ_{2}}(s) = s for all s ∈ S. This implies that φ_{2} is the restriction of D_{φ_{2}} to Γ and hence is in A_{S}^{1}(Γ). All of this shows that B ≤ A(Γ) and the result follows for S-split groups Γ. Let now Γ be a general Zariski-dense arithmetic subgroup of G which is separated in G. Using Lemma 4.3 we choose an arithmetic group Γ̃ ≥ Γ which is S-related to Γ. It is easy to see that Γ̃ is also Zariski-dense and separated in G. Using Lemma 4.5 it is straightforward to deduce that A(Γ) is of finite index in Aut(Γ) from the fact that the index of A(Γ̃) in Aut(Γ̃) is finite.

Proof of Proposition 4.1: We have established (i) already in the remarks preceeding our proposition. Let us prove next that ker(D) = Aut_{0}(Γ). From the statement of Theorem C we conclude that ker(D) ≤ Aut_{0}(Γ). Assume that φ is an element of Aut_{0}(Γ). Then there is a homomorphism v : Γ → Z(Γ) which is trivial on F(Γ) such that φ(γ) = v(γ) · γ for all γ ∈ Γ. It is then evident that D_{φ} and id_{G} satisfy the hypotheses of Lemma 2.9. We conclude that D_{φ} = id_{G}. It remains to prove that D(Aut(Γ)) is an arithmetic subgroup of 𝔄𝔲𝔱(G). We first do this in case Γ is S-split. Similar to the reduction map ρ_{U} : 𝔄𝔲𝔱(G) → 𝔄𝔲𝔱(U(G)) introduced above there is the reduction map ρ_{U} : Aut(Γ) → Aut(U(G, Γ)). It is injective when restricted to A_{S}^{1}(G, Γ). Hence we may as in (4.3) define a map

(4.31)  Θ : U(G, Γ) ⋊ ρ_{U}(A_{S}^{1}(G, Γ)) → Aut(Γ),  Θ : (u, ρ_{U}(φ)) ↦ In_{u} ∘ φ.

We furthermore have a natural homomorphism

(4.32)  μ : Aut(U(G, Γ)) → 𝔄𝔲𝔱(U(G)).

The map μ is injective since μ(φ) is an extension of φ ∈ Aut(U(G, Γ)) to a rational automorphism of U(G). It is straightforward to check that we obtain a commutative diagram

(4.33)  U(G, Γ) ⋊ ρ_{U}(A_{S}^{1}(G, Γ)) ⟶^{Θ} Aut(Γ) ↓^{id×μ}  ↓^{D} U(G) ⋊ ρ_{U}(𝔄_{S}^{1}(G, Γ)) ⟶^{Θ} 𝔄𝔲𝔱(G) .

By Lemma 4.6 the image of the left-upper corner gets mapped to an arithmetic subgroup of the left-lower corner. Since the lower Θ is a ℚ-defined rational map the image of the left-upper corner is an arithmetic subgroup of 𝔄𝔲𝔱(G). By Theorem 4.7 D(Aut(Γ)) contains this arithmetic group as a subgroup of finite index and hence is arithmetic. This finishes the proof of Proposition 4.1 in case of an S-split arithmetic group. To prove the general case let Γ_{1} ≥ Γ be a group which contains Γ of finite index and which is S-split and S-related to Γ. The existence of Γ_{1} is garantied by Lemma 4.3. We obtain a commutative diagram:

(4.34)  Aut(Γ_{1})_{Γ} ⟶^{D} 𝔄𝔲𝔱(G) ↓  ↓^{Id}_{G} Aut(Γ) ⟶^{D} 𝔄𝔲𝔱(G) .

where the homomorphism down-left is given by restriction. Lemma 4.5 establishes then the general result.

Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is separated in G. We shall construct now a ℚ-defined linear algebraic group 𝔄𝔲𝔱(Γ) which will contain Aut^{1}(Γ) as an arithmetic subgroup. Let (Γ/F(Γ))^{ab} be the abilianised quotient of Γ/F(Γ) and r := dim_{ℚ} ℚ ⊗_{ℤ} (Γ/F(Γ))^{ab}. Let 𝔄𝔲𝔱(G) act diagonally on Z(G)^{r} and define

(4.35)  𝔄𝔲𝔱(Γ) := Z(G)^{r} ⋊ 𝔄𝔲𝔱(G)

where 𝔄𝔲𝔱(G) acts diagonally on Z(G)^{r}. We fix elements s_{1}, …, s_{r} ∈ Γ such that their images in ℚ ⊗_{ℤ} (Γ/F(Γ))^{ab} are a ℚ-basis. We define

(4.36)  Σ : Aut^{1}(Γ) → 𝔄𝔲𝔱(Γ),  Σ : φ ↦ (u_{φ}(s_{1}), …, u_{φ}(s_{r}); D_{φ})

where u_{φ} and D_{φ} are as in Theorem C. For φ, ψ ∈ Aut(Γ) we find from Theorem C

(4.37)  u_{φ∘ψ}(γ) = D_{φ}(u_{ψ}(γ)) · u_{φ}(ψ(γ)).

If ψ is contained in Aut^{1}(Γ) then ψ(γ) · γ^{−1} ∈ U(G, Γ) for all γ ∈ Γ. This implies that Σ is a homomorphism.

Theorem 4.9: Let G be a reduced solvable-by-finite ℚ-defined linear algebraic group and Γ ≤ G(ℚ) a Zariski-dense arithmetic subgroup which is separated in G. Then the following hold.

(i) 𝔄𝔲𝔱(Γ) is a ℚ-defined linear algebraic group. (ii) The map Σ defined in (4.36) is injective and Σ(Aut^{1}(Γ)) is an arithmetic subgroup

of 𝔄𝔲𝔱(Γ).

Proof: (i): is obvious. (ii): The injectivity of Σ follows from Proposition 4.1 (ii). By Proposition 4.1 (ii) and Lemma 4.6 (ii) D(Aut^{1}(Γ)) is a arithmetic subgroup of 𝔄𝔲𝔱(G). Since Σ(Aut_{0}(Γ)) evidently is an arithmetic subgroup of Z(G)^{r} the result follows.

Proof of Theorem E: Theorem 4.9 together with Lemma 4.6 shows that Aut(Γ) contains an arithmetic subgroup of finite index. This does not apriori imply that Aut(Γ) is arithmetic (see [G-P1], [G-P2]). Let τ ∈ Aut(Γ). We shall construct a ℚ-defined rational homomorphism M_{τ} : 𝔄𝔲𝔱(Γ) → 𝔄𝔲𝔱(Γ) such that Σ(τφτ^{−1}) = M_{τ} (Σ(φ)) for all φ ∈ Aut^{1}(Γ). Lemma 3.3 then shows that Aut(Γ) is arithmetic. We deduce from (4.37) the formula

(4.38)  \begin{aligned}u_{τφτ^{−1}}(γ) & = D_{τ} (D_{φ}(u_{τ^{−1}}(γ))) · D_{τ} (u_{φ}(τ^{−1}(γ))) · u_{τ} (φτ^{−1}(γ)) \\ & = D_{τ} (D_{φ}(u_{τ^{−1}}(γ))) · D_{τ} (u_{φ}(τ^{−1}(γ))) · u_{τ} (τ^{−1}(γ)) \end{aligned}

For all φ ∈ Aut^{1}(Γ). We define

(4.39) 𝔴(τ) := (u_{τ^{−1}}(s_{1}), …, u_{τ^{−1}}(s_{r})), 𝔴̂(τ) := (u_{τ} (τ^{−1}(s_{1})), …, u_{τ} (τ^{−1}(s_{r}))).

These are two elements of Z(G)^{r}(ℚ). The automorphism τ^{−1} of Γ induces a ℚ-linear automorphism of the ℚ vector space ℚ ⊗_{ℤ} (Γ/F(Γ))^{ab} which we also denote by τ^{−1}. Hence there are rational numbers a_{ij} for 1 ≤ i, j ≤ r such that

(4.40)  τ^{−1}(ŝ_{i}) = ∑_{j=1}^{r} a_{ji} ŝ_{j}  (i = 1, …, r)

where ŝ_{i} stands for the image of s_{i} in ℚ ⊗_{ℤ} (Γ/F(Γ))^{ab}. Define now a ℚ-linear automorphism ℓ_{τ} : Z(G)^{r} → Z(G)^{r} by

(4.41)  ℓ_{τ} (u_{1}, …, u_{r}) := (∑_{j=1}^{r} a_{j1} u_{j}, …, ∑_{j=1}^{r} a_{jr} u_{j}) .

Here we have identified Z(G)^{r} with a ℂ-vector space. The automorphism ℓ_{τ} : Z(G)^{r} → Z(G)^{r} satisfies

(4.42)  ℓ_{τ} ((u_{φ}(s_{1}), …, u_{φ}(s_{r}))) = (u_{φ}(τ^{−1}(s_{1})), …, u_{φ}(τ^{−1}(s_{r})))

for all φ ∈ Aut^{1}(Γ). Define

(4.43)  M_{τ} ((𝔲; g)) := (D_{τ} ∘ g(𝔴(τ)) · D_{τ} (ℓ_{τ} (𝔲)) · 𝔴̂(τ); D_{τ} gD_{τ}^{−1})

for 𝔲 ∈ Z(G)^{r} and g ∈ 𝔄𝔲𝔱(G). The map M_{τ} : 𝔄𝔲𝔱(Γ) → 𝔄𝔲𝔱(Γ) satisfies Σ(τφτ^{−1}) = M_{τ} (Σ(φ)) for all φ ∈ Aut^{1}(Γ) by (4.38). It is obviously rational and defined over ℚ in the coordinates we have already chosen in Z(G)^{r} and 𝔄𝔲𝔱(G). Noticing D_{τ} (𝔴(τ)) · 𝔴̂(τ) = 1 it is straightforward to show that M_{τ} is also a group homomorphism.

Proof of Theorem E’: This result follows by first using Theorem 3.4 then Lemma 2.7 (iv) and finally Theorem E.

5. Non-arithmetic Polycyclic Groups

In this section we prove Theorem E. We start off with an arithmetic construction of some polycyclic groups which will supply the non-arithmetic examples needed for the proof of Theorem G. Let K be a number field and O_{K} its ring of integers. Assume that K has r_{1} real and 2r_{2} complex embeddings. Hence K is of degree n = r_{1} + 2r_{2} over ℚ. We write O_{K}^{∗} for the group of units of O_{K}. By Dirichlet’s unit theorem we have an isomorphism O_{K}^{∗} ≅ C × ℤ^{r_{K}} where C is a finite cyclic group and r_{K} = r_{1} + r_{2} − 1. We write O_{K}^{+} for the additive group of O_{K}. The group O_{K}^{∗} acts by multiplication on O_{K}^{+}. Let O_{K}^{+} ⋊ O_{K}^{∗} be the corresponding split extension. Let now G_{a}, G_{m} be the additive and multiplicative group considered as linear algebraic groups over K. Then G_{m} acts by multiplication on G_{a} and we again write G_{a}⋊G_{m} for the corresponding split extension. Let G_{K} := R_{K/ℚ}(G_{a} ⋊ G_{m}) = R_{K/ℚ}(G_{a}) ⋊ R_{K/ℚ}(G_{m}) be the base field restriction of this group to ℚ. Then G_{K} is a reduced solvable linear algebraic group defined over ℚ and have natural identifications

(5.1)  G_{K}(ℚ) = K^{+} ⋊ K^{∗},  G_{K}(ℤ) = O_{K}^{+} ⋊ O_{K}^{∗}.

Let A ≤ O_{K}^{∗} be a subgroup. Put

(5.2)  Γ_{K}(A) := O_{K}^{+} ⋊ A ≤ O_{K}^{+} ⋊ O_{K}^{∗}.

The group Γ_{K}(A) is polycyclic and its Fitting subgroup F(Γ_{K}(A)) is equal to the additive group of O_{K} which is naturally embedded in Γ_{K}(A). Hence we have F(Γ_{K}(A)) ≅ ℤ^{n}. The following lemma characterizes the subgroups A ≤ O_{K}^{∗} for which Γ_{K}(A) is an arithmetic group.

Proposition 5.1: Let K be an algebraic number field of degree n over ℚ and Then the group Γ_{K}(A) is arithmetic if and only if there is a ℚ-defined algebraic subgroup S ≤ R_{K/ℚ}(G_{m}) such that A is contained as a Zariski-dense arithmetic subgroup in S.

Proof: Let S ≤ R_{K/ℚ}(G_{m}) be a ℚ-defined algebraic subgroup. If A ≤ S is commensurable with S(ℤ) then Γ_{K}(A) is obviously an arithmetic subgroup of the ℚ-defined linear algebraic group R_{K/ℚ}(G_{a}) ⋊ S. Suppose now that the group Γ_{K}(A) constructed above is arithmetic. We then find by application of Theorem 3.4 a reduced solvable ℚ-defined linear algebraic group H so that there is an isomorphism ψ : Γ_{K}(A) → Γ where Γ is a Zariski-dense arithmetic subgroup of H(ℚ). We consider O_{K}^{+} and A to be identified with the corresponding subgroups of Γ_{K}(A). By Lemma 2.6 the Fitting subgroup of Γ is Γ ∩ U(H). We infer that ψ(O_{K}^{+}) = Γ ∩ U(H). Assume from now on that A ≠ {1}. It is then easy to see that A is a maximal commutative subgroup of Γ_{K}(A). Let S_{0} be the Zariski-closure of ψ(A) in H. This is a ℚ-defined commutative algebraic subgroup of H. It follows that Γ ∩ S_{1} is an arithmetic subgroup of S_{0}. Since ψ(A) is maximal commutative in Γ we have Γ ∩ S_{0} = ψ(A). This implies that ψ(A) is a Zariski-dense arithmetic subgroup of S_{0}(ℚ). From Lemma 2.6 and the remarks above we find that the Lie algebra 𝔘(H)(ℚ) is abelian of dimension n over ℚ and exp^{−1}(Γ ∩ U(H)) = exp^{−1}(ψ(O_{K}^{+})) is an abelian subgroup of 𝔘(H)(ℚ) which contains a ℚ-basis of this space. Choose a ℚ-basis B inside O_{K}^{+} of the ℚ-vector space K. This provides us with a rational ℚ-defined injective representation Λ : R_{K/ℚ}(G_{m}) → GL(n, ℂ). For k ∈ K^{∗} its image Λ(k) is the matrix corresponding to the multiplication by k. Note that Λ even gives a rational ℚ-defined isomorphism between R_{K/ℚ}(G_{m}) and the Zariski-closure of Λ(K^{∗}) ≤ GL(n, ℂ). The homomorphism exp^{−1} ∘ψ : O_{K}^{+} → 𝔘(H)(ℚ) can be extended to a ℚ-linear isomorphism Θ : K → 𝔘(H)(ℚ). Taking the basis Θ(B) for 𝔘(H)(ℚ) we obtain the rational ℚdefined representation α_{H} : H → GL(n, ℂ) (see (2.5)). The representations Λ, α_{H} evidently satisfy

(5.3)  Θ^{−1} ∘ α_{H}(ψ(a)) ∘ Θ = Λ(a)

for all a ∈ A. Put S := Θ^{−1}α_{H}(S_{0})Θ. The subgroup S < GL(n, ℂ) is defined over ℚ. The image of ψ(A) in S is Zariski-dense and by a result of Borel [B2] also arithmetic in S. Using (5.3) we see that Λ(A) is a Zariski-dense arithmetic subgroup S. This proves the proposition.

Before we proceed we report on some facts from the theory of commutative d-groups. Let \overline{ℚ} be the algebraic closure of ℚ in ℂ and 𝔾 its Galois group. For a ℚ-defined commutative d-group S let X(S) be the group of (\overline{ℚ}-defined) characters of S. If φ : S_{1} → S_{2} is a ℚ-defined rational homomorphism between two commotative ℚ-defined d-groups S_{1}, S_{2} we write X(φ) : X(S_{2}) → X(S_{1}) for the corresponding homomorphism between the character modules. All of this gives a contravariant functorial equivalence between the category of ℚ-defined d-groups and the category of continuos 𝔾-modules which are finitely generated as ℤ-modules. We write S_{1} ∼ S_{2} if S_{1}, S_{2} are isogenous. Then S_{1} ∼ S_{2} if and only if ℚ⊗_{ℤ} X(S_{1}) and ℚ⊗_{ℤ} X(S_{2}) are isomorphic as 𝔾-modules. For a commutative d-group S let dim(S) its dimension as an algebraic group. We have dim(S) = dim_{ℚ}(ℚ ⊗_{ℤ} X(S). Let L ⊆ ℂ be a subfield then S is called L-split if every element of X(S) is defined over L. It is easy to see that there is a minimal splitting field L_{S} for S. That is a subfield L_{S} ⊆ \overline{ℚ} such that S is L_{S}-split and such that L_{S} ⊆ L for every other field L ⊆ ℂ which has the property that S is L-split. Let K be a number field. We have the following equality of 𝔾-modules

(5.4)  X(R_{K/ℚ}(G_{m})) = ℤ[𝔾/𝕌(K)]

where 𝕌(K) ≤ 𝔾 is the open subgroup of finite index in 𝔾 with fixed-field K and ℤ[𝔾/𝕌(K)] is the corresponding permutation module. The d-group R_{K/ℚ}(G_{m}) is split over the Galois-closure K̂ of K in \overline{ℚ}. The norm map N_{K/ℚ} : K → ℚ defines a ℚdefined character of R_{K/ℚ}(G_{m}). We define R_{K/ℚ}^{1}(G_{m}) to be its kernel. The character module of R_{K/ℚ}^{1}(G_{m}) is isomorphic to ℤ[𝔾/𝕌(K)]/ℤ. We have R_{K/ℚ}^{1}(G_{m})(ℤ) = O_{K}^{1} where O_{K}^{1} stands for the group of units of norm 1 in O_{K}. In the following we show that Proposition 5.1 can really be used to construct nonarithmetic polycyclic groups.

Proposition 5.2: Let K be a totally real number field with [K : ℚ] = n ≥ 3. Then the following hold:

(i) There are up to commensurability only finitely many infinite cyclic subgroups A ≤ O_{K}^{∗} such that Γ_{K}(A) is arithmetic.

(i) If K is a Galois extension of ℚ and if [K : ℚ] is odd then the group Γ_{K}(A) is non-arithmetic for every infinite cyclic subgroup A ≤ O_{K}^{∗}.

Proof: To prove the proposition it is sufficient to discuss infinite cyclic subgroups A ≤ O_{K}^{1} which have the property that Γ_{K}(A) is arithmetic. Put T := R_{K/ℚ}^{1}(G_{m}) and let S ≤ T be a ℚ-defined algebraic subgroup so that A is arithmetic and Zariski-dense in S. The d-group T is ℝ-split since K is totally real. In fact we have an ℝ-defined rational isomorphism T (ℝ) ≅ (ℝ^{∗})^{n−1}. Moreover T (ℤ) is discrete in T (ℝ) and (by Dirichlet’s unit theorem) T (ℝ)/T (ℤ) is compact. This implies that S is ℝ-split and dim(S)=1. Let L_{S} be the splitting field of S. Since ℚ ⊗_{ℤ} X(S) is one dimensional we have [L_{S} : ℚ] ≤ 2. Since S(ℤ) is infinite S cannot be ℚ-split. Hence [L_{S} : ℚ] = 2. Let K̂ be the Galois-closure of K. Since S is K-split we find L_{S} ≤ K. This proves (ii) of our proposition. To prove (i) define S_{1} := R_{L_{S}/ℚ}(G_{m}) ≤ T. Since every irreducible one dimensional representation of 𝔾 occurs in ℚ⊗_{ℤ} X(T) at most once we have S = S_{1} and A is commensurable with S_{1}(ℤ). Finally we note that K̂ has only finitely many quadratic subfields.

Proposition 5.1 allows for the construction of many more non-arithmetic groups Γ_{K}(A) than those appearing in Proposition 5.2. For general (not totally real) number fields K the situation is not so simple. This is shown by the following proposition.

Proposition 5.3: Let K be a Galois extension of degree 6 over ℚ with Galois group isomorphic to the symmetric group S_{3}. Let K_{0} be its unique quadratic subfield. Assume that K_{0} is imaginary quadratic. Then Γ_{K}(A) is an arithmetic group for every subgroup A ≤ O_{K}^{∗}.

Proof: Since K is totally imaginary the group O_{K}^{∗} has a subgroup of finite index which is isomorphic to ℤ^{2}. From the well known decomposition of the group ring ℚ[S_{3}] into irreducible ideals we find that R_{K/ℚ}(G_{m}) ∼ T_{0} × T_{1} × T_{2} where T_{0} is isogenous to R_{K_{0}/ℚ}(G_{m}) and where T_{1} and T_{2} are ℚ-defined tori which are isomorphic over ℚ (see the remarks around 5.4). This implies that both T_{1}(ℤ) and T_{2}(ℤ) contain an infinite cyclic subgroup of finite index. Of course T_{0}(ℤ) is finite. Since T_{1} × T_{2} contains the one dimensional ℚ-defined tori { (t_{1}^{a}, t_{2}^{b}) t_{1} ∈ T_{1}, t_{2} ∈ T_{2} } for a, b ∈ ℕ the result follows from Proposition 5.1.

Proof of Theorem G: Let h ∈ ℕ with h ≥ 4 be given. By Dirichlet’s theorem on primes in arithmetic progressions there is a prime number p such that (p − 1)/2 is divisible by h − 1. It is then easy to see that the (h − 1)-th cyclotomic extension of ℚ contains a subfield which is totally real and of degree h − 1 over ℚ. We then apply Proposition 5.2.

It is known (see [Me]) that there are totally real number fields K of degree n ≥ 3 over ℚ such that the Galois group of the Galois-closure of K is isomorphic to the alternating group A_{n}. Theorem G can also be deduced (along the same lines as above) from the following proposition.

Proposition 5.4: Let K be a totally real number field of degree n ≥ 3 over ℚ. Assume that the Galois group of the Galois-closure of K is isomorphic to the alternating group A_{n}. Then Γ_{K}(A) is a non-arithmetic group for every infinite cyclic subgroup A ≤ O_{K}^{∗}.

Proof: Assume that A ≤ O_{K}^{∗} is an infinite cyclic subgroup such that Γ_{K}(A) is an arithmetic group. By Proposition 5.1 we find a ℚ-defined torus S ≤ R_{K/ℚ}(G_{m}) such that A is commensurable with S(ℤ). Similarly to the proof of Proposition 5.2 it can be argued that S is one dimensional and that the splitting field L_{S} of S is of degree 2 over ℚ. Also L_{S} is contained in the Galois-closure of K. This is a contradiction.

Proof of Theorem H: Let Γ be a polycyclic group of Hirsch-length ≤ 3. We have to prove that Γ is arithmetic. It is well known that Γ contains a torsion-free subgroup of finite index Γ_{0} so that Γ_{0}/F(Γ_{0}) is also torsion-free. Using Theorem C we need only to show that Γ_{0} is arithmetic. We make a case distinction on the structure of the Fitting subgroup. Case 1: h(F(Γ_{0})) = 3. In this case Γ_{0} is nilpotent. Every finitely generated nilpotent group is arithmetic by a result of Malcev [Mc]. Case 2: h(F(Γ_{0})) = 2. In this case F(Γ_{0}) is abelian, in fact F(Γ_{0}) ≅ ℤ^{2}. By our choice of Γ_{0} we have Γ_{0}/F(Γ_{0}) ≅ ℤ. Pick a σ ∈ Γ_{0} so that its image generates the quotient Γ_{0}/F(Γ_{0}). Put T := ⟨σ⟩. Then we have an isomorphism Γ_{0} = F(Γ_{0}) · T ≅ F(Γ_{0}) ⋊ T. Conjugation by σ on F(Γ_{0}) gives us a matrix A ∈ GL(2, ℤ). Put T_{1} := ^{}A̸⟩. Let the elements of T_{1} act linearly on ℤ^{2} and let Γ_{1} := ℤ^{2} ⋊ T_{1} be the corresponding split extension. The groups Γ_{0} and Γ_{1} are isomorphic. Let K be the ℚ-algebra generated by A in the 2 × 2 matrix ring. By Dirichlet’s unit theorem K is a real-quadratic extension field of ℚ. Let the multiplicative group G_{m} act on the additive group G_{a} by multiplication and let G_{a} ⋊ G_{m} be the corresponding split extension. It is easy to see that Γ_{1} is isomorphic to an arithmetic subgroup of the restriction of scalars R_{K/ℚ}(G_{a} ⋊ G_{m}) of G_{a} ⋊ G_{m}. The cases h(F(Γ_{0})) = 0, 1 are easily seen to be impossible.

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