Linear Representations of the Automorphism Group of a Free Group

Fritz Grunewald Mathematisches Institut Heinrich-Heine-Universität D-40225 Düsseldorf

e-mail: fritz@math.uni-duesseldorf.de

Alexander Lubotzky Einstein Institute of Mathematics The Hebrew University of Jerusalem Jerusalem, 91904, Israel e-mail: alexlub@math.huji.ac.il

May 19, 2006

Abstract Let F_{n} be the free group on n ≥ 2 elements and Aut(F_{n}) its group of automorphisms. In

this paper we present a rich collection of linear representations of Aut(F_{n}) arising through the action of finite index subgroups of it on relation modules of finite quotient groups of F_{n}. We show (under certain conditions) that the images of our representations are arithmetic groups.

2000 Mathematics Subject Classification: Primary 20F28, 20E05;

Secondary 20E36, 20F34, 20G05

Contents

1 Introduction 2

2 Relation modules and representations of subgroups of finite index in Aut(F_{n}) 6 2.1 The rational group ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 A result of Gaschütz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Representations of subgroups of finite index in Aut(F_{n}) . . . . . . . . . . . . . . 9

3 Derivations and the relation module 10 3.1 Fox calculus and cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2 Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) as right ℚ[G]-module . . . . . . . . . . . . . . . . . . . . . 13

4 Redundant presentations 16 4.1 Decomposition of the relation module . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2 Nielsen Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5 Arithmeticity in case of a redundant presentation 18

5.1 Elementary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.2 Constructing elements of G_{G,π}^{1}(ℤ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Cyclic groups 21

7 Completion of proofs 25

8 (SL) or not (SL)? 27 8.1 Reduction Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 8.2 Abelian and metabelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

9 Concluding remarks and suggestions for further research 31 9.1 Dependence on the presentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.2 Representations of IA(F_{n}) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 9.3 Representations of Γ(G, π) into affine groups . . . . . . . . . . . . . . . . . . . . 33 9.4 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1 Introduction

Let F_{n} be the free group on n ≥ 2 elements and Aut(F_{n}) its group of automorphisms. The latter is a much studied group, but very little seems to be known about its (finite dimensional) complex representation theory (see for example [12], [6], [29], [30], [22]). In fact, as far as we know the only representations studied (at least when n ≥ 3) are:

• The representation

ρ_{1} : Aut(F_{n}) → Aut(F_{n}/F′_{n}) ≅ GL(n, ℤ)  (1)

where F′_{n} is the commutator subgroup of F_{n}.

• The representations factoring through the homomorphisms

ρ_{i} : Aut(F_{n}) → Aut(F_{n}/F_{n}^{(i+1)})

where F_{n}^{(i+1)} (i = 0, 1, …) stands for the lower central series of F_{n}.

• Representations with finite image.

What is common to all of the above mentioned reprentations ρ is that if we denote by H := \overline{ρ(Aut(F_{n}))} the Zariski closure of ρ(Aut(F_{n})), then the semi-simple part of the connected component H^{∘} of H is either trivial or SL(n, ℂ). Moreover, ρ(IA(F_{n})) is always virtually solvable for these representations, where IA(F_{n}) is the kernel of ρ_{1}. In this paper we will show that the representation theory of Aut(F_{n}) is in fact much richer. Our main Theorem (Theorem 1.4 below) implies for example Theorem 1.1 Let n ≥ 2, k ≥ 1, h_{1} < … < h_{k}, m_{1}, …, m_{k} be natural numbers. Let ℚ(ζ_{m_{i}}) be the field of m_{i}-th roots of unity and ℤ(ζ_{m_{i}}) its ring of integers. There is a subgroup Γ ≤ Aut(F_{n}) of finite index and a representation

ρ : Γ → ∏_{i=1}^{k} SL((n − 1)h_{i}, ℚ(ζ_{m_{i}}))^{m_{i}}

such that ρ(Γ) is commensurable with ∏_{i=1}^{k} SL((n − 1)h_{i}, ℤ(ζ_{m_{i}}))^{m_{i}}.

Theorem 1.1 shows that Aut(F_{n}) has a representation ρ such that the connected component of \overline{ρ(Aut(F_{n}))} is isomorphic to

∏_{i=1}^{k} SL((n − 1)h_{i}, ℂ)^{m_{i}}

as above. In fact the proof shows that this is even true for \overline{ρ(IA(F_{n}))} and in particular the latter is very far from being virtually solvable (see Section 9.2 for more). Specifying m_{1} = … = m_{k} = 1 in Theorem 1.1, we deduce

Corollary 1.2 Let n ≥ 2, k ≥ 1 be natural numbers. There is a subgroup Γ ≤ Aut(F_{n}) of finite index and a representation

ρ : Γ → ∏_{i=1}^{k} SL((n − 1)i, ℤ)

such that ρ(Γ) is of finite index in ∏_{i=1}^{k} SL((n − 1)i, ℤ).

Specialising Theorem 1.1 even further to n = 3, k = 1, h_{1} = m_{1} = 1 we get that Aut(F_{3}) has a subgroup of finite index which can be mapped onto a subgroup of finite index in SL(2, ℤ). This implies

Corollary 1.3 The automorphism group Aut(F_{3}) is large, that is it has a subgroup of finite index which can be mapped onto a free nonabelian group. In particular Aut(F_{3}) does not have Kazdhan’s property (T).

The corollary is easy for Aut(F_{2}), see the discussion in Section 7, but it is a well known open problem for larger n. Our solution for n = 3 does not indicate what should be the answer for n > 3. We shall describe now how the representations to be considered here arise. Let G be a finite group and π : F_{n} → G a surjective homomorphism of the free group F_{n} onto G. Let R be the kernel of π. We define

Γ(R) := { φ ∈ Aut(F_{n})  φ(R) = R }  (2)

and

Γ(G, π) := { φ ∈ Γ(R)  φ induces the identity on F_{n}/R }.  (3)

These are subgroups of finite index in Aut(F_{n}). We let

R̄ := R/R′  (4)

be the relation module of the presentation π : F_{n} → G. The action of F_{n} on R by conjugation leads to an action of G on R̄.

The structure of R̄ as a G-module is described by Gaschütz’ theory (see Section 2.2 and the references therein). It says in particular that

ℚ ⊗_{ℤ} R̄ ≅ ℚ ⊕ ℚ[G]^{n−1}  (5)

as a module over the rational group ring ℚ[G] (see also (18)). Our work can be described as an equivariant Gaschütz’ theory where we try to describe the action of Γ(G, π) on R̄: Every φ ∈ Γ(R) induces a linear automorphism φ̄ of R̄ and (as an application of the theorem of Gaschütz) Γ(G, π) consists exactly of those elements φ ∈ Γ(R) for which φ̄ is G-equivariant. Already now we can see from (5) why the number n−1 plays such an important role in Theorem 1.1. To be more precise: The relation module R̄ is a finitely generated free abelian group, let t be its ℤ-rank. Then

G_{G,π} := Aut_{G}(ℂ ⊗_{ℤ} R̄) ≤ GL(t, ℂ)  (6)

is a ℚ-defined algebraic subgroup of GL(t, ℂ), in fact G_{G,π} is the centraliser of the group G acting on ℂ ⊗ℤ R̄ through matrices with rational entries. Let

G_{G,π}^{1} ≤ SL(t, ℂ)  (7)

be the kernel of all ℚ-defined homomorphisms from the complex algebraic group ℂ ⊗ G_{G,π} to the multiplicative group. This is a ℚ-defined subgroup of G_{G,π}. We shall describe it in more detail in Section 2.2. We define

G_{G,π}(ℤ) := { ϕ ∈ G_{G,π}  ϕ(R̄) = R̄ }.  (8)

This is an arithmetic subgroup of the ℚ-defined algebraic group G_{G,π} which contains the arithmetic subgroup

G_{G,π}^{1}(ℤ) := { ϕ ∈ G_{G,π}^{1}  ϕ(R̄) = R̄ }  (9)

of G_{G,π}^{1}. Following the definitions we obtain an integral linear representation

ρ_{G,π} : Γ(G, π) → G_{G,π}(ℤ),  ρ_{G,π}(φ) = φ̄ (φ ∈ Γ(G, π)).  (10)

Examples show (see Section 6) that in general ρ_{G,π} is not onto, neither is its image of finite index in G_{R,π}(ℤ). However, our main result shows that, at least for redundant presentations, the image of ρ_{G,π} captures the whole semi-simple part of G_{R,π}(ℤ). More precisely:

Theorem 1.4 Assume n is a natural number with n ≥ 4. Let π : F_{n} → G be a redundant presentation of the finite group G. Then ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1} is of finite index in the arithmetic group G_{G,π}^{1}(ℤ).

A presentation π : F_{n} → G is called redundant if there is a basis x_{1}, …, x_{n} of F_{n} such that π(x_{n}) = 1. This is equivalent to the kernel of π containing at least one primitive element (that is a member of a basis) of F_{n}. In general we do not know whether ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1}(ℤ) is of finite index in ρ_{G,π}(Γ(G, π)). Section 8 contains criteria under which this is going to happen. Choosing the finite group in various ways we obtain a rich variety of linear representations of subgroups of finite index in Aut(F_{n}) with arithmetic image groups. Let us start with some simple examples. Example 1: Let G = C_{2} be the group of order 2 and π : F_{n} → G be any surjective homomorphism. Since we generally have assumed n ≥ 2, the representation is redundant. As an abelian group we have R̄ ≅ ℤ^{2(n−1)+1}. Also ℚ ⊗_{ℤ} R̄ can be decomposed into the +1-eigenspace of a generator of G and the corresponding −1-eigenspace. We obtain a decomposition ℚ ⊗_{ℤ} R̄ = ℚ^{n} ⊕ ℚ^{n−1} which has to be respected by ρ_{G,π}(Γ(G, π)). Thus we obtain a representation

ρ_{G,π} : Γ(G, π) → GL(n, ℚ) × GL(n − 1, ℚ)

which we show to have an image commensurable with SL(n, ℤ) × SL(n − 1, ℤ). The first factor is the familiar one arising from the representation (1). But the second is somewhat less expected. It already gives rise to a representation which is suitable for proving Corollary 1.3. Example 2: Let p be a prime number, ζ_{p} ∈ ℂ a non trivial p-th root of unity. Let K := ℚ(ζ_{p}) be the p-th cyclotomic field and ℤ(ζ_{p}) its ring of integers. Taking G = C_{p} the cyclic group of order p we have R̄ ≅ ℤ^{p(n−1)+1}. Now p(n − 1) + 1 = (p − 1)(n − 1) + n and we get a representation

ρ_{G,π} : Γ(G, π) → GL(n, ℚ) × GL(n − 1, K)

which we show to have an image commensurable with SL(n, ℤ)×SL(n−1, ℤ(ζ_{p})). This example is treated in Section 6 along with the case of a general cyclic group. Let G be now an arbitrary finite group and π : F_{n} → G a redundant epimorphism. Decompose the group algebra ℚ[G] as

ℚ[G] = ℚ ⊕ ∏_{i=2}^{ℓ} M(h_{i}, D_{i})

where the D_{i} are division algebras. Theorem 1.4 implies that there is a homomorphism ρ from a finite index subgroup Γ of Aut(F_{n}) into a subgroup of ∏_{i=2}^{ℓ} GL((n − 1)h_{i}, D_{i}) where the image contains an arithmetic subgroup of

Λ := ∏_{i=2}^{ℓ} SL((n − 1)h_{i}, R_{i})

where R_{i} is an order in D_{i}. In general we do not know whether ρ(Γ) ∩ Λ is of finite index in ρ(Γ). It will be especially interesting if it is not! In this case ρ(Γ)/(ρ(Γ)∩Λ) would be an infinite abelian group and this would imply that Aut(F_{n}) does not have Kazdhan property (T). We have some results in the opposite direction showing, for example, that if G is metabelian then ρ(Γ) is in fact commensurable to Λ. In any event, whenever ρ(Γ) ∩ Λ is of finite index or not, it becomes of finite index after we divide by the center of ∏_{i=2}^{ℓ} GL((n − 1)h_{i}, D_{i}). This shows that in all cases an arithmetic group commensurable to Λ is a quotient of some finite index subgroup of Aut(F_{n}). Thus Theorem 1.4 provides a very rich class of arithmetic subgroups which are quotients of (finite index subgroups) of Aut(F_{n}). Theorem 1.1 is obtained from the main result by some special choices of the finite group G and by an application of Margulis super-rigidity (see Section 7). The present paper opens up a large number of interesting problems. Among others it shows that the classical Torelli subgroup (i.e. IA(F_{n})) is just a first example in a series of infinitely many. The kernel of ρ_{G,π}, for any (redundant) homomorphism π : F_{n} → G of F_{n} to a finite group G, can be considered as a natural generalisation of IA(F_{n}) which corresponds to the case where G is the trivial group. Just as IA(F_{n}), the kernel of ρ_{G,π} is residually torsion-free nilpotent and, at least when π is redundant, the image of ρ_{G,π} is an arithmetic group hence is finitely presented, which implies that ker(ρ_{G,π}) is a finitely generated normal subgroup. Is ker(ρ_{G,π}) a finitely generated group? In case of the classical Torelli group this is a result of Magnus (see [23]).

Let us now describe the method of the proof of the main result and the layout of the paper: In Section 2, we describe in detail the rational relation module ℚ ⊗_{ℤ} R̄ together with G_{G,π} which is the centraliser of G when acting on the relation module. In Section 3, we describe the homomorphism from Γ(G, π) to G_{G,π}, which is given by the action of Γ(G, π) on ℚ ⊗ℤ R̄. We compute the action of various elements of Γ(G, π) on ℚ ⊗_{ℤ} R̄ in Section 4 using results on Fox calculus from Section 3. A careful choice of elements from Γ(G, π) allows us to produce sufficiently many unipotent elements of G_{G,π}(ℤ) to appeal to results of Vaserstein [33] (see also Venkataramana [35] for generalisations) to deduce in Section 5 that these elements generate a finite index subgroup. Section 6 is devoted to a detailed study of the case when G is a cyclic group. In this case we give fairly complete results. Some of them are used in Section 7 for the proof of Theorem 1.1. In Section 8 we discuss the question whether the image of Γ(G, π) is commensurable with G_{G,π}^{1}(ℤ) or an infinite extension of it. We solve it for cyclic groups and also for metabelian groups. Some remarks, computational results and open problems are given in Section 9. In a subsequent paper [16] we apply similar methods to study linear representations of the mapping class group. Acknowledgement: The authors are grateful to A. Rapinchuk for a useful conversation which led to the results about abelian groups in Section 8. They thank the Minerva-Landau Center for its support and especially the Institute for Advanced Study in Princeton, where this work was done, for its hospitality in the academic year 2005/2006. The second author thanks the NSF, the Monell Foundation and the Ellentuck Fund for support.

2 Relation modules and representations of subgroups of finite index in Aut(F_{n})

In this section we collect some results on the structure of the relation modules of finite groups. We also consider integrality properties of the linear reprentations of subgroups of finite index in Aut(F_{n}) introduced in the introduction.

2.1 The rational group ring

Here G is a finite group and ℚ[G] is its rational group ring. Our basic reference for the structural results we need is the book of Curtis and Reiner [4] (see also [32]), in particular Sections IV and XI therein. We start to set up some notation. Let N be an irreducible left ℚ[G]-module. We write

I_{N}(M)

for the N-isotypic component of the left ℚ[G]-module M. We also choose representatives N_{1}, …, N_{ℓ} for the isomorphism classes of irreducible left ℚ[G]-modules and abreviate

I_{i}(M) := I_{N_{i}}(M) (i = 1, …, ℓ)

for a left ℚ[G]-module M. We take N_{1} to be the one dimensional trivial module. Viewing ℚ[G] as a left ℚ[G]-module we obtain a direct product decomposition

ℚ[G] = ∏_{i=1}^{ℓ} I_{i}(ℚ[G]).  (11)

In this case every I_{i}(ℚ[G]) is also a right submodule of ℚ[G]. Furthermore every I_{i}(ℚ[G]) is an ideal of ℚ[G] having its own unity. The decomposition (11) is a decomposition of ℚ[G] as a direct product of ℚ-algebras. We let P_{i} ∈ ℚ[G] (i = 1, …, ℓ) be the right projector of ℚ[G] onto I_{i}(ℚ[G]), that is we have

RP: AP_{i} = A for all A ∈ I_{i}(ℚ[G]) and AP_{i} = 0 if A is in the direct product of the subrings I_{j}(ℚ[G]) with I_{i}(ℚ[G]) excluded.

Notice that we also have

LP: P_{i}A = A for all A ∈ I_{i}(ℚ[G]) and P_{i}A = 0 if A is in the direct product of the subrings I_{j}(ℚ[G]) with I_{i}(ℚ[G]) excluded.

In fact we have P_{i} ∈ I_{i}(ℚ[G]) for i = 1, …, ℓ and P_{i} can be thought of as the unit element in I_{i}(ℚ[G]). Given two left ℚ[G]-modules M_{1}, M_{2} we define Hom_{G}(M_{1}, M_{2}) to be the vector space of ℚ[G]- module homomorphisms from M_{1} to M_{2}. The ℚ-algebra of ℚ[G]-module endomorphisms of M_{1} is denoted by End_{G}(M_{1}). Let us write S^{op} for the opposite ring of an associative ring S. Mapping an element A ∈ I_{i}(ℚ[G]) to the multiplication from the right by A defines ℚ-algebra isomorphisms

I_{i}(ℚ[G])^{op} → End_{G}(I_{i}(ℚ[G]))  (i = 1, …, ℓ).  (12)

The existence of the isomorphisms (12) implies that

dim_{ℚ}(Hom_{G}(I_{i}(ℚ[G])^{f}, I_{i}(ℚ[G])) = fdim_{ℚ}(I_{i}(ℚ[G]))  (13)

for all f ∈ ℕ and i = 1, …, ℓ. The algebras

D_{i} := End_{G}(N_{i}) (i = 1, …, ℓ)

of left ℚ[G]-module endomorphism of the irreducible modules N_{i} are finite dimensional division algebras with cyclotomic number fields as center. We set

h_{i} := dim_{D_{i}}(N_{i}) (i = 1, …, ℓ).

Then the algebra I_{i}(ℚ[G]) is isomorphic to a h_{i} × h_{i} matrix algebra M(h_{i}, D_{i}) over D_{i}, that is

I_{i}(ℚ[G]) ≅ M(h_{i}, D_{i})  (i = 1, …, ℓ).  (14)

Using (12) we get isomorphisms of algebras

End_{G}(I_{i}(ℚ[G])) ≅ M(h_{i}, D_{i}^{op})  (i = 1, …, ℓ).  (15)

We shall further discuss integral versions of the isomorphisms (11) and (14). Our reference here is [4] Section XI or [5] Section XX. Let K be a number field with ring of integers O and S a finite dimensional K-algebra. A subring R of S is called an order if it contains a K-basis of S and if there is a subring O_{0} of finite index in O such that R is a finitely generated O_{0}-module (subrings are always assumed to contain the identity element of the bigger ring). Let h be a natural number and S = M(h, D) where D is a finite dimensional division algebra over K with center K. We write End_{S}(S^{m}) (m ∈ ℕ) for the algebra of S-left module endomorphisms of the S-left module S^{m}. We obtain an algebra isomorphism

Θ : End_{S}(S^{m}) → M(m, S)^{op} = M(m, S^{op}) = M(m, M(h, D^{op})).  (16)

Here we identify M(m, S)^{op} in the usual way with M(m, S^{op}) and then with M(m, M(h, D^{op})). We call a subgroup Λ of the additive group of S^{m} a lattice if Λ contains a K-basis of S^{m} and if there is a subring O_{0} of finite index in O such that Λ is a finitely generated O_{0}-module. We write End_{S}(S^{m})_{Λ} for the ring of those endomorphisms stabilising Λ. The following is well known (see [4], [5]).

Lemma 2.1 Let Λ ≤ S^{m} (m ∈ ℕ) be a lattice. There is a sublattice Λ̃ ≤ Λ (of finite index in Λ) and an order R in D such that

Θ(End_{S}(S^{m})Λ̃) ⊂ M(m, M(h, R^{op})).

Let Aut_{S}(S^{m})_{Λ} be the group of invertible elements in End_{S}(S^{m})_{Λ}. Note that Aut_{S}(S^{m})Λ̃ is commensurable with Aut_{S}(S^{m})_{Λ} in particular Aut_{S}(S^{m})_{Λ} ∩ Aut_{S}(S^{m})Λ̃ has finite index in Aut_{S}(S^{m})_{Λ} and we obtain a representation

Θ : Aut_{S}(S^{m})_{Λ} ∩ Aut_{S}(S^{m})Λ̃ → GL(m, M(h, R^{op})) = GL(mh, R^{op}).  (17)

2.2 A result of Gaschütz

Next we need to report on a result of Gaschütz (see [13], [15] and [19]). For this, let G be a finite group and π : F_{n} → G a surjective homomorphism of a rank n free group F_{n} onto G. Let R be the kernel of π and R̄ := R/R′ the corresponding relation module with its action (from the left) of G. The result of Gaschütz says that there is a ℚ[G]-module isomorphism

κ : ℚ ⊕ ℚ[G]^{n−1} → ℚ ⊗_{ℤ} R̄.  (18)

where ℚ is to be the trivial one dimensional ℚ[G]-module. We fix a ℚ[G]-module isomorphism κ once for all and identify ℚ ⊗ℤ R̄ with ℚ ⊕ ℚ[G]^{n−1}. In particular we obtain dimℚ(ℚ ⊗ℤ R̄) = 1 + |G|(n − 1), which is also clear from the formula of Reidemeister and Schreier. We deduce

I_{1}(ℚ ⊗_{ℤ} R̄) = I_{1}(ℚ[G])^{n} = ℚ^{n},  (19)
I_{i}(ℚ ⊗_{ℤ} R̄) = I_{i}(ℚ[G])^{n−1}  (i = 2, …, ℓ).  (20)

From (13) we get

Lemma 2.2 Let G be a finite group and π : F_{n} → G a surjective homomorphism. Let R̄ := R/R′ be the corresponding relation module. Then we have:

dim_{ℚ}(Hom_{G}(I_{1}(ℚ ⊗_{ℤ} R̄), I_{1}(ℚ[G]))) = n
dim_{ℚ}(Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G]))) = (n − 1)dim_{ℚ}(I_{i}(ℚ[G])

The result of Gaschütz enables us further to decompose the algebraic group G_{G,π} and to describe the components. We define for i = 1, …, ℓ:

G_{G,π,i} := { ϕ ∈ G_{G,π}  ϕ|_{ℂ⊗I_{j}(ℚ⊗_{ℤ}R̄)} = Id_{ℂ⊗I_{j}(ℚ⊗_{ℤ}R̄)} for j ≠ i } .  (21)

Note that the groups G_{G,π,i} centralise each other and have pairwise trivial intersection. It is also easy to see that we have an internal product decomposition:

G_{G,π} = ∏_{i=1}^{ℓ} G_{G,π,i}

of ℚ-defined algebraic groups. We write GL(d, S) (d ∈ ℕ) for the group of invertible d×d-matrices with entries in an associative ring S. Following the discussion of the previous section we infer that the identification made using (18) defines isomorphisms

Σ_{i} : G_{G,π,i}(ℚ) → GL(n − 1, I_{i}(ℚ[G])^{op}) = GL((n − 1)h_{i}, D_{i}^{op})

for i = 2, … ℓ and also an isomorphism

Σ_{1} : G_{G,π,1}(ℚ) → GL(n, I_{1}(ℚ[G])^{op}) = GL(n, ℚ).

Together they define an isomorphism

Σ : G_{G,π}(ℚ) → GL(n, ℚ) × ∏_{i=2}^{ℓ} GL((n − 1)h_{i}, D_{i}^{op}).

Suppose that i is one of the numbers 1, …, ℓ and let K_{i} be the center of I_{i}(ℚ[G])^{op}. As already mentioned, this is a cyclotomic number field. The group GL((n − 1)h_{i}, D_{i}^{op}) is the group of K_{i}- rational points of a K_{i}-defined linear algebraic group H_{i}. The above discussion shows that the ℚ-defined algebraic group G_{G,π,i} is equal to the base field restriction of H_{i} from K_{i} to ℚ. For this concept see [28]. We shall add a description of the groups G_{G,π}^{1} introduced in the introduction. Every element g of GL((n − 1)h_{i}, D_{i}^{op}) (i = 1, …, ℓ) acts by multiplication on D_{i}^{(n−1)h_{i}}. Taking a K_{i}-vector space identification D_{i} = K_{i}^{d_{i}} we associate to g a square matrix of size (n − 1)h_{i}d_{i}. The determinant of this matrix is called the reduced norm Nrd(g) of g. The group of elements of reduced norm 1 is conventionally denoted by SL((n − 1)h_{i}, D_{i}^{op}). We have

G_{G,π,i}^{1}(ℚ) = { g ∈ G_{G,π,i}(ℚ) Nrd(Σ_{i}(g)) = 1 }.

2.3 Representations of subgroups of finite index in Aut(F_{n})

In this section we describe a matrix version of the representation (10). To do this we use Lemma 2.1 which constructs an integral matrix representation of a subgroup of finite index in Γ(G, π) which is compatible with the representations Σ_{i} (i = 1, …, ℓ) and Σ from the previous section. Let G be a finite group and π : F_{n} → G a surjective homomorphism of the free group F_{n} onto G. Let R be the kernel of π and R̄ the corresponding relation module. Let further Γ(G, π) be the subgroup of Aut(F_{n}) defined in (3). In the sequel we shall often use the following fact without further notice.

Lemma 2.3 An element φ ∈ Γ(R) is in Γ(G, π) if and only if the map φ̄ : R̄ → R̄ induced by φ is G-equivariant.

Proof: If φ is in Γ(G, π) then for every w ∈ F_{n} we have a r_{w} ∈ R with φ(w) = wr_{w}. Hence φ(wrw^{−1}) = wr_{w}φ(r)r_{w}^{−1}w^{−1} holds for every r ∈ R. The latter is equal to wφ(r)w^{−1} modulo R′ which implies that φ̄ : R̄ → R̄ is G-equivariant.

If φ̄ : R̄ → R̄ is G-equivariant then for every w ∈ F_{n} the congruence φ(w)φ(r)φ(w)^{−1} ≡ wφ(r)w^{−1} holds modulo R′ for every r ∈ R. This means that w and φ(w) act the same on R̄. The result of Gaschütz implies that the action of G on R̄ is faithful, that is no element except the identity of G acts as the identity. We conclude that φ(w)w^{−1} is in R.

Keeping the notations set up in subsections 2.1, 2.2 we define

Λ_{i} := R̄ ∩ I_{i}(ℚ ⊗_{ℤ} R̄) (i = 1, …, ℓ).

Each Λ_{i} is a ℤ[G]-submodule of R̄∩I_{i}(ℚ⊗_{ℤ} R̄) and ∏_{i=1}^{ℓ} Λ_{i} ≤ R̄ is a finite index ℤ[G]-submodule. Notice that the Λ_{i} is a sublattice of I_{i}(ℚ ⊗_{ℤ} R̄) = M(h_{i}, D_{i})^{n−1} (i = 2, …, ℓ) in the sense of the previous section, so is Λ_{1} in I_{1}(ℚ ⊗_{ℤ} R̄) = ℚ^{n}. Applying Lemma 2.1 we choose orders R_{i} ⊂ D_{i} and finite index additive subgroups Λ̃_{i} ≤ Λ_{i} (i = 1, …, ℓ). We define

Γ̂(G, π) ≤ Γ(G, π)  (22)

to be the simultaneous stabiliser of all the Λ̃_{i} ≤ Λ_{i} (i = 1, …, ℓ). Notice further that Γ̂(G, π) is of finite index in Γ(G, π). Using the various representations Θ from Lemma 2.1 we obtain representations

σ_{G,π,1} : Γ̂(G, π) → GL(n, ℤ), σ_{G,π,i} : Γ̂(G, π) → GL((n − 1)h_{i}, R_{i}^{op})

(i = 2, …, ℓ), which come by projection from the analogously defined representation

σ_{G,π} : Γ̂(G, π) → GL(n, ℤ) × ∏_{i=2}^{ℓ} GL(n − 1, M(h_{i}, R_{i}^{op})).

Notice that the diagram

\xymatrix{Γ̂(G, π) \ar[r] \ar[d]^{σ_{G,π}} & G_{G,π}(ℚ) \ar[dl]^{Σ} \\ GL(n, ℚ) × ∏_{i=2}^{ℓ} GL(n − 1, M(h_{i}, D_{i}^{op}))}

is commutative.

3 Derivations and the relation module

This section contains a brief discussion of the correspondences between Fox-derivatives, the relation module of a finite group and cohomology. We apply these results to obtain a certain canonical system of right module generators of Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) where G is a finite group and R̄ is the relation module coming from a presentaion π : F_{n} → G.

3.1 Fox calculus and cohomology

For n ∈ ℕ, let F_{n} be the free group with basis x_{1}, …, x_{n}. We write ℤ[F_{n}], ℚ[F_{n}] for the group rings of F_{n} over the integers or the rational numbers, respectively. Let

\frac{∂}{∂x_{i}} : F_{n} → ℤ[F_{n}] (i = 1, … n)

be the Fox-derivatives (see [3] for a more elaborate exposition). They are defined by the rules

\frac{∂x_{j}}{∂x_{i}} = δ_{ij},  \frac{∂w_{1}w_{2}}{∂x_{i}} = \frac{∂w_{1}}{∂x_{i}} + w_{1} \frac{∂w_{2}}{∂x_{i}}  (23)

for i, j = 1, … n and w_{1}, w_{2} ∈ F_{n}. As usual, δ_{ij} denoting the Kronecker symbol (δ_{ij} = 1 if i = j and δ_{ij} = 0 if i ≠ j). For later use we note some consequences of (23). For i = 1, …, n and e ∈ ℕ we have

\frac{∂x_{i}^{e}}{∂x_{i}} = ∑_{k=0}^{e−1} x_{i}^{k},  \frac{∂x_{i}^{−e}}{∂x_{i}} = − ∑_{k=1}^{e} x_{i}^{−k}.  (24)

Let further w := v_{1}x_{i}^{e_{1}}v_{2}x_{i}^{e_{2}} · . . . · v_{k}x_{i}^{e_{k}}v_{k+1} (e_{1}, …, e_{k} ∈ ℤ) be an element of F_{n} with the subwords v_{j} (j = 1, …, k + 1) not involving x_{i}, then

\frac{∂w}{∂x_{i}} = v_{1} \frac{∂x_{i}^{e_{1}}}{∂x_{i}} + v_{1}x_{i}^{e_{1}}v_{2} \frac{∂x_{i}^{e_{2}}}{∂x_{i}} + … + v_{1}x_{i}^{e_{1}} · v_{2}x_{i}^{e_{2}} · . . . · v_{k} \frac{∂x_{i}^{e_{k}}}{∂x_{i}} .  (25)

Let G be a finite group generated by its elements g_{1}, …, g_{n} (n ∈ ℕ) and let

π : F_{n} → G π(x_{1}) = g_{1}, …, π(x_{n}) = g_{n}

be the corresponding surjective group homomorphism. The homomorphism π : F_{n} → G extends by linearity to surjective ring homomorphisms:

π : ℤ[F_{n}] → ℤ[G] π : ℚ[F_{n}] → ℚ[G].

Let R ≤ F_{n} be the kernel of π and set R̄ := R/R′ to be the commutator factor group of R. Let ¯ : R → R/R′ stand for the quotient homomorphism. The free group F_{n} acts by conjugation on R. Since its subgroup R acts trivially on R̄ we get an induced action of G on R̄ which we denote by g · u (g ∈ G, u ∈ R̄). We generally write the composition in R̄ additively. The free abelian group R̄ then obtains the structure of a left ℤ[G]-module which extends to a left ℚ[G]-module structure on ℚ ⊗ℤ R̄. We have

Lemma 3.1 Let G be a group and π : F_{n} → G a homomorphism. Let R be the kernel of π. Then

π (\frac{∂r}{∂x_{i}} ) = 0

for all r ∈ R′ and i = 1, … n.

Proof: Note first that (23) implies \frac{∂w^{−1}}{∂x_{i}} = −w^{−1} \frac{∂w}{∂x_{i}} for all w ∈ F_{n} and i = 1, … n. It is enough to show the result for single commutators r = r_{1}r_{2}r_{1}^{−1}r_{2}^{−1}. A repeated application of (23) implies that

\frac{∂r}{∂x_{i}} = \frac{∂r_{1}}{∂x_{i}} + r_{1} \frac{∂r_{2}}{∂x_{i}} − r_{1}r_{2}r_{1}^{−1} \frac{∂r_{1}}{∂x_{i}} − r_{1}r_{2}r_{1}^{−1}r_{2}^{−1} \frac{∂r_{2}}{∂x_{i}}

for all i = 1, … n. As π(r_{1}) = π(r_{2}) = 1 the result follows.

The Fox-derivatives \frac{∂}{∂x_{i}} now induce maps

∂_{i} : R̄ → ℤ[G] (i = 1, …, n)

defined as follows. For every u ∈ R̄ choose w_{u} ∈ R with w̄_{u} = u and define

∂_{i}(u) := π (\frac{∂w_{u}}{∂x_{i}} )  (i = 1, …, n).  (26)

We have Lemma 3.2 The maps ∂_{1}, … ∂_{n} do not depend on the choices of the preimages w_{u} (u ∈ R̄). They are ℤ[G]-module homomorphisms when ℤ[G] acts on itself from the left.

Proof: The first statement follows from Lemma 3.1. For the second note that

\frac{∂grg^{−1}}{∂x_{i}} = \frac{∂g}{∂x_{i}} + g \frac{∂r}{∂x_{i}} − grg^{−1} \frac{∂g}{∂x_{i}}

for all g, r ∈ F_{n} and all i = 1, … n. When r is in R, π(r) = 1 and the projection of the right hand side into ℤ[G] is equal to π(g)π(\frac{∂r}{∂x_{i}}).

We also write ∂_{i} : ℚ ⊗_{ℤ} R̄ → ℚ[G] for the induced ℚ[G]-module homomorphisms. Let t be the ℤ-rank of the free abelian group R̄. Note that t = 1 + |G|(n − 1) and also t = dim_{ℚ}(ℚ ⊗_{ℤ} R̄). Let B := (u_{1}, …, u_{t}) be a ℚ-basis of ℚ ⊗_{ℤ} R̄. We consider here

J_{B} := ( \begin{matrix}∂_{1}(u_{1}) & … & ∂_{n}(u_{1}) \\ ⋮ & & ⋮ \\ ∂_{1}(u_{t}) & … & ∂_{n}(u_{t}) \end{matrix} ) ,  (27)

which is a t × n matrix with entries in ℚ[G]. Given a left ℚ[G]-module M and s ∈ ℕ we write

M^{s} := {( m_{1} ; ⋮ ; m_{s} ) m_{1}, …, m_{s} ∈ M}

for the corresponding ℚ[G]-module of column vectors. The matrix J_{B} induces a ℚ-linear map J_{B} : M^{n} → M^{t}. Notice for later use that given a ℚ[G]-submodule N ≤ M we have J_{B}(N^{n}) ⊆ N^{t}. Given the left ℚ[G]-module M we write

Der(G, M) := { d : G → M d(gh) = gd(h) + d(g) for all g, h ∈ G }

for the vector space of derivations from G to M. Using the generators g_{1}, …, g_{n} of our finite group G we define for d ∈ Der(G, M)

L(d) := ( d(g_{1}) ; ⋮ ; d(g_{n}) ) ∈ M^{n}.

We have

Lemma 3.3 The map L : Der(G, M) → M^{n} is injective and its image is equal to Ker(J_{B}). Hence L defines a ℚ-vector space isomorphism between Der(G, M) and Ker(J_{B}).

Proof: Let d be a derivation in the kernel of L. We infer that d(g_{1}) = … = d(g_{n}) = 0. The derivation rule and the fact that the g_{1}, …, g_{n} generate G implies that d is identically zero. This shows that L is injective. Let D : F_{n} → M be a derivation where F_{n} acts on M through the homomorphism π : F_{n} → G. Since both sides are derivations and agree on the x_{i}’s we find

D(w) = \frac{∂w}{∂x_{1}} D(x_{1}) + … + \frac{∂w}{∂x_{n}} D(x_{n})  (28)

for all w ∈ F_{n}. Given a derivation d : G → M, define an induced derivation D : F_{n} → M by setting D(w) := d(π(w)). Applying (28) we find for every r ∈ R

0 = d(1) = D(r) = ∑_{i=1}^{n} \frac{∂r}{∂x_{i}} D(x_{i}) = ∑_{i=1}^{n} ∂_{i}(r)d(g_{i}).

Applying this to r = u_{1}, …, u_{t} shows that the image of L is contained in the kernel of J_{B}. Let now

m := ( m_{1} ; ⋮ ; m_{n} ) ∈ M^{n}

be an n-tuple of elements of M. It is well known that there is a derivation D : F_{n} → M with D(x_{1}) = m_{1}, …, D(x_{n}) = m_{n}. If m is an element of the kernel of J_{B} then (28) implies that D(r) = 0 for all r ∈ R. The derivation D then induces a derivation d : G → M with the property d(g_{1}) = m_{1}, …, d(g_{n}) = m_{n}. This shows that every element of the kernel of J_{B} is in the image of L.

Corollary 3.4 Let M be a ℚ[G]-module then the kernel of the map J_{B} : M^{n} → M^{t} is zero if M is a trivial module and has ℚ-vector space dimension equal to dim_{ℚ}(M) if M does not have a trivial submodule.

Proof: Given an element m ∈ M we let d_{m} : G → M be the corresponding inner derivation d_{m}(g) = (g − 1)m. Let

IDer(G, M) := { d_{m} m ∈ M } ⊆ Der(G, M)

be the space of inner derivations. It is well known that the first cohomology group H^{1}(G, M) = Der(G, M)/IDer(G, M) is zero (G is a finite group). Note also that the map μ : M → IDer(G, M), defined by μ(m) := d_{m} (m ∈ M) is surjective. If M is a trivial module then μ is identically zero, that implies IDer(G, M) = {0} and hence also Der(G, M) = {0}. We find from Lemma 3.3 that dim_{ℚ}(ker(J_{B})) = 0. If M does not contain the trivial module then μ is injective, that implies dim_{ℚ}(IDer(G, M)) = dim_{ℚ}(M). We find from Lemma 3.3 that dim_{ℚ}(ker(J_{B})) = dim_{ℚ}(Der(G, M)) = dim_{ℚ}(M).

The results of this section will be applied in the next section in case of the relation modules.

3.2 Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) as right ℚ[G]-module

We keep here our notation from before, that is G is a finite group generated by its elements g_{1}, …, g_{n} (n ∈ ℕ) and π : F_{n} → G is the homomorphism defined by π(x_{1}) = g_{1}, …, π(x_{n}) = g_{n}. Furthermore R is the kernel of π and R̄ is the corresponding relation module. As the main result we shall show that the ∂_{1}, …, ∂_{n} ∈ Hom_{G}(ℚ ⊗ℤ R̄, ℚ[G]), which were constructed in the previous subsection generate Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) if this space is considered as a ℚ[G]-module from the right in the following way. Let M be a left ℚ[G]-module. For f ∈ Hom_{G}(M, ℚ[G]) and A ∈ ℚ[G] we define f^{A} ∈ Hom_{G}(M, ℚ[G]) by

f^{A}(x) := f(x)A  (x ∈ M).  (29)

Thus the ℚ-vector space Hom_{G}(M, ℚ[G]) becomes a right ℚ[G]-module. We shall prove here:

Proposition 3.5 The maps ∂_{1}, …, ∂_{n} (defined in (26)) generate Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) as right ℚ[G]-module.

To prepare for the proof we decompose the matrix J_{B} into submatrices. To do this we choose the basis B adapted to the decomposition

ℚ ⊗_{ℤ} R̄ = ∏_{i=1}^{ℓ} I_{i}(ℚ ⊗_{ℤ} R̄).

That is we choose bases B_{i} = (u_{i1}, … u_{it_{i}}) (t_{i} := dim_{ℚ}(I_{i}(ℚ ⊗_{ℤ} R̄)) for i = 1, …, ℓ and put them together to obtain a ℚ-vector space basis

B = (u_{1} = u_{11}, …, u_{t} = u_{ℓt_{ℓ}})

of ℚ ⊗_{ℤ} R̄. Consider the matrices

J_{B}^{P_{i}} := ( \begin{matrix}∂_{1}(u_{1})P_{i} & … & ∂_{n}(u_{1})P_{i} \\ ⋮ & & ⋮ \\ ∂_{1}(u_{t})P_{i} & … & ∂_{n}(u_{t})P_{i} \end{matrix} ) = ( \begin{matrix}∂_{1}^{P_{i}}(u_{1}) & … & ∂_{n}^{P_{i}}(u_{1}) \\ ⋮ & & ⋮ \\ ∂_{1}^{P_{i}}(u_{t}) & … & ∂_{n}^{P_{i}}(u_{t}) \end{matrix} )  (30)

for i = 1, …, ℓ. Notice that according to property LP of the right projector P_{i} (see section 2.1) the maps

J_{B}, J_{B}^{P_{i}} : I_{i}(ℚ[G])^{n} → I_{i}(ℚ[G])^{t}

are the same. Furthermore we have ∂_{j}(u) ∈ I_{i}(ℚ[G]) (j = 1, …, n) for u ∈ I_{i}(ℚ ⊗_{ℤ} R̄). This follows since each ∂_{j} is a ℚ[G]-module homomorphism which has to respect isotypic components. Thus property RP of the right projector P_{i} implies that all entries of J_{B}^{P_{i}} are equal to zero except those in the submatrix

J_{B_{i}} := ( \begin{matrix}∂_{1}(u_{i1})P_{i} & … & ∂_{n}(u_{i1})P_{i} \\ ⋮ & & ⋮ \\ ∂_{1}(u_{it_{i}})P_{i} & … & ∂_{n}(u_{it_{i}})P_{i} \end{matrix} )  (31)

hence the kernels of the two maps J_{B} : I_{i}(ℚ[G])^{n} → I_{i}(ℚ[G])^{t} and J_{B_{i}} : I_{i}(ℚ[G])^{n} → I_{i}(ℚ[G])^{t_{i}} are the same. Note finally that we have

J_{B_{i}} = ( \begin{matrix}∂_{1}^{P_{i}}(u_{i1}) & … & ∂_{n}^{P_{i}}(u_{i1}) \\ ⋮ & & ⋮ \\ ∂_{1}^{P_{i}}(u_{it_{i}}) & … & ∂_{n}^{P_{i}}(u_{it_{i}}) \end{matrix} )

for all i = 1, …, ℓ.

Proof of Proposition 3.5: We have the decomposition

Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) = ∏_{i=1}^{ℓ} Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G]))

of Hom_{G}(ℚ⊗_{ℤ} R̄, ℚ[G]) as a right ℚ[G]-module. Since the ∂_{1}^{P_{i}}, …, ∂_{n}^{P_{i}} are zero on I_{j}(ℚ⊗_{ℤ} R̄) for j ≠ i it is enough to show that they generate Hom_{G}(I_{i}(ℚ⊗_{ℤ} R̄), I_{i}(ℚ[G])) as a right ℚ[G]-module. Consider the ℚ-linear map Φ_{i} : I_{i}(ℚ[G])^{n} → Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G])) given by

Φ_{i} (( m_{1} ; ⋮ ; m_{n} )) := ∂_{1}^{P_{i}m_{1}} + … + ∂_{n}^{P_{i}m_{n}}.

Notice that I_{i}(ℚ[G]) · ℚ[G] = I_{i}(ℚ[G]) and hence the image of Φ_{i} is the right ℚ[G]-submodule of Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G])) generated by the ∂_{1}^{P_{i}}, …, ∂_{n}^{P_{i}}. The key observation now is

kernel(J_{B_{i}}) = kernel(Φ_{i})  (32)

which is obvious by what is explained above. Notice that in fact (m_{1}, …, m_{n})^{t} is in the kernel of Φ_{i} if and only if ∂_{1}^{P_{i}m_{1}}(u_{ik}) + … + ∂_{n}^{P_{i}m_{n}}(u_{ik}) = 0 holds for all k = 1, …, t_{i}. We can treat now the case i = 1 (the case of the trivial module). Here we have by (19) that dim_{ℚ}(I_{1}(ℚ ⊗_{ℤ} R̄)) = n and the kernel of Φ_{1} is trivial (by Corollary 3.4). Since we have dim_{ℚ}(Hom_{G}(I_{1}(ℚ ⊗_{ℤ} R̄), I_{1}(ℚ[G]))) = n (by Lemma 2.2) the map Φ_{1} has to be surjective. In the cases i > 1 we have dim_{ℚ}(kernel(Φ_{i})) = dim_{ℚ}(I_{i}(ℚ[G])) by (32) and Corollary 3.4. By Lemma 2.2 and (13) we have dim_{ℚ}(Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G])) = (n − 1)dim_{ℚ}(I_{i}(ℚ[G])) which implies that Φ_{i} has to be surjective.

As a corollary to Proposition 3.5 we note

Corollary 3.6 The maps ∂_{1}^{P_{i}}, …, ∂_{n}^{P_{i}} generate Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G])) as right I_{i}(ℚ[G])- module for every i = 1, …, ℓ.

We shall give now a reformulation of the last corollary which will be needed later. Let Id_{i,k} ∈ I_{i}(ℚ ⊗_{ℤ} R̄) (k = 1, …, n − 1) be the images under κ_{i} (see (20), (19)) of the unit elements of of the components of I_{i}(ℚ ⊗_{ℤ} R̄)^{n−1} respectively of I_{1}(ℚ ⊗_{ℤ} R̄)^{n} with k = 1, …, n. Using the identification (15) we think of f ∈ Hom_{G}(I_{i}(ℚ ⊗_{ℤ} R̄), I_{i}(ℚ[G])) being identified with the collection of matrices f(Id_{i,1}) = A_{i,1}, …, f(Id_{i,n−1}) = A_{i,n−1} ∈ M(h_{i}, D_{i}^{op}) or f(Id_{1,1}) = A_{1,1}, …, f(Id_{i,n}) = A_{1,n} ∈ M(h_{1}, D_{1}^{op}) = ℚ.

Corollary 3.7 Let i be one of the numbers 2, …, ℓ. Given A_{1}, …, A_{n−1} ∈ M(h_{i}, D_{i}^{op}) there are B_{1}, …, B_{n} ∈ ℚ[G] such that

∂_{1}^{P_{i}B_{1}} + … + ∂_{n}^{P_{i}B_{n}}(Id_{i,k}) = A_{k} (k = 1, …, n − 1).

Given A_{1}, …, A_{n} ∈ M(h_{1}, D_{1}^{op}) there are B_{1}, …, B_{n} ∈ ℚ[G] such that

∂_{1}^{P_{1}B_{1}} + … + ∂_{n}^{P_{1}B_{n}}(Id_{1,k}) = A_{k} (k = 1, …, n).

Remark: We sketch here another (more conceptual) proof of Proposition 3.5. Features of the proof given above will play a role in the sequel. Using H^{2}(G, ℚ ⊗_{ℤ} R̄) = 0 we obtain an injective homomorphism

ϵ : F_{n}/R′ → (ℚ ⊗_{ℤ} R̄) ⋊ G

from F_{n}/R′ to the semidirect product on the right hand side. On R/R′ ≤ F_{n}/R′ the homomorphism ϵ induces the inclusion of R/R′ into ℚ ⊗_{ℤ} R̄ ≤ (ℚ ⊗_{ℤ} R̄) ⋊ G. The second component of ϵ coincides with π. Associate to f ∈ Hom_{G}(ℚ ⊗_{ℤ} R̄, ℚ[G]) the homomorphism τ_{f} : (ℚ ⊗_{ℤ} R̄) ⋊ G → ℚ[G] ⋊ G (τ_{f}((u, g)) = (f(u), g)). Composition with ϵ creates a homomorphism

λ_{f} : F_{n} → F_{n}/R′ → ℚ[G] ⋊ G, λ_{f}(w) = (d_{f}(w), π(w)), (w ∈ F_{n})

The map d_{f} : F_{n} → ℚ[G] is a derivation. Hence we have for w ∈ F_{n}

d_{f}(w) = \frac{∂w}{∂x_{1}} d_{f}(x_{1}) + … + \frac{∂w}{∂x_{n}} d_{f}(x_{n}).

Restricting this formula to R/R′ we have expressed f in the desired way.

4 Redundant presentations

In this section, we present a special decomposition of the relation module R̄ for redundant presentations of finite groups. We then describe the action of various Nielsen automorphisms of F_{n} on R̄ with respect to this decomposition. Let F_{n} (n ∈ ℕ, n ≥ 2) be the free group with basis x_{1}, …, x_{n−1}, y. Let G be a finite group generated by its elements g_{1}, …, g_{n−1} and let

π : F_{n} → G  π(x_{1}) = g_{1}, …, π(x_{n−1}) = g_{n−1}, π(y) = 1  (33)

be the surjective group homomorphism corresponding to these data. We keep the notation introduced above of objects related to the presentation π of G. That is R is the kernel of π and R̄ = R/R′ is the relation module, etc.. We shall construct certain Nielsen-like elements in Γ̂(G, π) (see 22) which will allow us to find many unipotent elements in the image of the representation

σ_{G,π} : Γ̂(G, π) → GL(n, ℤ) × ∏_{i=2}^{ℓ} GL(n − 1, M(h_{i}, R_{i}^{op}))  (34)

defined in Section 2.3.

4.1 Decomposition of the relation module

With the notation fixed as indicated above we define

R_{old} := { x ∈ ⟨x_{1}, …, x_{n−1}⟩ π(x) = 1 } and R̄_{old} := R_{old}/R′_{old}.

The injection R_{old} → R defines a ℤ[G]-module homomorphism R̄_{old} → R̄. Since y ∈ R, we can consider the ℤ[G]-submodule of R̄ generated by ȳ ∈ R̄.

Lemma 4.1 The homomorphism R̄_{old} → R̄ is injective and so is the homomorphism ℤ[G] → R̄ defined by A ↦ A · ȳ. Identifying R̄_{old} with its image in R̄ we obtain a direct sum decomposition

R̄ = R̄_{old} ⊕ ℤ[G] · ȳ.

as a ℤ[G]-module.

Proof: Let us temporarily write R̂_{old} for the image of R_{old} in R̄. We first show that R̄ = R̂_{old} + ℤ[G] · ȳ. Given any w ∈ F_{n}, by looking at the image of w modulo the normal closure of y in F_{n}, we see that w can be written in a unique way as w = w_{1}w_{2} with w_{1}, w_{2} ∈ F_{n} such that w_{1} does not involve the letter y whereas w_{2} is a product of conjugates of powers of y. If w is an element of R then w_{1} has to be in R_{old}. This implies R̄ = R̂_{old} + ℤ[G] · ȳ. We infer that ℚ ⊗_{ℤ} R̄ = ℚ ⊗_{ℤ} R̂_{old} + ℚ[G] · ȳ. We now count dimensions using the result of Gaschütz. As R̄ and R̄_{old} are free over ℤ, the lemma follows.

4.2 Nielsen Automorphisms

We study here the effect of certain Nielsen automorphisms on the relation module R̄. We define elements α_{i} ∈ Aut(F_{n}) (i = 1, …, n − 1) by

α_{i}(x_{i}) := yx_{i}.  (35)

Our convention here is that values not given are identical to the argument, that is we assume α_{i}(x_{j}) := x_{j}, for j ≠ i and α_{i}(y) = y. For X ∈ ⟨x_{1}, …, x_{n−1}⟩ we define elements β_{X} ∈ Aut(F_{n}) by

β_{X}(y) := XyX^{−1}  (36)

and finally for U ∈ R_{old} we define γ_{U} ∈ Aut(F_{n}) by

γ_{U}(y) := Uy.  (37)

The α_{i}, β_{X}, γ_{U} extend to automorphisms of F_{n}. We have

Lemma 4.2 The α_{i} (i = 1, …, n − 1), β_{X} (X ∈ ⟨x_{1}, …, x_{n−1}⟩) and γ_{U} (U ∈ R_{old}) are in Γ(G, π) and the following formulas hold for the corresponding automorphisms induced on R̄:

ᾱ_{i}(u) = u + ∂_{i}(u) · ȳ for u ∈ R̄_{old} and  ᾱ_{i}(ȳ) = ȳ.  (38)
β̄_{X}(u) = u, for u ∈ R̄_{old} and  β̄_{X}(ȳ) = π(X) · ȳ,  (39)
γ̄_{U}(u) = u, for u ∈ R̄_{old} and  γ̄_{U}(ȳ) = Ū + ȳ,  (40)

Proof: It is first of all clear from our definitions that all the α_{i}, β_{X}, γ_{U} are elements of Γ(R) := { φ ∈ Aut(F_{n}) φ(R) = R }. Moreover all of them induce the identity on F_{n}/R and hence are even in Γ(G, π). It follows (see Section 2.3) that they act as ℤ[G]-automorphisms on R̄. Next we prove formula (38). Fix i ∈ {1, …, n − 1} and define for e ∈ ℕ

W_{e} := y · x_{i}yx_{i}^{−1} · . . . · x_{i}^{e−1}yx_{i}^{1−e}, W_{−e} := x_{i}^{−1}y^{−1}x_{i} · . . . · x_{i}^{−e}y^{−1}x_{i}^{e} ∈ F_{n}

and add W_{0} := 1. We then have α_{i}(x_{i}^{e}) = W_{e} · x_{i}^{e} for all e ∈ ℤ. For u ∈ R̄_{old} choose w ∈ R_{old} with w̄ = u. We decompose w ∈ ⟨x_{1}, …, x_{n}⟩ as w := v_{1}x_{i}^{e_{1}}v_{2}x_{i}^{e_{2}} · . . . · v_{k}x_{i}^{e_{k}}v_{k+1} (e_{1}, …, e_{k} ∈ ℤ) such that the v_{j} do not involve x_{i}. Define further

V_{1} := v_{1}, V_{2} := v_{1}x_{i}^{e_{1}}v_{2}, …, V_{k} := v_{1}x_{i}^{e_{1}}v_{2} … x_{i}^{e_{k}}v_{k}.

We then have

α_{i}(w) = V_{1}W_{e_{1}}V_{1}^{−1} · V_{2}W_{e_{2}}V_{2}^{−1} · . . . · V_{k}W_{e_{2}}V_{k}^{−1} · w.

Switching to the additive notation and using formulas (24) we find

ᾱ_{i}(u) = u + π (v_{1} \frac{∂x_{i}^{e_{1}}}{∂x_{i}} + v_{1}x_{i}^{e_{1}}v_{2} \frac{∂x_{i}^{e_{2}}}{∂x_{i}} + … + v_{1}x_{i}^{e_{1}}v_{2}x_{i}^{e_{2}} · . . . · v_{k} \frac{∂x_{i}^{e_{k}}}{∂x_{i}} ) · ȳ

and formula (38) follows from (25) and (26). To prove formula (39) notice that β_{X} is the identity on R̄_{old}, subsequently (39) is clear. All statements made about the automorphisms γ_{U} (U ∈ R_{old}) are evident. Notice that the map γ̄_{U} depends only on the image Ū of U in R̄_{old}.

For later use we introduce now a notation for some elements in Γ(G, π) obtained by composing the α_{i} and β_{X}. For every g ∈ G we choose X_{g} ∈ ⟨x_{1}, …, x_{n−1}⟩ with π(X_{g}) = g and define β_{g} := β_{X_{g}}. We infer from formulas (38) and (39)

β̄_{g}ᾱ_{i}^{m}β̄_{g^{−1}}(u + A · ȳ) = u + (∂_{i}(u)mg + A) · ȳ (u ∈ R̄_{old}, A ∈ ℤ[G])

for all g ∈ G, m ∈ ℤ. Given an element B = m_{1}g_{1} + … + m_{k}g_{k} ∈ ℤ[G] (m_{1}, …, m_{k} ∈ ℤ, g_{1}, …, g_{k} ∈ G) we define η_{B,i} ∈ Aut(F_{n}) by

η_{B,i} := β_{g_{1}}α_{i}^{m_{1}}β_{g_{1}^{−1}} ∘ … ∘ β_{g_{k}}α_{i}^{m_{k}}β_{g_{k}^{−1}}.  (41)

for i = 1, …, n − 1. By Lemma 4.2 it follows that η_{B,i} ∈ Γ(G, π) for i = 1, …, n and B ∈ ℤ[G]. Using the notation of Section 3.2 we have

η̄_{B,i}(u + A · ȳ) = u + (∂_{i}^{B}(u) + A) · ȳ (u ∈ R̄_{old}, A ∈ ℤ[G])

for i = 1, …, n − 1 and B ∈ ℤ[G]. This formula makes it clear that

η̄_{m_{1}B_{1}+m_{2}B_{2},i} = η̄_{B_{1},i}^{m_{1}} ∘ η̄_{B_{2},i}^{m_{2}}

holds for all i = 1, …, n − 1 and B_{1}, B_{2} ∈ ℤ[G] and m_{1}, m_{2} ∈ ℤ. Using the representation (1) we set

Aut^{+}(F_{n}) := { ϕ ∈ Aut(F) det(ρ_{1}(ϕ)) = 1 }.

The subgroup Aut^{+}(F_{n}) is of index 2 in Aut(F_{n}). For n ≥ 3 it is a perfect group, that is Aut^{+}(F_{n}) is equal to its own commutator subgroup. We note that all elements η̄_{B,i} are in Aut^{+}(F_{n}).

5 Arithmeticity in case of a redundant presentation

In this section we prove Theorem 1.4: The decomposition of R̄ to its isotypic components shows that our main goal is to prove that the image of Γ̂(G, π) in (34) contains a finite index subgroup of each component

SL(n − 1, M(h_{i}, R_{i}^{op})) = SL((n − 1)h_{i}, R_{i}^{op}).

To prove this we will show that this image contains sufficiently many unipotent elements with nonzero entries in the bottom strip of h_{i} rows and also in the most right hand strip of h_{i} columns of this SL((n−1)h_{i}, R_{i}^{op}). Then we can appeal to a theorem of Vaserstein which ensures that the elementary matrices with entries from a non-zero two-sided ideal of R_{i} generate a finite index subgroup of the arithmetic group SL((n − 1)h_{i}, R_{i}^{op}).

5.1 Elementary matrices

This section sets up certain notations and facts about elementary matrices. Let S be an associative ring with unity. We denote by GL(n, S) (n ∈ ℕ) the group of invertible n × n matrices with entries in S. We write I_{n} for the identity matrix. Given 1 ≤ i, j ≤ n with i ≠ j and s ∈ S we define E_{ij}(s) ∈ GL(n, S) to be the corresponding elementary matrix, that is E_{ij}(s) is equal to I_{n} except for the (i, j) position, where we put s. For 1 ≤ i, j ≤ n with i ≠ j, 1 ≤ k, l ≤ n with k ≠ l and s_{1}, s_{2} ∈ S the Steinberg relations between the elementary matrices are

[E_{ij}(s_{1}), E_{kℓ}(s_{2})] := \{ \begin{matrix}1 & for j ≠ k, & i ≠ l, \\ E_{il}(s_{1}s_{2}) & for j = k, & i ≠ l, \\ E_{kj}(−s_{2}s_{1}) & for j ≠ k, & i = l, \end{matrix}  (42)

where [E_{ij}(s_{1}), E_{kl}(s_{2})] = E_{ij}(s_{1})E_{kl}(s_{2})E_{ij}(s_{1})^{−1}E_{kl}(s_{2})^{−1}. For every admissible pair i, j we have E_{ij}(s_{1})E_{ij}(s_{2}) = E_{ij}(s_{1} + s_{2}) for all s_{1}, s_{2} ∈ S. Let 𝔞 ⊂ S be an ideal of S. Given natural numbers m_{1}, m_{2} ∈ ℕ we denote by M(m_{1}, m_{2}; 𝔞) the space of m_{1} × m_{2}-matrices with entries in 𝔞. Suppose now that m_{1} + m_{2} = n. For A ∈ M(m_{1}, m_{2}; 𝔞) we define

H(A) := ( \begin{matrix}I_{m_{2}} & 0 \\ A & I_{m_{1}} \end{matrix} ) , V (A) := ( \begin{matrix}I_{m_{2}} & A^{t} \\ 0 & I_{m_{1}} \end{matrix} )

Here A^{t} stands for the transpose of the matrix A. Suppose that A := (a_{ij}) where i = 1, …, m_{1}, j = 1, …, m_{2} with a_{ij} ∈ 𝔞, then we have

H(A) = ∏_{i=1}^{m_{1}} ∏_{j=1}^{m_{2}} E_{m_{2}+i j}(a_{ij}), V (A) = ∏_{i=1}^{m_{1}} ∏_{j=1}^{m_{2}} E_{j m_{2}+i}(a_{ij}).

Given m_{1}, m_{2} ∈ ℕ with m_{1} + m_{2} = n and an ideal 𝔞 ≤ S we define

H(m_{1}, m_{2}; 𝔞) := { H(A) A ∈ M(m_{1}, m_{2}; 𝔞) },
V (m_{1}, m_{2}; 𝔞) := { V (A) A ∈ M(m_{1}, m_{2}; 𝔞) }.

Both H(m_{1}, m_{2}; 𝔞) and V (m_{1}, m_{2}; 𝔞) are abelian subgroups of GL(n, S). We apply this notation in case of the orders M(h_{i}, R_{i}) of the simple factor rings I_{i}(ℚ[G]) of the rational group ring ℚ[G]. So let K be a number field and D a finite dimensional division algebra over K with center K. Let R be a order in D. It is well known that every two-sided ideal 𝔞 ≤ R is additively finitely generated and has finite index in R if it is non-zero. We have

Proposition 5.1 Let n ∈ ℕ with n ≥ 3 and let m_{1}, m_{2} ∈ ℕ satisfy m_{1} + m_{2} = n. Let 𝔞 be a non-zero two-sided ideal of R Then H(m_{1}, m_{2}; 𝔞) and V (m_{1}, m_{2}; 𝔞) generate a subgroup of finite index in SL(n, R).

Proof: Let U be the sugroup generated by H(m_{1}, m_{2}; 𝔞) and V (m_{1}, m_{2}; 𝔞). Using the Steinberg relations (42) we conclude that there is a non-zero two-sided ideal 𝔟 (= 𝔞^{2}) in R such that every E_{ij}(b) (b ∈ 𝔟, i ≠ j) is in U. The main result of [33] implies that U has finite index in SL(n, R).

5.2 Constructing elements of G_{G,π}^{1}(ℤ)

Let F_{n} (n ∈ ℕ, n ≥ 3) be the free group with basis x_{1}, …, x_{n−1}, y, G a finite group with generators g_{1}, …, g_{n−1} and let

π : F_{n} → G π(x_{1}) = g_{1}, …, π(x_{n−1}) = g_{n−1}, π(y) = 1

be the redundant presentation resulting from the data. Notice the assumption n ≥ 3, if n = 2 then G is cyclic. We enclude this case into the next section. Given i = 2, …, ℓ and an ideal 𝔞 of the order R_{i}^{op}, let us define

H_{i}(𝔞) := H_{i}(h_{i}, (n − 2)h_{i}; 𝔞) ≤ GL(n, ℤ) × ∏_{i=2}^{ℓ} GL(n − 1, M(h_{i}, R_{i}^{op}))  (43)

to be H(h_{i}, (n − 2)h_{i}; 𝔞) ≤ GL((n − 1)h_{i}, R_{i}^{op}) = GL(n − 1, M(h_{i}, R_{i}^{op})) but considered as a subgroup of the big direct product on the right hand side of (43) (with all components of the factors different from the i-th factor equal to the identity matrix). In case i = 1 we have O_{1} = ℤ and we define H_{1}(𝔞) = H_{1}(1, n − 1; 𝔞) with n − 1 replaced by n. Let us also define

V_{i}(𝔞) := V_{i}(h_{i}, (n − 2)h_{i}; 𝔞) ≤ GL(n, ℤ) × ∏_{i=2}^{ℓ} GL(n − 1, M(h_{i}, R_{i}^{op}))  (44)

in a similar way. We have

Lemma 5.2 Let i be one of the 1, …, ℓ and g ∈ H_{i}(R_{i}^{op}). There is an e ∈ ℕ such that g^{e} is in σ_{G,π}(Γ̂(G, π)) where σ_{G,π} is as in (34).

Proof: We prove the statement for i ≥ 2, for i = 1 the proof is only different in notation. Let i ≥ 2 be fixed and consider g ∈ H_{i}(R_{i}^{op}). There are matrices A_{1}, …, A_{n−2} ∈ M(h_{i}, R_{i}^{op}) such that the i-th component of g_{i} of g is equal to

g_{i} = ( \begin{matrix}I_{h_{i}} & 0 & 0 & … & 0 & 0 \\ 0 & I_{h_{i}} & 0 & … & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & I_{h_{i}} & 0 \\ A_{1} & A_{2} & A_{3} & … & A_{n−2} & I_{h_{i}} \end{matrix} ) ,

all other components being equal to the identity matrix in the corresponding component group. By Corollary 3.7 there are B_{1}, …, B_{n−1} ∈ ℚ[G] such that

∂_{1}^{P_{i}B_{1}} + … + ∂_{n−1}^{P_{i}B_{n−1}}(Id_{i,k}) = A_{k}

for k = 1, …, n − 2. Choose e ∈ ℕ to have sufficiently many divisors so that eP_{i}B_{1}, …, eP_{i}B_{n−2} are in ℤ[G]. Put

γ := η_{eP_{i}B_{1},1} ∘ … ∘ η_{eP_{i}B_{n−1},n−1}

with the η defined in (41). Increase now (if necessary) e to ensure γ ∈ Γ̂(G, π) (see (22)). We have σ_{G,π}(γ) = g^{e} and the lemma is proved.

Lemma 5.3 Let i be one of the 1, …, ℓ and g ∈ V_{i}(R_{i}^{op}). There is an e ∈ ℕ such that g^{e} is in σ_{G,π}(Γ̂(G, π)).

Proof: We prove the statement for i ≥ 2, again for i = 1 the proof is only different in notation. Let i ≥ 2 be fixed and consider g ∈ V_{i}(R_{i}^{op}). There are matrices A_{1}, …, A_{n−2} ∈ M(h_{i}, R_{i}^{op}) such that the i-th component of g_{i} of g is equal to

g_{i} = ( \begin{matrix}I_{h_{i}} & 0 & 0 & … & 0 & A_{1} \\ 0 & I_{h_{i}} & 0 & … & 0 & A_{2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & … & I_{h_{i}} & A_{n−2} \\ 0 & 0 & 0 & … & 0 & I_{h_{i}} \end{matrix} ) ,

The matrices A_{1}, …, A_{n−2} can be considered as elements of M(h_{i}, R_{i}), after all M(h_{i}, R_{i}^{op}) and M(h_{i}, R_{i}) are the same sets. Using the convention of Section 2.3 we consider u := (A_{1}, …, A_{n−2}) as an element of M(h_{i}, D_{i})^{n−2} ⊆ ℚ ⊗_{ℤ} R̄_{old} (see (20)). Then there is e ∈ ℕ such that eu ∈ R̄_{old}. We choose U ∈ R_{old} with Ū = eu. We take e ∈ ℕ to have sufficiently many divisors so that γ̄_{U} (see (37)) is in Γ̂(G, π). We have γ̄_{U} = g^{e} and the lemma is proved.

Since every H_{i}(R_{i}^{op}) or V_{i}(R_{i}^{op}) (i = 1, …, ℓ) is a finitely generated abelian group the two previous lemmas immediately imply

Proposition 5.4 There is e ∈ ℕ such that

∏_{i=1}^{ℓ} H_{i}(eR_{i}) ≤ σ_{G,π}(Γ̂(G, π)) and ∏_{i=1}^{ℓ} V_{i}(eR_{i}) ≤ σ_{G,π}(Γ̂(G, π)).  (45)

Finally we can state

Theorem 5.5 Let n ∈ ℕ satisfy n ≥ 4, then

σ_{G,π}(Γ̂(G, π)) ∩ SL(n, ℤ) × ∏_{i=2}^{ℓ} SL((n − 1)h_{i}, R_{i}^{op}))

is of finite index in SL(n, ℤ) × ∏_{i=2}^{ℓ} SL((n − 1)h_{i}, R_{i}^{op})).

Proof: Just use Proposition 5.4 together with Proposition 5.1.

6 Cyclic groups

In this section we present a detailed study of the case when G is a cyclic group. We restrict ourselves to a particular type of (redundant) presentation, but following the discussion in Section 9.1 this is in fact the general case. An important feature is that we are able to present a set of generators of Γ(G, π), in the case n ≥ 3. For a general group G we are not able to do this. As a consequence we can determine the precise image of Γ(G, π) (not only up to commensurability) acting on its invariant pieces of the relation module. Some of the results here will be needed in Section 7, but we believe that this section is of independent interest. In fact, our method can be observed here in a very simple, almost elementary situation.

Let us call the attention of the reader to the fact that our results concerning cyclic groups, in fact, cover the general case of G being abelian. This follows since every irreducible representation of an abelian group factors through a cyclic quotient, see the discussion in Section 8.2. Let the free group F_{n} (n ≥ 2) be generated by x, y_{1}, …, y_{n−1}. Let us introduce the following elements of Aut(F_{n}). Our convention here is again that values not given are identical to the argument.

• δ_{i}(x) := y_{i}x and ϵ_{i}(x) := xy_{i} for i = 1, …, n − 1,

• φ_{i}(y_{i}) := xy_{i} and ψ_{i}(y_{i}) := y_{i}x for i = 1, …, n − 1,

• λ_{ij}(y_{i}) := y_{j}y_{i} and ν_{ij}(y_{i}) := y_{i}y_{j} for i, j = 1, …, n − 1 with i ≠ j.

A theorem of Nielsen (see [23]) asserts that these elements generate Aut^{+}(F_{n}) where Aut^{+}(F_{n}) is the kernel of the homomorphism ρ_{1} : Aut(F_{n}) → GL(n, ℤ) followed by the determinant. If Γ ≤ Aut(F_{n}) is a subgroup, we define Γ^{+} := Γ ∩ Aut^{+}(F_{n}). Let us further introduce

• κ_{jk}(x) := x[y_{j}, y_{k}], κ_{ijk}(y_{i}) := y_{i}[y_{j}, y_{k}] for 1 ≤ i, j, k ≤ n − 1 with i ≠ j, i ≠ k,

• τ_{ij}(y_{i}) := y_{i}[x, y_{j}] for 1 ≤ i, j ≤ n − 1 with i ≠ j.

The set T_{n} consisting of the κ_{jk}, κ_{ijk}, τ_{ij} (with indices as above), the δ_{i}^{−1} ∘ ϵ_{i}, φ_{i}^{−1} ∘ ψ_{i} for i = 1, …, n − 1 and the λ_{ij} ∘ ν_{ij}^{−1} for 1 ≤ i, j ≤ n − 1 with i ≠ j generates the group IA(F_{n}) by another theorem of Nielsen (see [23]). Notice that T_{n} is contained in Γ(G, π) for every presentation π of an abelian group G. Let us also introduce the following notation concerning matrix groups. We write

Γ^{1}(n, m) ≤ SL(n, ℤ)

for the subgroup of SL(n, ℤ) consisting of those elements having a first row which is congruent to (1, 0, …, 0) modulo m ∈ ℕ. We need

Lemma 6.1 Let n and m be natural numbers with n ≥ 3. The group Γ^{1}(n, m) is generated by the elementary matrices

E_{ij}(1) (i, j = 1, …, n, i ≠ j, i ≠ 1), E_{1j}(m) (j = 2, …, n).

This lemma is an elementary exercise using the euclidean algorithm in ℤ. We first treat the cyclic group of order 2 which we call C_{2}. We have singled out this case since it is completely elementary and also plays a special role in the proof of Corollary 1.3. Let g be a generator of C_{2}. We choose the presentation

π : F_{n} → C_{2}, π(x) = g, π(y_{1}) = … = π(y_{n−1}) = 1.

The corresponding relation module R̄ has the following elements as a ℤ-basis.

\overline{x^{2}}, ȳ_{1}, …, ȳ_{n−1}, \overline{xy_{1}x^{−1}} = gȳ_{1}, …, \overline{xy_{n−1}x^{−1}} = gȳ_{n−1}.

The ℚ-vector space ℚ ⊗_{ℤ} R̄ decomposes as ℚ ⊗_{ℤ} R̄ = V_{1} ⊕ V_{−1} where V_{1}, V_{−1} are the ±1 eigenspaces of g respectively. Set R̄_{1} := R̄ ∩ V_{1} and R̄_{−1} := R̄ ∩ V_{−1}. Notice that R̄_{1} + R̄_{−1} is of finite index in R̄. Introduce

v_{i} := ȳ_{i} + \overline{xy_{i}x^{−1}}, w_{i} := ȳ_{i} − \overline{xy_{i}x^{−1}}  (i = 1, …, n − 1).  (46)

Then \overline{x^{2}}, v_{1}, …, v_{n−1} is a ℤ-basis of R̄_{1} and w_{1}, …, w_{n−1} is a ℤ-basis of R̄_{−1}. Since Γ(C_{2}, π) leaves R̄_{1} and R̄_{−1} invariant we obtain, with the above ℤ-bases being chosen, representations

σ_{1} : Γ^{+}(C_{2}, π) → GL(n, ℤ),  σ_{−1} : Γ^{+}(C_{2}, π) → GL(n − 1, ℤ).  (47)

With all these data given we have:

Proposition 6.2 Let n ≥ 2 be a natural number. The following hold.

(i) The group Γ^{+}(C_{2}, π) is generated by the automorphisms δ_{i}, ϵ_{i} (i = 1, …, n − 1), the λ_{ij} and ν_{ij} (i, j = 1, …, n − 1, i ≠ j), the φ_{i}^{2} (i = 1, …, n − 1) and the elements of the set T_{n} introduced above. (ii) The image of σ_{1} is equal to Γ^{1}(n, 2). (iii) The index of Γ^{+}(C_{2}, π) in Aut^{+}(F_{n}) is 2^{n} − 1. (iv) The representation σ_{−1} is surjective onto GL(n − 1, ℤ).

(v) The group σ_{−1}(IA(F_{3})) ≤ GL(2, ℤ) is generated by the matrices

( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) ,  ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ,  ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) ,  ( \begin{matrix}1 & 0 \\ 2 & 1 \end{matrix} ) .

The proof of this proposition is elementary but repeated below in the more general case, so we skip it here. Items (ii), (iv) and (v) follow from formulas describing the action of the given generators of Γ^{+}(C_{2}, π) on the bases in R̄_{1} and R̄_{−1}. An important ingredient is Lemma 6.1. Note that Γ^{1}(2, 2) is generated by the elementary matrices

E_{21}(1) = ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) , E_{12}(2) = ( \begin{matrix}1 & 2 \\ 0 & 1 \end{matrix} ) .

For Γ^{1}(2, m) (m ≥ 3) the analogous statement is not true. Proposition 6.2 already implies Corollary 1.3. We turn now to the case of a general cyclic group C_{m} = ⟨g⟩ (m ∈ ℕ). We choose the presentation

π : F_{n} → C_{m}, π(x) = g, π(y_{1}) = … = π(y_{n−1}) = 1.

The corresponding relation module R̄ has the following 1 + m(n − 1) elements as a ℤ-basis

\overline{x^{m}}, \overline{x^{k}y_{1}x^{−k}}, …, \overline{x^{k}y_{n−1}x^{−k}} (k = 0, …, m − 1).

The rational group ring of C_{m} decomposes as

ℚ[C_{m}] = ∏_{d|m} ℚ(ζ_{d}).  (48)

Here ζ_{d} ∈ ℂ is a primitive d-th root of unity. We consider each of the ℚ(ζ_{d}) as a C_{m}-module letting g act by multiplication with ζ_{d}. The ring of integers ℤ(ζ_{d}) is then a ℤ[C_{m}]-submodule of ℚ(ζ_{d}). We mimic the decomposition (48) inside the relation module by introducing

v_{i}(d, m) := ∑_{k=0}^{m−1} Tr(ζ_{d}^{k}) \overline{x^{k}y_{i}x^{−k}}  (49)

for i = 1, …, n−1. Here Tr(ζ_{d}^{k}) is the trace of ζ_{d}^{k} taken from ℚ(ζ_{d}) to ℚ. Notice that the v_{i}(d, m) are generalisations of the generators in (46). Let R̄_{d} be the C_{m}-submodule of R̄ generated by the v_{i}(d, m) for i = 1, …, n − 1 if d > 1 and define R̄_{1} to be the C_{m}-submodule of R̄ generated by the v_{i}(1, m) for i = 1, …, n − 1 together with \overline{x^{m}}. By setting

Λ_{d}(v_{i}(d, m)) := (0, …, 0, ∑_{k=0}^{m−1} Tr(ζ_{d}^{k})ζ_{d}^{k}, 0, …, 0) (i = 1, …, n − 1)

(with the non-zero entry in the i-th component) for d > 1 and extending C_{m}-linearly we obtain well defined additive homomorphisms

Λ_{d} : R̄_{d} → ℤ(ζ_{d})^{n−1} (d|m, d > 1).

The following are true

• the homomorphism Λ_{d} is C_{m}-equivariant and injective (onto an ideal in ℤ(ζ_{d})),

• the ℤ-rank of R̄_{d} is (n − 1)φ(d) for d > 1 and n for d = 1,

• R̄_{d} ∩ R̄_{d′} = ⟨0⟩ for distinct divisors d, d′ of m,

• the (direct) sum of all R̄_{d} (d|m) has finite index in R̄,

• each R̄_{d} (d|m) is left invariant by Γ(C_{m}, π).

As above we transport the action of Γ(C_{m}, π) on R̄_{d} to the image of λ_{d} and obtain our representations

σ_{d} : Γ(G, π) → GL(n − 1, ℚ(ζ_{d})) (d|m, d > 1).

To control the image of σ_{d} we first show:

Proposition 6.3 Let n and m be natural numbers with n ≥ 3. The group Γ^{+}(C_{m}, π) is generated by the automorphisms δ_{i}, ϵ_{i} (i = 1, …, n − 1), the λ_{ij} and ν_{ij} (i, j = 1, …, n − 1, i ≠ j) and the φ_{i}^{m}, (i = 1, …, n − 1) and the elements of the set T_{n} introduced above.

Proof: Since every element of Γ(C_{m}, π) has to leave the normal closure of x^{m} in F_{n} invariant and has to fix x modulo R we see that ρ_{1}(Γ^{+}(C_{m}, π)) has to be contained in Γ^{1}(n, m) (for ρ_{1} see (1)). Examining the action of elements in the statement of the proposition on F_{n}/F′_{n} and using Lemma 6.1 we find that ρ_{1}(Γ^{+}(C_{m}, π)) = Γ^{1}(n, m). Now our set of generators contains enough elements to generate IA(F_{n}) by the theorem of Nielsen and also enough elements to generate the image of Γ^{+}(G, π) in SL(n, ℤ) by Lemma 6.1. The proposition follows.

Finally we can state:

Proposition 6.4 Let n and m be natural numbers with n ≥ 2. The image of the representation σ_{d} (d|m, d > 1) is GL^{+}(n − 1, ℤ(ζ_{d})) where GL^{+}(n − 1, ℤ(ζ_{d})) is the subgroup consisting of those elements in GL(n − 1, ℤ(ζ_{d})) which have a power of ζ_{d} as determinant.

Proof: Assume first that n ≥ 3. Evaluating σ_{d} on the generators from Proposition 6.3 we see that σ_{d}(Γ^{+}(C_{m}, π)) is contained in GL^{+}(n−1, ℤ(ζ_{d})). We now show that all elementary matrices E_{ij}(ζ_{d}^{k}) (i, j = 1, … n − 1, k ∈ ℤ) are in the image of σ_{d}. This is done by evaluating σ_{d} on some of the generators from Proposition 6.3. We report only a special case: Take τ_{ij} for a pair (i, j) with i ≠ j. We have τ_{ij}(y_{i}) = y_{i}[x, y_{j}]. This leads to

τ̄_{ij}(v_{i}(d, m)) = v_{i}(d, m) + gv_{j}(d, m) − v_{j}(d, m)

which in turn shows that E_{ij}(ζ_{d} − 1) is in the image of σ_{d}. Since E_{ij}(1) is also present (use λ_{ij}) we have shown that E_{ij}(ζ_{d}) is contained in the image of σ_{d}. After more elementary considerations like that we can show that all the elementary matrices with entries in ℤ(ζ_{d}) are in the image of σ_{d}. These matrices generate SL(n − 1, ℤ(ζ_{d})): For n ≥ 4 this follows from results in [33], for n = 3 and d ≥ 5 we apply the main result of [34], in case n = 3 and d ≤ 4 the ring ℤ(ζ_{d}) is euclidean, in case n = 2 nothing has to be proved. As a last step we use the φ_{i}^{−1} ∘ ψ_{i} (i = 1, …, n − 1) to show that the image of σ_{d} contains a matrix with determinant equal to ζ_{d}. This finishes the proof for n ≥ 3. Let us finally discuss the case n = 2 which seems trivial, but in fact is not. Note that GL(1, ℤ(ζ_{d})) contains elements of infinite order whenever d ≥ 5, d ≠ 6 holds. Since we do not have generators for Γ^{+}(2, m) (m ≥ 3) at hand we cannot directly rule out the possibility that such an element lies in the image of σ_{d}. We argue as follows. Let φ ∈ Γ^{+}(2, m) be such that for some divisor d of m the image σ_{d}(φ) has a determinant of infinite order. We extend φ to an element of ψ of Γ^{+}(3, m) by setting ψ(x) := φ(x), ψ(y_{1}) := φ(y_{1}), ψ(y_{2}) := y_{2}. Since the determinants of σ_{d}(φ) and σ_{d}(ψ) coincide we have finished also the case n = 2.

Proposition 6.4 allows us to conclude the following result concerning the abelianisations of the images of Γ(C_{m}, π) under the representations σ_{d} (d|m, d > 1).

Corollary 6.5 Let n and m be natural numbers with n ≥ 2. Then σ_{d}(Γ(C_{m}, π)) (d|m, d > 1) has finite abelianisation.

Proof: It is well known that GL^{+}(n − 1, ℤ(ζ_{d})) has finite abelianisation for all d ∈ ℕ: For n = 2 this fact is obvious, for n = 3 and d = 2, 3, 4, 6 a presentation of SL(2, ℤ(ζ_{d})) (see [8]) can be used, for d = 5 or d ≥ 7 this fact is contained in [31]. For other values of n, d the group GL^{+}(n − 1, ℤ(ζ_{d})) has Kazdhan’s property (T). Since σ_{d}(Γ(C_{m}, π)) contains σ_{d}(Γ^{+}(C_{m}, π)) as a subgroup of finte index the result follows.

We finally remark that

σ_{d}(IA(F_{n})) ≤ GL^{+}(n − 1, ℤ(ζ_{d})) (d|m, d > 1)

is equal to the full congruence subgroup of GL^{+}(n − 1, ℤ(ζ_{d})) with respect to the ideal of ℤ(ζ_{d}) generated by ζ_{d} − 1. This can be shown by a more detailed analysis of the arguments involved in the proof of Proposition 6.4.

7 Completion of proofs

In this section we complete the proofs of all the results promised in the introduction. Theorem 5.5 proves Theorem 1.4. Both theorems were formulated only for n ≥ 4 but our method applies also to the case n = 3. The only obstacle for n = 3 is that congruence elementary matrices in SL(2) = SL(n − 1) do not always generate a finite index subgroup, for example the matrices

( \begin{matrix}1 & 3 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 3 & 1 \end{matrix} )

do not generate a subgroup of finite index in SL(2, ℤ). By the result of Vaserstein [34] this is a real obstacle (in the case of commutative rings of endomorphisms) only for SL(2, ℤ) and SL(2) of a ring of integers in an imaginary quadratic number field. Thus our results still work for n = 3 in many cases. For example if the group algebra ℚ[G] of the finite group G has (except for the trivial module) no irreducible module N with dim_{D}(N) = 1 (D = End_{G}(N)). If there are such modules N but if in each case D is a number field which is not ℚ or imaginary quadratic we are also fine. The other cases remain open.

Proof of Theorem 1.1: n = 2: In this case we just consider the (surjective) representation

ρ_{1} : Aut(F_{2}) → Aut(F_{2}/F′_{2}) ≅ GL(2, ℤ).

It is well known that GL(2, ℤ) contains a free subgroup of any rank (≥ 2) of finite index. Hence any finitely generated group is the image of a subgroup of finite index in Aut(F_{2}). Note that the groups appearing in Theorem 1.1 as image groups are finitely generated since they are arithmetic groups. n = 3: Here we use the representation σ_{−1} : Γ^{+}(C_{2}, π) → GL(2, ℤ) from Section 6 to find for every r ∈ ℕ with r ≥ 2 a subgroup of finite index in Aut(F_{3}) which can be mapped onto the free group of rank r. We then argue as in the case n = 2. n ≥ 4: We first consider the case k = 1. For h, m ∈ ℕ we have to find a subgroup Γ ≤ Aut(F_{n}) of finite index and a representation

ρ : Γ → SL((n − 1)h, ℚ(ζ_{m}))^{m}

such that ρ(Γ) is commensurable with SL((n − 1)h, ℤ(ζ_{m}))^{m}. To do this we take the group

G := S_{h+1} × C_{m} × C_{m}.

As an input for Theorem 5.5 we need suitable ℚ[G]-modules. We let P M_{h+1} be the standard h + 1 dimensional ℚ[S_{h+1}]-permutation module and let M_{h} be the kernel of the augmentation map from P M_{h+1} to ℚ. The ℚ[S_{h+1}]-module M_{h} is irreducible and has ℚ-dimension h. We put N := ℚ(ζ_{m}) ⊗ℚ M_{h} and consider N as a ℚ-vector space. Here, as before, ζ_{m} ∈ ℂ is a primitive m-th root of unity. Let ϵ : C_{m} × C_{m} → ⟨ζ_{m}⟩ be a surjective homomorphism. We turn the ℚ-vector space N = ℚ(ζ_{m}) ⊗ℚ M_{h} into a ℚ[G]-module N_{ϵ} by letting S_{h+1} act on M_{h} and letting g ∈ C_{m} × C_{m} act by multiplication by ϵ(g) on ℚ(ζ_{m}). The following are clear for every surjective homomorphism ϵ : C_{m} × C_{m} → ⟨ζ_{m}⟩.

• dimℚ(N_{ϵ}) = φ(m)h,

• End_{G}(N_{ϵ}) = ℚ(ζ_{m}),

• dim_{ℚ(ζ_{m})}(N_{ϵ}) = h,

• if ϵ_{1}, ϵ_{2} : C_{m} × C_{m} → ⟨ζ_{m}⟩ are two surjective homomorphisms with distinct kernels then N_{ϵ_{1}} and N_{ϵ_{2}} are not isomorphic as ℚ[G]-modules.

Our group G can be generated by 3 elements: take g_{1}, g_{2}, g_{3} ∈ S_{h+1} such that g_{3} and the commutator of g_{1}, g_{2} generate S_{h+1} and consider the elements (g_{1}, 1, 0), (g_{2}, 0, 1), (g_{3}, 0, 0) in G. Let π : F_{n} → G be a redundant presentation of G (remember the assumption is n ≥ 4). Since the number of surjective homomorphisms ϵ : C_{m} × C_{m} → ⟨ζ_{m}⟩ with pairwise distinct kernels is bigger or equal to m we may apply Theorem 5.5 to obtain a subgroup Γ ≤ Aut(F_{n}) and a representation

σ : Γ → GL((n − 1)h, ℚ(ζ_{m}))^{m}

such that σ(Γ) ∩ SL((n − 1)h, ℤ(ζ_{m}))^{m} is of finite index in SL((n − 1)h, ℤ(ζ_{m}))^{m}. Here we do not know whether σ(Γ) ∩ SL((n − 1)h, ℤ(ζ_{m}))^{m} is of finite index in σ(Γ). However, there is a subgroup Δ ≤ GL((n − 1)h, ℤ(ζ_{m}))^{m} of finite index such that SL((n − 1)h, ℤ(ζ_{m}))^{m} ≤ Δ and such that Δ ≤ Z · SL((n − 1)h, ℤ(ζ_{m}))^{m} with a subgroup Z ≤ GL((n − 1)h, ℤ(ζ_{m}))^{m} which is central and satisfies Z ∩SL((n−1)h, ℤ(ζ_{m}))^{m} = ⟨1⟩. Let Γ_{0} ≤ Γ be the inverse image of Δ under σ. The representation

σ_{0} : Γ_{0} → Δ/Z = SL((n − 1)h, ℤ(ζ_{m}))^{m}

satisfies the requirements for k = 1. Another way of looking at the last construction is to project a finite index subgroup of the image of σ(Γ) into PGL(n − 1, ℤ(ζ_{m})) which contains PSL(n − 1, ℤ(ζ_{m})) as a finite index subgroup. We then intersect this subgroup in PGL(n − 1, ℤ(ζ_{m})) with PSL(n − 1, ℤ(ζ_{m})) and pull the resulting subgroup of PSL(n − 1, ℤ(ζ_{m})) back to obtain a finite index subgroup of Γ which is mapped into SL(n − 1, ℤ(ζ_{m})). We turn now to the general case. Let Γ_{1}, …, Γ_{k} ≤ Aut(F_{n}) be the subgroups of finite index and

σ_{i} : Γ_{i} → SL((n − 1)h_{i}, ℤ(ζ_{m_{i}}))^{m_{i}} (i = 1, …, k)

be the representations constructed above. Set Γ := Γ_{1} ∩ … ∩ Γ_{k} and consider the representation

σ : Γ → ∏_{i=1}^{k} SL((n − 1)h_{i}, ℤ(ζ_{m_{i}}))^{m_{i}}

which maps an element γ ∈ Γ to the tuple (σ_{1}(γ), …, σ_{k}(γ)). The image of Γ projects onto an arithmetic subgroup in each of the factors of the direct product. We finish the proof by applying the following lemma.

Lemma 7.1 Let H_{1}, …, H_{k} be pairwise non-isogeneous ℚ-defined simple linear algebraic groups which all have real rank greater or equal to 2. Let

Γ ≤ ∏_{i=1}^{k} H_{i}^{m_{i}}(ℚ)

be a subgroup such that all projections into the factors H_{i}^{m_{i}}(ℚ) are arithmetic groups. Then Γ is an arithmetic subgroup of ∏_{i=1}^{k} H_{i}^{m_{i}}, that is Γ is commensurable with ∏_{i=1}^{k} H_{i}^{m_{i}}(ℤ).

This lemma is proved using Margulis-superrigidity. See [20] item (iv) on pages 328-329 or [17] Section 2.

8 (SL) or not (SL)?

In this section we show some results concerning the question whether ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1} is of finite index in ρ_{G,π}(Γ(G, π)) or not (for the notation see the introduction). The relevance of this question and the notation are explained in the introduction. We change here from the consideration of the rational relation module to its complex version. This brings certain technical advantages and still reflects the problem. Let us set up some notation for this section. Let G be a finite group and π : F_{n} → G an epimorphism with kernel R. We define Γ(G, π), G_{G,π} and G_{G,π}^{1} as in the introduction and consider the action of Γ(G, π) now on the complex relation module ℂ ⊗ℤ R̄. The result of Gaschütz now says that

ℂ ⊗_{ℤ} R̄ = ℂ ⊕ ℂ[G]^{n−1}

as ℂ[G]-modules. If Q is an irreducible ℂ[G]-module and M is an arbitrary ℂ[G]-module we keep the notation I_{Q}(M) for the Q-isotypic component inside M. We hope that this creates no confusion with the I_{N}(M) of Section 2.1 where N, M are ℚ[G]-modules. Let Q be an non-trivial irreducible ℂ[G]-module of complex dimension h_{Q}. Let G_{G,π,Q} be the stabiliser of I_{Q}(ℂ ⊗ℤ R̄) inside G_{G,π} (see 21). By our usual procedure we obtain a representation

ρ_{G,π,Q} : Γ(G, π) → G_{G,π,Q} = GL(n − 1, M(h_{Q}, ℂ)) = GL((n − 1)h_{Q}, ℂ).  (50)

We call the pair (G, Q) of type (SL) if ρ_{G,π,Q}(Γ(G, π)) ∩ SL((n − 1)h_{Q}, ℂ) is of finite index in ρ_{G,π,Q}(Γ(G, π)) for every n ∈ ℕ and every epimorphism π : F_{n} → G. We say the finite group G is of type (SL) if for all non-trivial irreducible ℂ[G]-module Q the pair (G, Q) is of type (SL). Note that, since Γ(G, π) is finitely generated, ρ_{G,π,Q}(Γ(G, π)) ∩ SL((n − 1)h_{Q}, ℂ) is of finite index in ρ_{G,π,Q}(Γ(G, π)) if and only if ρ_{G,π,Q}(Γ(G, π)) contains no element with a determinant of infinite order. Note that the above implies that once we find a finite group G which is not of type (SL), we have found a subgroup Γ of finite index in some Aut(F_{n}) which has an epimorphism onto ℤ. Let G be a finite group G and N_{2}, …, N_{ℓ} its non-trivial irreducible ℚ[G]-modules. Let π : F_{n} → G be a surjective homomorphism. The following are equivalent:

• For all i = 2, …, ℓ the intersection

σ_{G,π,i}(Γ(G, π)) ∩ SL((n − 1)h_{i}, R_{i}^{op})

has finite index in σ_{G,π,i}(Γ(G, π)) ≤ GL((n − 1)h_{i}, R_{i}^{op}).

• For all non-trivial irreducible ℂ[G]-modules Q the intersection

ρ_{G,π,Q}(Γ(G, π)) ∩ SL((n − 1)h_{Q}, ℂ)

is of finite index in ρ_{G,π,Q}(Γ(G, π)).

This is seen by introducing the decomposition of the complexifications ℂ ⊗ℚ N_{i} into irreducible ℂ[G]-modules. There are some non-trivial irreducible ℂ[G]-modules Q which are obviously of type (SL):

Proposition 8.1 Let G be a finite group and Q a non-trivial irreducible ℂ[G]-module. If there is an (irreducible) ℚ[G]-module N with Q = ℂ ⊗ℚ N then (G, Q) is of type (SL).

Proof: We note that under the above assumptions the endomorphism algebra D := End_{G}(N) is a division algebra over the rational numbers with ℚ as center. In this case SL(m, R) (i.e. the group of elements of reduced norm 1 in M(m, R)) is of finite index in SL(m, R) for any order R ≤ D and any m ∈ ℕ.

In case of the symmetric groups S_{m} (m ∈ ℕ) it is known, that every ℂ[S_{m}]-module satisfies the assumptions of Proposition 8.1, see [32], Section 13. This implies:

Corollary 8.2 All the symmetric groups S_{m} are of type (SL).

In case of the alternating group A_{5} Proposition 8.1 applies to the irreducible ℂ[A_{5}]-modules of dimensions 4 and 6 but not to the two 3-dimensional irreducible ℂ[A_{5}]-modules. They fit together to form a 6-dimensional irreducible ℚ[A_{5}]-module N with End_{G}(N) = ℚ(\sqrt{5}).

8.1 Reduction Steps

After the definitions above we discuss some reduction steps which will allow us to recognise pairs (G, Q) of type (SL).

Lemma 8.3 Let G be a finite group with normal subgroup H_{0} and quotient H = G/H_{0}. Let Q be a non-trivial irreducible ℂ[G]-module on which H_{0} acts trivially. If (H, Q) is of type (SL) then (G, Q) also has this property.

Proof: Let π : F_{n} → G (n ≥ 2) be any epimorphism and let R_{G} be the kernel of π. Let π_{0} : F_{n} → G → H be the resulting presentation of H and R_{H} its kernel. Obviously R_{G} is a subgroup of finite index in R_{H}. It is also easily seen that Γ(G, π) is a subgroup of finite index of Γ(H, π_{0}). The inclusion of R_{G} into R_{H} gives rise to a surjective ℂ-linear map

Θ : ℂ ⊗_{ℤ} R̄_{G} → ℂ ⊗_{ℤ} R̄_{H}.

This linear map is equivariant for the induced actions of G and Γ(G, π). By restriction we obtain a map

Θ_{Q} : I_{Q}(ℂ ⊗_{ℤ} R_{G}) → I_{Q}(ℂ ⊗_{ℤ} R_{H})  (51)

which is surjective and H-equivariant. Gaschütz’ result implies that Θ_{Q} is in fact an (Γ(G, π)- equivariant) isomorphism. We have to show that det(ρ_{G,π,Q}(φ)) is of finite order for any φ ∈ Γ(G, π). We compute this determinant on the right hand side of (51) where our assumption implies this property.

Lemma 8.4 Let G be a finite group with normal subgroup H. Let Q be a non-trivial irreducible ℂ[G]-module which is induced from H, that is, Q is of the form Q = Ind_{H}^{G}(Q_{0}) where Q_{0} is an irreducible ℂ[H]-module. If (H, Q_{0}) is of type (SL) then (G, Q) also has this property.

Proof: Let π : F_{n} → G (n ≥ 2) be any epimorphism and let R be the kernel of π. The subgroup π^{−1}(H) ≤ F_{n} is a free group on m = 1 + [G : H](n − 1) generators, we denote it by F_{m}. We then have a commutative diagram

\xymatrix{1 \ar[r] & R \ar[r] & F_{n} \ar[r]^{π} & G \ar[r] & 1 \\ 1 \ar[r] & R \ar[u]^{=} \ar[r] & F_{m} \ar[u] \ar[r]^{π_{0}} & H \ar[u] \ar[r] & 1}

where π_{0} is the restriction of π to F_{m}. Note that R = ker(π) = ker(π_{0}). An element φ ∈ Γ(G, π) stabilises the subgroup F_{m} ≤ F_{n} and as an automorphism of F_{m} it lies in Γ(H, π_{0}). On the vector space ℂ ⊗ℤ R̄ the linear maps induced by φ as an element of Γ(G, π) and by φ as an element of Γ(H, π_{0}) coincide. We denote both by φ̄. The complex relation module ℂ ×ℤ R̄ now has a structure as a ℂ[G]-module and a structure of a ℂ[H]-module. In fact the second is just the restriction of the first. Let Q = Q_{0} ⊕ Q_{1} ⊕ … ⊕ Q_{r} be the decomposition of Q into irreducible ℂ[H]-modules We now prove that

I_{Q}(ℂ ⊗_{ℤ} R̄) = I_{Q_{0}}(ℂ ⊗_{ℤ} R̄) ⊕ I_{Q_{1}}(ℂ ⊗_{ℤ} R̄) ⊕ … ⊕ I_{Q_{r}}(ℂ ⊗_{ℤ} R̄).  (52)

On the left hand side the isotypic component of the ℂ[G]-module Q is taken whereas on the right hand side we see the the isotypic component of the corresponding ℂ[H]-modules. Clearly the left hand side of (52) is contained in the right hand side. Note that, since H is normal in G the module Q is also induced from any of the Q_{0}, …, Q_{r}. The Frobenius reciprocity law ([32], chapter 7) implies that none of the Q_{0}, …, Q_{r} can occur as an irreducible constituent of the restriction of an irreducible ℂ[G]-module different from Q. We have to show that det(ρ_{G,π,Q}(φ)) is of finite order for any φ ∈ Γ(G, π). We compute this determinant on the right hand side of (52) where our assumption implies this property.

8.2 Abelian and metabelian groups

This section contains results which describe the image of ρ_{G,π} for abelian and then for metabelian groups G up to commensurability. In Section 6 we have proved a very precise result in this direction for cyclic groups G. This allowed us to conclude for n ≥ 2 that the image of Γ(G, π) (under certain linear representations) has finite abelianisation. We present here the following much stronger result for n ≥ 3. This result is due to A. Rapinchuk (it appears in a somewhat hidden form in [30], page 150), we thank him for the permission to include it.

Proposition 8.5 Let n ≥ 3 be a natural number, G a finite abelian group G and let π : F_{n} → G be any surjective homomorphism. Then Γ(G, π) has finite abelianisation.

Proof: Since G is abelian we have F′_{n} ≤ R where R is, as always, the kernel of π. Consequently we have IA(F_{n}) ≤ Γ(G, π). Writing Γ̃(G, π) := ρ_{1}(Γ(G, π)) ≤ GL(n, ℤ) for the image of ρ_{1}(Γ(G, π)) in GL(n, ℤ) we have the exact sequence of groups

1 → IA(F_{n}) → Γ(G, π) → Γ̃(G, π) → 1

Note that Γ̃(G, π) is a subgroup of finite index in GL(n, ℤ). Since Γ(G, π) is a finitely generated group we need only show that

H^{1}(Γ(G, π), ℚ) = Hom(Γ(G, π), ℚ) = 0

We shall do that by applying the edge-term sequence of the Hochschild-Serre spectral sequence for the above short exact sequence of groups. We shall use that there is an exact sequence of ℚ-vector spaces

H^{1}(Γ̃(G, π), ℚ) → H^{1}(Γ(G, π), ℚ) → H^{1}(IA(F_{n}), ℚ)^{Γ̃(G,π)}  (53)

First of all the left hand side of (53) is 0 since n ≥ 3 and GL(n, ℤ) has Kazdhan’s property (T). Next we evaluate the right hand side of (53). By a theorem of Bachmuth ([1] and also [11]) there is a Aut(F_{n})-equivariant isomorphism

H^{1}(IA(F_{n}), ℤ) → Hom_{ℤ}(F_{n}/F′_{n}, F′_{n}/γ_{2}(F_{n}))

where γ_{2}(F_{n}) := [[F_{n}, F_{n}], F_{n}] is the third term of the lower central series of F_{n}. Let GL^{±}(n, ℂ) be the subgroup of GL(n, ℂ) consisting of those elements having determinant ±1. Let ℂ^{n} be the standard GL^{±}(n, ℂ)-module of dimension n and Λ^{2}(ℂ^{n}) be its outer square. Consider GL^{±}(n, ℤ) as a subgroup of GL^{±}(n, ℂ). By identifying F′_{n}/γ_{2}(F_{n})) with Λ^{2}(F_{n}/F′_{n}) = Λ^{2}(ℤ^{n}) we obtain an GL^{±}(n, ℤ)-equivariant isomorphism

ℂ ⊗_{ℤ} Hom_{ℤ}(F_{n}/F′_{n}, F′_{n}/γ_{2}(F_{n})) → Hom(ℂ^{n}, Λ^{2}(ℂ^{n})).

It is well known (see [11], page 427) that the right hand side splits as a GL^{±}(n, ℂ)-module as

Hom(ℂ^{n}, Λ^{2}(ℂ^{n})) = ℂ^{n} ⊕ V_{n}

where V_{n} is a GL^{±}(n, ℂ)-module of dimension dimℂ(V_{n}) = n(n+1)(n−2)/2 which is irreducible even as SL(n, ℂ)-module. Hence the space of SL(n, ℂ)-invariants in Hom(ℂ^{n}, Λ^{2}(ℂ^{n})) is 0. Since the Zariski closure of Γ̃(G, π) contains SL(n, ℂ) the group Homℤ(F_{n}/F′_{n}, F′_{n}/γ_{2}(F_{n}))^{Γ̃(G,π)} is 0. This shows that the right hand side of (53) is 0. Together with the above the proposition is proved.

Proposition 8.5 is clearly not true for n = 2, since the left-most term in (53) contributes to the H^{1}(Γ(G, π), ℚ). See also the tables in Section 9.4. As an immediate corollary from Proposition 8.5 we get:

Corollary 8.6 All finite abelian groups are of type (SL).

This corollary can also be deduced from Corollary 6.5 (in fact this corollary is needed in the case n = 2) and Lemma 8.3. But the statement incorporated in Proposition 8.5 (for n ≥ 3) is in fact stronger. A metabelian group is a group G with abelian commutator subgroup G′. We prove:

Proposition 8.7 All metabelian finite groups G are of type (SL)

Proof: We use induction on the order of G and the fact proved above that all finite abelian groups have property (SL). Let Q be a non-trivial irreducible ℂ[G]-module. We may assume by induction and Lemma 8.3 that G acts faithfully on Q. Case 1: the commutator subgroup is not contained in the center Z(G) of G. The restriction of the module Q to G′ cannot be isotypic. Otherwise the abelian subgroup G′ would act by scalar matrices on Q and G′ would have to be contained in the center of G. Now Proposition 24 of [32] says that Q is either isotypic or induced. It follows that Q is induced from a proper subgroup H of G containing G′. By our construction H is normal in G. Hence this case is finished by induction and application of Lemma 8.4. Case 2: the commutator subgroup is contained in the center Z(G) of G. In this case G is nilpotent of class 2. Unless G is abelian (the case treated above) we find g ∈ G with g ∉ Z(G). The subgroup A := ⟨G′, g⟩ generated by G′ and G is abelian, normal in G and not contained in Z(G). We then proceed as in case 1.

9 Concluding remarks and suggestions for further research

9.1 Dependence on the presentation

The results of the current paper can be considered as a first step toward a systematic study of the relation module of a finite group as a Γ(G, π)-module, i.e. an equivariant Gaschütz’s theory. Here we gave a quite satisfactory answer in the case of redundant presentations. It is not clear to what extend this represents the general case.

For a fixed finite group G let us look at all the possible epimorphisms from the free group F_{n} to G and R(n, G) the set of their kernels. The automorphism group Aut(F_{n}) acts on this set. It is easy to see that the equivariant Gaschütz theory that we are trying to develop here depends only on the orbits of Aut(F_{n}) on R(n, G) and not on the actual presentation. Various authors studied the transitivity properties of the action of Aut(F_{n}) on R(n, G). It is not transitive in general (a first example was given by B. H. Neumann [25]) but it is in some interesting cases. For example, it is transitive if G is a cyclic group and n ≥ 2. Thus our results in Section 6 give the full picture for all relation modules of cyclic groups. It is also transitive if n > 2log_{2}(|G|) or if G is solvable and n > d(G), where d(G) denotes the minimal number of generators of G (see [7]). An old conjecture of Wiegold predicts transitivity for finite simple groups if n ≥ 3, in which case again n > d(G) holds. For some partial results see [14], [9] and the references in [27]. It was even suggested in [27] that for every finite group G it is transitive if n > d(G). In [9] it is shown that if a finite group G has spread 2 (i.e. for every g_{1}, g_{2} ∈ G with g_{1} ≠ 1 ≠ g_{2} there is h ∈ G such that ⟨g_{1}, h⟩ = ⟨g_{2}, h⟩ = G) then Aut(F_{n}) acts transitively on the set of kernels of all redundant presentations. Guralnick and Shalev [18] proved that almost all finite simple groups have spread 2. Theorem 1.4 is not true in general if the presentation is not redundant, at least when n = 2, see the A_{5}-example in Section 9.4. The story for n ≥ 3 may be different as Aut(F_{2}) behaves in many ways differently from Aut(F_{n}) for n ≥ 3.

9.2 Representations of IA(F_{n})

A well known theorem of Formanek and Procesi (see [12]) asserts that the automorphism group Aut(F_{n}) has, for n ≥ 3, no faithful linear representation over any field. Their proof suggests a stronger Conjecture: For every linear representation ρ of Aut(F_{n}) the image ρ(Inn(F_{n})) is virtually solvable. As far as we know, in all previously constructed cases of linear representations even ρ(IA(F_{n})) had this property. It should be mentioned that in all the representations ρ studied in this paper, ρ(Inn(F_{n})) is finite but ρ(IA(F_{n})) is far from being virtually solvable and, in fact, our results show that ρ(IA(F_{n})) can be Zariski dense in very large semi-simple groups. Our reprentations seem to be the first known with this property. We have:

Proposition 9.1 Let n ≥ 3, k ≥ 1, h_{1} < … < h_{k}, m_{1}, …, m_{k} be natural numbers. Let ℚ(ζ_{m_{i}}) be the field of m_{i}-th roots of unity and ℤ(ζ_{m_{i}}) its ring of integers. There is a subgroup Γ ≤ IA(F_{n}) of finite index and a representation

ρ : Γ → ∏_{i=1}^{k} SL((n − 1)h_{i}, ℚ(ζ_{m_{i}}))^{m_{i}}

such that ρ(Γ) is commensurable with ∏_{i=1}^{k} SL((n − 1)h_{i}, ℤ(ζ_{m_{i}}))^{m_{i}}.

Proof: The proof of Theorem 1.1 shows that there is a subgroup of finite index Γ_{0} ≤ Aut(F_{n}) and a representation

ρ_{0} : Γ_{0} → SL(n, ℤ) × ∏_{i=1}^{k} SL((n − 1)h_{i}, ℚ(ζ_{m_{i}}))^{m_{i}}  (54)

such that the image ρ_{0}(Γ_{0}) is the internal direct product of its intersections with the factors in (54). These intersections in turn have finite index in the corresponding factor. Using the fact that a normal subgroup of a subgroup of finite index in SL(n, ℤ) (n ≥ 3) is either finite or has finite index we conclude the proof of the prposition.

Another easy consequence of Theorem 1.1 concerns representations of the the outer automorphism group Out(F_{n}). We have:

Proposition 9.2 Let n ≥ 2, k ≥ 1, h_{1} < … < h_{k}, m_{1}, …, m_{k} be natural numbers. Let ℚ(ζ_{m_{i}}) be the field of m_{i}-th roots of unity and ℤ(ζ_{m_{i}}) its ring of integers. There is a subgroup Γ ≤ Out(F_{n}) of finite index and a representation

ρ : Γ → ∏_{i=1}^{k} SL((n − 1)h_{i}, ℚ(ζ_{m_{i}}))^{m_{i}}

such that ρ(Γ) is commensurable with ∏_{i=1}^{k} SL((n − 1)h_{i}, ℤ(ζ_{m_{i}}))^{m_{i}}.

Here we only have to note that the image of the inner automorphisms under any of the representations from Theorem 1.1 is finite. This finite subgroup of the image can be avoided by going to a subgroup of finite index in Γ. As a consequence of Propositions 9.2, 9.2 we get:

Corollary 9.3 The groups IA(F_{3}) and Out(F_{3}) are large.

Our result from Theorem 1.1 can conviniently be summarised in the language of the pro-algebraic completion A(Aut(F_{n})) of Aut(F_{n}). For a definition and discussion of properties of the proalgebraic completion (also called the Hochschild-Mostow group) of a group see [21] and [2]. We have

Proposition 9.4 Let n ≥ 2 be a natural number and S(Aut(F_{n})) be the semisimple part of the connected component of the identity of the pro-algebraic completion A(Aut(F_{n})). Then for every h ∈ ℕ, the group SL((n − 1)h, ℂ) appears infinitely many times as a factor in S(Aut(F_{n})). That is, there is a surjective homomorphism

S(Aut(F_{n})) → ∏_{h=1}^{∞} ∏_{i=1}^{∞} SL((n − 1)h, ℂ).

Propositions 9.2, 9.2 imply similar results for the groups IA(F_{n}), Out(F_{n}) (n ≥ 3).

9.3 Representations of Γ(G, π) into affine groups

In this section we give another construction of a linear representation of subgroups of finite index in Aut(F_{n}). This representation takes values in affine groups (which have a non-trivial unipotent radical). The construction originates from the proof of Gaschütz’ result as presented in [19]. Besides giving new types of image groups, it is of importance for the computer algorithms which we have used to create the computational results described in the next subsection. Let G be a finite group and π : F_{n} → G a surjective homomorphism of the free group F_{n} onto G. Let R be the kernel of π and R̄ the corresponding relation module. Let further Γ(G, π) be the subgroup of Aut(F_{n}) defined in (3). Given a group H and a commutative ring S we write S[H] for the corresponding group ring and I(S[H]) for its augmentation ideal. If H_{0} ≤ H is a subgroup we define I(S[H], H_{0}) to be the two-sided ideal of S[H] generated by the h − 1 for h ∈ H_{0}. Gaschütz’ result (18) is proved (in [19]) by considering the exact sequence

0 → ℚ ⊗ℤ R̄ → I(ℚ[F_{n}])/(I(ℚ[F_{n}], R) · I(ℚ[F_{n}])) → I(ℚ[G]) → 0  (55)

of G-equivariant homomorphisms. The right hand map is induced by π : F_{n} → G while the left hand map comes from the map from R to I(ℚ[F_{n}]) which sends r ∈ R to r − 1. The free group F_{n} acts on the middle term in (55) by multiplication from the left. This leads to an action of the finite group G on this term. We now let Γ(G, π) act on the left hand term as before, on the middle term by its action on F_{n} and trivially on ℚ[G]. It is straightforward to see that the sequence (55) is then Γ(G, π)-equivariant. We obtain a representation

η_{G,π} : Γ(G, π) → Hom_{G}(I(ℚ[G]), ℚ ⊗_{ℤ} R̄) ⋊ G_{G,π}(ℚ).  (56)

The semi-direct product in (56) is formed with respect to the action of G_{G,π}(ℚ) on ℚ ⊗ℤ R̄. Our methods show:

Theorem 9.5 Assume n is a natural number with n ≥ 4. Let π : F_{n} → G be a redundant presentation of the finite group G. Then

η_{G,π}(Γ(G, π)) ∩ Hom_{G}(I(ℚ[G]), ℚ ⊗ℤ R̄) ⋊ G_{G,π}^{1}(ℚ)

is of finite index in the arithmetic group Hom_{G}(I(ℤ[G]), R̄) ⋊ G_{G,π}^{1}(ℤ).

Theorem 9.5 can then be used to prove

Theorem 9.6 Let n ≥ 2, h, m be natural numbers. Let ℚ(ζ_{m}) be the field of m-th roots of unity and ℤ(ζ_{m}) its ring of integers. There is a subgroup Γ ≤ Aut(F_{n}) of finite index and a representation

η : Γ → (ℚ(ζ_{m})^{(n−1)h} ⋊ SL((n − 1)h, ℚ(ζ_{m})))^{m}

such that η(Γ) is commensurable with (ℤ(ζ_{m})^{(n−1)h} ⋊ SL((n − 1)h, ℤ(ζ_{m})))^{m}.

9.4 Computational results

An important feature of our methods is that subgroups of quite high indices in Aut(F_{n}) may be computationally located. We describe here some of the results of our computer calculations. The computations where done using the computer algebra system MAGMA. We shall describe our approach now in detail for n = 2. Let F_{2} := ⟨x, y⟩ be the free group. The automorphism group Aut^{+}(F_{2}) is generated by α, β which are given by

α(x) = y^{−1}, α(y) = x β(x) = x^{−1}y^{−1}, β(y) = x.

The group Aut^{+}(F_{2}) is finitely presented, a presentation is (see [24])

Aut^{+}(F_{2}) = ⟨ α, β  α^{4} = β^{3} = α^{2}β^{2}α^{2}βαβα^{2}β^{2}α = 1 ⟩.  (57)

Let π : F_{2} → G be a presentation of a finite group. If φ ∈ Aut^{+}(F_{2}) is given by its values on x, y it is easy to check whether φ is in Γ(G, π) or not. Using a random process we generate a list of words φ_{1}, …, φ_{k} ∈ Γ(G, π) in the automorphisms α, β. We then wait until

Δ := ⟨ φ_{1}, …, φ_{k} ⟩

has finite index in Aut^{+}(F_{2}). This is checked using a Todd-Coxeter algorithm using the presentation (57). By this approach we first of all find the following table. The first column contains the finite group G, which we always think of being given as a permutation group. The third and fourth column contain the images x, y under the surjective homomorphism π : F_{2} → G. The fifth column gives the index of Δ ≤ Γ(G, π) in Aut^{+}(F_{2}). We give the abelianised group Δ^{ab} in the last column.

G  |G|  π(x)  π(y)  [Aut^{+}(F_{2}) : Δ]  Δ^{ab}
C_{2}  2  (1,2)  (1)  3  ℤ^{2} × C_{2} × C_{4}
C_{3}  3  (1,2,3)  (1)  8  ℤ × C_{3}^{2}
C_{4}  4  (1,2,3,4)  (1)  12  ℤ^{2} × C_{4}
C_{2} × C_{2} 4  (1,2)  (3,4)  6  ℤ^{2} × C_{2}^{3}
C_{5}  5  (1,2,3,4,5)  (1)  24  ℤ^{3} × C_{5}
C_{6}  6  (1,2,3,4,5,6)  (1)  24  ℤ^{3} × C_{6}
S_{3}  6  (1,2,3)  (1,2)  18  ℤ^{2} × C_{2}
C_{7}  7  (1,2,3,4,5,6,7)  (1)  48  ℤ^{5} × C_{7}
D_{4}  8  (1,2,3,4)  (1,4)(2,3)  24  ℤ^{3} × C_{2}
Q_{8}  8 (1,7,2,8)(3,6,4,5) (1,4,2,3)(5,7,6,8)  24  ℤ^{2} × C_{4}
D_{5}  10  (1,2,3,4,5)  (1,5)(2,4)  30  ℤ^{2} × C_{2}
A_{4}  12  (1,2,3)  (1,2)(3,4)  96  ℤ^{3}
S_{3} × C_{2} 12  (1,3)(4,5)  (1,2)  36  ℤ^{3} × C_{2}
Sm(12, 1) 12  σ_{1}  σ_{2}  72  ℤ^{3} × C_{2}
A_{5}  60  (1,2,3,4,5)  (1,2,3)  1080  ℤ^{17}

The notation for the finite groups is: C_{n} is the cyclic group of order n, D_{n} is the dihedral group of order 2n, Q_{8} is the quaternion group of order 8 and Sm(12, 1) is the non-abelian group of order 12 not isomorphic to A_{4} or S_{3} × C_{2}. The two permutations σ_{1}, σ_{2} are:

σ_{1} := (1, 8, 4, 11)(2, 9, 5, 12)(3, 7, 6, 10), σ_{2} := (1, 9, 4, 12)(2, 7, 5, 10)(3, 8, 6, 11).

Proceeding with our computer calculations, we then evaluated the φ_{1}, …, φ_{k} (which now generate a subgroup of finite index in Γ(G, π)) on the middle term of the sequence (55). Locating the isotypic component of an irreducible ℚ[G]-module N inside all three terms of (55) we were able to compute ρ_{G,π,N}(φ_{i}) (i = 1, …, k) as matrices. Computing determinants we could decide whether ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1}(ℤ) is of finite index in ρ_{G,π}(Γ(G, π)) or not. We have considered at least one presentation π : F_{2} → G for every group G with |G| ≤ 60 which can be generated by 2 elements and have found that ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1}(ℤ) is of finite index in ρ_{G,π}(Γ(G, π)). Of course we have run experiments on many more groups (like PSL(2, 7)) but always found a similar result. Let us report some more on the particularly interesting case G = A_{5}. Let N be the irreducible ℚ[G]-module of dimension 6. We have End_{G}(N) = ℚ(\sqrt{5}). Its ring of integers is O = ℤ((1 + \sqrt{5})/2). Identifying the isotypic component of N in ℚ[G] with M(3, ℚ(\sqrt{5})) we obtain a representation

ρ : Γ(G, π) → GL(1, M(3, ℚ(\sqrt{5}))) = GL(3, ℚ(\sqrt{5})).

Running the programm described above we have found

Proposition 9.7 For every presentation π : F_{2} → G = A_{5} there is a subgroup Δ ≤ Γ(G, π) of finite index such that ρ_{A_{5},π}(Δ) is contained up to change of bases in the subgroup of SL(3, O) consisting of elements which have (1, 0, 0) as a first row.

Proposition 9.7 shows that Theorem 1.4 is not true for n = 2 and non-redundant presentations. It is not clear what Proposition 9.7 suggests toward the general case. It may suggest that Theorem 1.4 is not true for non-redundant presentations but it may also be that n = 2 is exceptional. We have developed similar programms also for the cases n = 3 and n = 4. They lead to the following tables. The notation is the same as in the above table for n = 2. The presentations used for n = 4 take the value 1 on the fourth generator of the free group.

Subgroups in Aut(F_{3})

G  |G|  π(x)  π(y)  π(z) [Aut(F_{3}) : Δ]  Δ^{ab}
C_{2}  2  (1,2)  (1)  (1)  7  C_{2}^{3}
C_{3}  3  (1,2,3)  (1)  (1)  26  C_{6}
S_{3}  6  (1,2,3)  (1,2)  (1)  168  C_{2}^{3}
D_{4}  8  (1,3)  (1,4,3,2)  (1)  336  C_{2}^{8}
Q_{8}  8 (1,7,2,8)(3,6,4,5) (1,4,2,3)(5,7,6,8) (1)  336  C_{2}^{7}
D_{5}  10  (1,2,3,4,5)  (1,5)(2,4)  (1)  840  C_{2}^{5}
A_{4}  12  (1,2,3)  (1,2)( 3,4)  (1)  1560  C_{6}
S_{3} × C_{2} 12  (1,3)(4,5)  (1,2)  (1)  1008  C_{2}^{9}
Sm(12, 1) 12  σ_{1}  σ_{2}  (1)  1344  C_{2}^{2} × C_{4}
A_{5}  60  (1,2,3,4,5)  (1,2,3)  (1)  200160  ?

Subgroups in Aut(F_{4})

G  |G| [Aut(F_{4}) : Δ]  Δ^{ab}
C_{2}  2  15  C_{2}^{2}
C_{3}  3  80  C_{6}
S_{3}  6  80  C_{6}
D_{4}  8  3360  C_{2}^{4}
Q_{8}  8  840  C_{2}^{6}
D_{5}  10  930  C_{2}^{3}
A_{4}  12  1680  C_{3} × C_{6}
S_{3} × C_{2} 12  2730  C_{2}^{4}
Sm(12, 1) 12  3120  C_{2}^{2} × C_{4}
A_{5}  60  213098  ?

We have run experiments on many finite groups G and presentations π : F_{n} → G, we have always found that ρ_{G,π}(Γ(G, π)) ∩ G_{G,π}^{1}(ℤ) is of finite index in ρ_{G,π}(Γ(G, π)).

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