Maximal holonomy of infra-Nilmanifolds with ${frak so}(3)widetildetimesBbb R^3$-geometry

(1)

MAXIMAL HOLONOMY OF

INFRA-NILMANIFOLDS WITH so(3) e ×R

³

-GEOMETRY

Kyung Bai Lee and Joonkook Shin

Abstract. Let so(3) e ×R

³

be the 6-dimensional nilpotent Lie group with group operation (s, x)(t, y) = (s + t + xy

^t

− yx

^t

, x + y). We prove that the maximal order of the holonomy groups of all infra- nilmanifolds with so(3) e ×R

³

-geometry is 16.

1. Introduction

Let so(n) be the additive group of skew-symmetric matrices. This is the Lie algebra of the special orthogonal group SO(n). There is a bilinear map

I : R

ⁿ

× R

ⁿ

−→ so(n) defined as follows: For

x =



 

 x

₁

x

₂

·

· · x

_n



 

 , y =



 

 y

₁

y

₂

·

· · y

_n



 

 ,

I(x, y) = xy

^t

− yx

^t

,

where ()

^t

denotes transpose of the matrix (). With this I, the set so(n)×

R

ⁿ

gets a group structure given by

(s, x)(t, y) = (s + t + I(x, y), x + y).

Received June 30, 2005.

2000 Mathematics Subject Classification: 20H15, 20F18, 20E99, 53C55.

Key words and phrases: almost Bieberbach group, holonomy group, infra-nilmanifold.

The second author was supported in part by KOSEF grant R01-2000-000-00005-

0(2004).

(2)

This group is denoted by

R

ⁿ

= so(n) e ×R

ⁿ

.

Then R

ⁿ

is a simply connected nilpotent Lie group, of nilpotency class 2, with the center so(n). Note that so(n) is viewed as a commutative Lie group isomorphic to R

ⁿ⁽ⁿ⁻¹⁾²

(not as a Lie algebra). It fits the short exact sequences of Lie groups

0 → so(n) → R

ⁿ

→ R

ⁿ

→ 1.

Let M be an infra-nilmanifold with R

ⁿ

-geometry; that is, M = Π\R

ⁿ

, where Π is a torsion free, discrete, cocompact subgroup of R

ⁿ

oC for some compact subgroup C of Aut(R

ⁿ

). Such a subgroup Π is called an almost Bieberbach group (=AB group). It is well known that Π contains a cocompact lattice Γ of R

ⁿ

of finite index, and the quotient group Π/Γ is called the holonomy group of M .

Our aim is to understand those infra-nilmanifolds modelled on the 6-dimensional nilpotent Lie group

R

³

= so(3) e ×R

³

.

In particular, we are interested in finding the possible maximal order for the holonomy groups. We shall prove that the holonomy group of maximal order is D

₈

× Z

₂

, of order 16.

2. The automorphism group of R

ⁿ

Since the center, Z(R

ⁿ

) = so(n), is a characteristic subgroup of R

ⁿ

, every automorphism of R

ⁿ

restricts to an automorphism of so(n). Con- sequently an automorphism of R

ⁿ

induces an automorphism on the quotient group R

ⁿ

. Thus there is a natural homomorphism Aut(R

ⁿ

) → Aut(so(n)) × Aut(R

ⁿ

), θ 7→ (ˆ θ, ¯ θ). Here Aut(so(n)) is the group of linear automorphisms of the R-vector space so(n), not the Lie algebra automorphisms.

Lemma 2.1. Aut(R

ⁿ

) → Aut(R

ⁿ

) = GL(n, R) is surjective. More- over, the exact sequence Aut(R

ⁿ

) → GL(n, R) → 1 splits.

Proof. First we define

J : GL(n, R) −→ Aut(so(n)) by

(2.1) J(C)(s) = CsC

^t

(3)

for C ∈ GL(n, R) and s ∈ so(n). Then clearly, J is a homomorphism.

We denote J(C) by b C.

We claim that, for any C ∈ GL(n, R),

(2.2) (s, x) 7→ ( b Cs, Cx)

is an automorphism of R

ⁿ

. From

( b Cs, Cx) · ( b Ct, Cy) = ( b Cs + b Ct + I(Cx, Cy), Cx + Cy)

= ( b C(s + t) + I(Cx, Cy), C(x + y)) ( b C(s + t + I(x + y)), C(x + y)) = ( b C(s + t) + b CI(x + y), C(x + y)), for the assignment (2.2) to be an automorphism, it is necessary and sufficient that

(2.3) C(I(x, y)) = I(Cx, Cy) b holds for all x, y ∈ R

ⁿ

. Now this follows from

C(xy

^t

− yx

^t

)C

^t

= Cx(Cy)

^t

− Cy(Cx)

^t

for all x, y ∈ R

ⁿ

.

Theorem 2.2 (Structure of Aut(R

ⁿ

)).

Aut(R

ⁿ

) = Hom(R

ⁿ

, so(n)) o GL(n, R),

where Hom(R

ⁿ

, so(n)) is the additive group of linear transformations of vector spaces, and an element (η, C) ∈ Hom(R

ⁿ

, so(n)) o GL(n, R) acts by

(η, C)(s, x) = ( b Cs + η(x), Cx).

Proof. Let θ ∈ Aut(R

ⁿ

). Then we have the following commutative diagram of exact sequences

1 −−−−→ so(n) −−−−→ R

ⁿ

−−−−→ R

ⁿ

−−−−→ 1

  y

θ^ˆ

  y

θ

  y

θ^¯

1 −−−−→ so(n) −−−−→ R

ⁿ

−−−−→ R

ⁿ

−−−−→ 1

Thus θ(s, x) = (ˆ θ(s) + η(s, x), ¯ θ(x)) for (s, x) ∈ R

ⁿ

, where η : R

ⁿ

→ so(n). Since θ is a homomorphism, one can show that η is a homomorphism, i.e.,

η((s, x)(t, y)) = η(s, x) + η(t, y).

In particular, (ˆ θ(s), 0) = θ(s, 0) = (ˆ θ(s) + η(s, 0), 0) implies that η(s, 0)

= 0 for all s ∈ so(n), and thus η(s, x) = η((s, 0)(0, x)) = η(s, 0) +

η(0, x) = η(0, x). Hence η ∈ Hom(R

ⁿ

, so(n)).

(4)

Let’s find out the kernel of the surjective homomorphism of Lemma 2.1:

Aut(R

ⁿ

) → GL(n, R), θ 7→ ¯ θ.

Suppose that θ ∈ Aut(R

ⁿ

) with ¯ θ = id

_Rⁿ

. Then ˆ¯ θ = id

_so(n)

(see the equality (2.1)) and thus θ(s, x) = (s + η(x), x) for some η ∈ Hom(R

ⁿ

, so(n)). Conversely given η ∈ Hom(R

ⁿ

, so(n)), define θ ∈ Aut(R

ⁿ

) by θ(s, x) = (s + η(x), x). Clearly this θ lies in the kernel of the homomorphism. Hence we have a short exact sequence

1 → Hom(R

ⁿ

, so(n)) → Aut(R

ⁿ

) → GL(n, R) → 1.

By Lemma 2.1, this sequence is split.

Note that

Hom(R

ⁿ

, so(n)) o GL(n, R) ⊂ Hom(R

ⁿ

, so(n)) o (Aut(so(n)) × GL(n, R)) (η, A) 7→ (η, ( b A, A))

and the action of GL(n, R) on Hom(R

ⁿ

, so(n)) is

^A

η(x) = ˆ A · η(A

⁻¹

x).

The group operation on R

ⁿ

o GL(n, R) is given by ((s, x), A)((t, y), B)

= ((s, x) ·

^A

(t, y), AB)

= ((s, x) · ( ˆ At, Ay), AB)

= ((s + ˆ At + I(x, Ay), x + Ay), AB).

3. The structure of AB-groups for R

ⁿ

Let Π ⊂ R

ⁿ

o Aut(R

ⁿ

) be an AB-group. Then it is well known that

Γ = Π ∩R

ⁿ

, the pure translations in Π, is the maximal normal nilpotent

subgroup, and Φ = Π/Γ, the holonomy group of Π, is finite. Since Γ is

a lattice of R

ⁿ

, Z = Γ ∩ Z(R

ⁿ

) is a lattice of Z(R

ⁿ

) = so(n), and Γ/Z

is a lattice of R

ⁿ

/Z(R

ⁿ

) = R

ⁿ

. Thus Z = Γ ∩ Z(R

ⁿ

) ∼ = Z

ⁿ⁽ⁿ⁻¹⁾²

and

Γ/Γ ∩ Z(R

ⁿ

) ∼ = Z

ⁿ

.

(5)

Consider the following natural commutative diagram:

1 1

  y

  y so(n) −−−−→

=

so(n)

  y

1 −−−−→ R

ⁿ

−−−−→ R

ⁿ

o Aut(R

ⁿ

) −−−−→ Aut(R

ⁿ

) −−−−→ 1

  y

⁼

1 −−−−→ R

ⁿ

−−−−→ R

ⁿ

o Aut(R

ⁿ

) −−−−→ Aut(R

ⁿ

) −−−−→ 1

  y

1 1

Recall from Theorem 2.2 that an element

(η, A) ∈ Aut(R

ⁿ

) = Hom(R

ⁿ

, so(n)) o GL(n, R) acts on (s, x) ∈ R

ⁿ

by

(η, A)(s, x) = ( ˆ As + η(x), Ax).

Thus GL(n, R) acts on so(n) via the homomorphism b : GL(n, R) → Aut(so(n)), and GL(n, R) acts on R

ⁿ

by matrix multiplication GL(n, R)

×R

ⁿ

→ R

ⁿ

.

Let Q = Π/Z. Then the above diagram induces the following commutative diagram:

(3.1)

1 1

  y

  y Z −−−−→

=

Z

  y

1 −−−−→ Γ −−−−→ Π −−−−→ Φ −−−−→ 1

  y

⁼

1 −−−−→ Z

ⁿ

−−−−→ Q −−−−→ Φ −−−−→ 1

  y

1 1

(6)

Here an element (η, A) ∈ Φ ⊂ Aut(R

ⁿ

) = Hom(R

ⁿ

, so(n)) o GL(n, R) acts on Z

ⁿ

by A, and on Z by ˆ A.

Recall from [4, Proposition 2] that a virtually free abelian group 1 → Z

ⁿ

→ Q → Φ → 1 is a crystallographic group if and only if the centralizer of Z

ⁿ

in Q has no torsion elements. Suppose an element of Φ acts trivially on Z

ⁿ

. Then it is of the form (η, A) ∈ Aut(R

ⁿ

), where A = I ∈ GL(n, R). Then ˆ A = I also, see equation (2.1). Since Hom(R

ⁿ

, so(n)) is isomorphic to the additive group of n×

ⁿ⁽ⁿ⁻¹⁾₂

matrices (≈ R

ⁿ⁽ⁿ⁺¹⁾²

), it has no elements of finite order. Consequently, (η, A) ∈ Aut(R

ⁿ

) is trivial. This shows that Φ acts effectively on Z

ⁿ

. It follows that Q is naturally an n-dimensional crystallographic group.

The finite group Φ must be in a maximal compact subgroup O(n) of Aut(R

ⁿ

) = Hom(R

ⁿ

, so(n)) o GL(n, R).

Proposition 3.1. Let Q ⊂ R

ⁿ

o O(n) be a crystallographic group, and Q

⁰

⊂ R

ⁿ

o GL(n, R) be an affine crystallographic group isomorphic to Q. Then there exists an AB-group Π obtained from Q if and only if there exists an AB-group Π

⁰

obtained from Q

⁰

.

Proof. Let

Q = h(v

₁

, I), (v

₂

, I), . . . , (v

_n

, I), (a

₁

, A

₁

), (a

₂

, A

₂

), . . . , (a

_p

, A

_p

)i, Q

⁰

= h(v

₁⁰

, I), (v

⁰₂

, I), . . . , (v

⁰_n

, I), (a

⁰₁

, A

⁰₁

), (a

⁰₂

, A

⁰₂

), . . . , (a

⁰_p

, A

⁰_p

)i and (d, D) ∈ R

ⁿ

o GL(n, R) with

(v

⁰_i

, I) = (d, D)

⁻¹

(v

_i

, I)(d, D), i = 1, 2, . . . , n (a

⁰_j

, A

⁰_j

) = (d, D)

⁻¹

(a

_j

, A

_j

)(d, D), j = 1, 2, . . . , p.

Suppose there exist t

₁

, t

₂

, . . . , t

_n

, s

₁

, s

₂

, . . . , s

_p

∈ so(n) such that Π = h(t

₁

, v

₁

, I), (t

₂

, v

₂

, I), . . . , (t

_n

, v

_n

, I),

(s

₁

, a

₁

, A

₁

), (s

₂

, a

₂

, A

₂

), . . . , (s

_p

, a

_p

, A

_p

)i

is an AB-group. Then clearly, conjugate of Π by (0, d, D) ∈ R

ⁿ

o Aut(R

ⁿ

) is an AB-group Π

⁰

obtained from Q

⁰

. The converse is similar.

From this Proposition, we need not put our abstract crystallographic group into R

ⁿ

o O(n), but it is enough to use any convenient affine embedding.

Lemma 3.2. Let v

₁

, v

₂

, . . . , v

_n

be a basis of R

ⁿ

. Then the set {I(v

_i

,

v

_j

) | i < j} forms a lattice of so(n).

(7)

Proof. First we set notation. Let E

_ij

be the skew-symmetric matrix whose (i, j) entry is 1, (j, i) entry is −1, and 0 elsewhere. Suppose v

_i

= e

_i

for all i = 1, 2, . . . , n. Then I(e

_i

, e

_j

) = e

_i

e

^t_j

− e

_j

e

^t_i

= E

_ij

. Therefore {I(e

_i

, e

_j

) | i < j} = {E

_ij

| i < j} is a basis for the vector space so(n). For a general basis {v

₁

, v

₂

, . . . , v

_n

}, let C be the matrix whose jth column is v

_j

. Then I(v

_i

, v

_j

) = J(C)(E

_ij

), and hence the set {I(v

_i

, v

_j

) | i < j} forms a basis of so(n).

This lemma tells us that the lattice Z

ⁿ

of R

ⁿ

generated by {v

₁

, v

₂

, . . . , v

_n

} yields a lattice generated by {I(v

_i

, v

_j

) | i < j}, which must be contained in Z in the above diagram. However, the Z in the diagram can be finer than the lattice generated by {I(v

_i

, v

_j

) | i < j}.

Proposition 3.3. Let Q ⊂ R

ⁿ

o SO(n) be a crystallographic group generated by

(v

₁

, I), (v

₂

, I), . . . , (v

_n

, I), (a

₁

, A

₁

), (a

₂

, A

₂

), . . . , (a

_p

, A

_p

),

where the subgroup h(v

₁

, I), (v

₂

, I), . . . , (v

_n

, I)i is the maximal normal free abelian subgroup of Q. If there exists an AB-group obtained from Q, then there is an AB-group Π

_(s₁_,s₂_,...,s_p₎

generated by

(0, v

₁

, I), (0, v

₂

, I), . . . , (0, v

_n

, I), (s

₁

, a

₁

, A

₁

), (s

₂

, a

₂

, A

₂

), . . . , (s

_p

, a

_p

, A

_p

), for some s

₁

, s

₂

, . . . , s

_p

∈ so(n).

Proof. Let Π ⊂ R

ⁿ

o Aut(R

ⁿ

) be an AB-group obtained from Q.

Then

Π ∩ so(n) = h(v

₁

, 0, I), (v

₂

, 0, I), . . . , (v

n(n−1) 2

, 0, I)i for some v

₁

, v

₂

, . . . , v

n(n−1)

2

∈ so(n). Let us take any pre-images in Π as follows:

(w

₁

, v

₁

, I), (w

₂

, v

₂

, I), . . . , (w

_n

, v

_n

, I), (t

₁

, a

₁

, A

₁

), (t

₂

, a

₂

, A

₂

), . . . , (t

_p

, a

_p

, A

_p

)

for some w

₁

, w

₂

, . . . , w

_n

, t

₁

, t

₂

, . . . , t

_p

∈ so(n). Since [(w

_i

, v

_i

, I), (w

_j

, v

_j

, I)] = (2I(v

_i

, v

_j

), 0, I), the group generated by

(w

₁

, v

₁

, I), (w

₂

, v

₂

, I), . . . , (w

_n

, v

_n

, I)

becomes a lattice of R

ⁿ

already (see Lemma 3.2), and hence the group generated by

(w

₁

, v

₁

, I), (w

₂

, v

₂

, I), . . . , (w

_n

, v

_n

, I),

(t

₁

, a

₁

, A

₁

), (t

₂

, a

₂

, A

₂

), . . . , (t

_p

, a

_p

, A

_p

)

(8)

(without (v

₁

, 0, I), (v

₂

, 0, I), . . . , (v

n(n−1) 2

, 0, I)) will be a subgroup of Π of finite index, and is an AB-group (since it is torsion free). Let us call this smaller group Π

⁰

.

Let η ∈ Hom(R

ⁿ

, so(n)). Consider the product in R

ⁿ

o Aut(R

ⁿ

):

((0, 0), η)

⁻¹

((w

_i

, v

_i

), I)((0, 0), η) = ((w

_i

− η(v

_i

), v

_i

), I).

Certainly, there exists η ∈ Hom(R

ⁿ

, so(n)) such that η(v

_i

) = w

_i

for all i = 1, 2, . . . , n.

Consequently, the group Π

⁰⁰

obtained by conjugating Π

⁰

by (0, 0, η) is another AB-group, satisfying the condition w

₁

= w

₂

= · · · = w

_n

= 0 (without changing the second and third slots). Thus we have obtained an AB-group Π

⁰⁰

= Π

_(s₁_,s₂_,...,s_p₎

⊂ R

ⁿ

o Aut(R

ⁿ

) generated by

(0, v

₁

, I), (0, v

₂

, I), . . . , (0, v

_n

, I), (s

₁

, a

₁

, A

₁

), (s

₂

, a

₂

, A

₂

), . . . , (s

_p

, a

_p

, A

_p

) for some s

₁

, s

₂

, . . . , s

_p

∈ so(n).

The Proposition says that: If there is an AB-group Π constructed from Q, then Π

_(s₁_,s₂_,...,s_p₎

is conjugate to a subgroup of Π of finite index, which is another AB-group constructed from Q. Conversely, if Π

_(s₁_,s₂_,...,s_p₎

is an AB-group (i.e., is torsion free), then we are done.

Therefore, the existence/non-existence of construction is solely deter- mined by the group of the form Π

_(s₁_,s₂_,...,s_p₎

; that is, whether there exist s

₁

, s

₂

, . . . , s

_p

for which Π

_(s₁_,s₂_,...,s_p₎

is torsion free.

4. The group R

³

Our group R

³

= so(3) e ×R

³

has group law

(s, x)(t, y) = (s + t + I(x, y), x + y), where

I(x, y) = xy

^t

− yx

^t

=



 0 x

₁

y

₂

− x

₂

y

₁

x

₁

y

₃

− x

₃

y

₁

x

₂

y

₁

− x

₁

y

₂

0 x

₂

y

₃

− x

₃

y

₂

x

₃

y

₁

− x

₁

y

₃

x

₃

y

₂

− x

₂

y

₃

0 

 .

(9)

If we identify so(3) with R

³

by



 0 z −y

−z 0 x

y −x 0



 ←→



 x y z



 , then clearly,

I(x, y) = x × y (cross product)

so that R

³

= so(3) × R

³

= R

³

× R

³

has the group operation (s, x)(t, y) = (s + t + x × y, x + y).

In our case, b C = J(C) has a very simple description. Let e

_i

∈ R

ⁿ

whose ith component is 1 and 0 elsewhere. From the condition (2.3), we have

C(I(e b

_i

, e

_j

)) = I(Ce

_i

, Ce

_j

)

= I(c

_i

, c

_j

)

for all i, j, where c

_i

is the ith column of the matrix C. Note that I(e

_i

, e

_j

) is the skew-symmetric matrix whose (i, j) entry is 1, (j, i) entry is −1, and 0 elsewhere.

By the identification of so(3) with R

³

as above, the above equality becomes

C(e b

₁

) = c

₂

× c

₃

C(e b

₂

) = c

₃

× c

₁

C(e b

₃

) = c

₁

× c

₂

so that

C = det(C)(C b

⁻¹

)

^t

.

For each 3-dimensional crystallographic group Q, we shall check if there exists an AB-group Π constructed from Q; that is, a torsion free Π ⊂ R

³

o O(3) ⊂ R

³

o Aut(R

³

) fitting the short exact sequence

1 −→ Z −→ Π −→ Q −→ 1.

This is the key notion for our arguments and construction. We have a complete classification of 3-dimensional crystallographic groups (Q’s in the above statement). We shall use the representations of the 3- dimensional crystallographic groups given in the book [1].

Every Q has an explicit representation Q −→ R

³

o GL(3, Z) (not into R

³

o O(3)) in this book.

Our goal is to determine which Q will give rise to a torsion free Π

that fits the diagram (3.1). When Q is torsion free, then Π will be

(10)

automatically torsion free, but when Q contains a torsion subgroup Q

₀

, we need to check whether the lift Q

₀

to Π will be torsion free.

Let Π

₀

be the lift of Q

₀

to Π. Thus

1 −−−−→ Z −−−−→ Π

₀

−−−−→ Q

₀

−−−−→ 1 (4.1)

is exact. If Π is torsion free, then so is Π

₀

, and is a 3-dimensional Bieberbach group. Therefore,

Proposition 4.1. Any finite subgroup of Q is one of the holonomy groups 1, Z

₂

, Z

₃

, Z

₄

, Z

₆

or Z

₂

× Z

₂

of 3-dimensional crystallographic groups.

Proposition 4.2. There is no AB-group Π ⊂ R

³

o Aut(R

³

) constructed from Q given in 7/4/3/02, 7/3/2/02, 7/2/1/03, 7/2/3/02, (all holonomy group of order 24).

Proof. Let

Q = h(e

₁

, I), (e

₂

, I), (e

₃

, I), (a, A), (b, B), (c, C), (d, D)i, and let, for some s, t, u, v ∈ so(3),

t

₁

= (0, e

₁

, I), t

₂

= (0, e

₂

, I), t

₃

= (0, e

₃

, I), α = (s, a, A), β = (t, b, B), γ = (u, c, C), δ = (v, d, D).

By Proposition 3.3, there is no AB-group from Q, if and only if, for any s, t, u, v ∈ so(3), the group Π generated by t

₁

, t

₂

, t

₃

, α, β, γ, δ

(4.2) Π

_(s,t,u,v)

= ht

₁

, t

₂

, t

₃

, α, β, γ, δi

has a nontrivial torsion element. Now let Z = Π ∩ so(3). Then there is no AB-group from Q if and only if the following holds: for any s, t, u, v ∈ so(3), there exists a nontrivial torsion element of the form

zt

ⁿ₁¹

t

ⁿ₂²

t

ⁿ₃³

α

^p

β

^q

γ

^r

δ

^o

where z ∈ Z and n

_i

, p, q, r, o, k ∈ Z with 0 ≤ p < order of A, etc.

(7/4/3/02). For any choice of s, t, u, v ∈ so(3), α

⁻¹

γ

³

αγ is a torsion element of order 3.

(7/3/2/02). Let z = (16(1, −1, −1), 0, I). Then z ∈ Z. For any choice of s, t, u, v ∈ so(3), zα

⁻¹

γ

³

αγ is a torsion element of order 3.

(7/2/1/03). For any choice of s, t, u, v ∈ so(3), α

⁻¹

γ

³

αγ is a torsion element of order 3.

(7/2/3/02). For any choice of s, t, u, v ∈ so(3), α

⁻¹

γ

³

αγ is a torsion

element of order 3.

(11)

5. A construction of an AB-group with holonomy order 16 Theorem 5.1 (Existence). There exists an almost Bieberbach group Π ⊂ R

³

o Aut(R

³

) whose holonomy group is D

₈

× Z

₂

, of order 16.

Proof. Consider the 3-dimensional crystallographic group Q (Case 4/7/1/10) generated by the following elements:

(e

₁

, I), (e

₂

, I), (e

₃

, I), (a, A), (b, B), (c, C), (d, D), where I is the identity matrix, e

_i

is the unit vector with 1 in the ith coordinate, and

(a, A) =





¹

2



 1 1 0



 ,



 −1 0 0

0 −1 0

0 0 1







 ,

(b, B) =





¹

2



 1 0 0



 ,



 0 −1 0

1 0 0

0 0 1







 ,

(c, C) =





¹

2



 0 1 1



 ,



 −1 0 0

0 1 0

0 0 −1







 ,

(d, D) =





¹

2



 0 0 0



 ,



 −1 0 0

0 −1 0

0 0 −1







 .

Note that

A

²

= I, B

²

= A, C

²

= I, D

²

= I, [B, C] = I, [B, D] = I, [C, D] = I and that Q has the holonomy group Φ of order 16.

We consider the subgroup Π of (so(3) ˜ ×R

³

) o SO(3) = (R

³

×R ˜

³

) o SO(3) generated by

t

₁

= (0, e

₁

, I), t

₂

= (0, e

₂

, I), t

₃

= (0, e

₃

, I), α =







 0 0

14



 , a, A



 , β =







 0 0

18



 , b, B



 ,

γ =









14

0

14



 , c, C



 , δ =









14 14 14



 , d, D



 .

(12)

Let Z = Π ∩ so(3). We know Z is a lattice of so(3) from Lemma 3.2.

By calculation, we see that Z is actually generated by the vectors

z

₁

=





14 14

0 

 , z

₂

=



 −

¹₄

14

0 

 , z

₃

=



 0 0

12



 .

In particular, Z is discrete and hence Π is discrete. [Notice that we are abusing the notation: z

_i

= (z

_i

, 0, I) ∈ (so(3) ˜ ×R

³

) o SO(3)].

Next we claim that Π is torsion free. Every element of Π can be written as

z

₁^`¹

z

₂^`²

z

₃^`³

t

ⁿ₁¹

t

ⁿ₂²

t

ⁿ₃³

α

^p

β

^q

γ

^r

δ

^s

,

where `

₁

, `

₂

, `

₃

, n

₁

, n

₂

, n

₃

, p, q, r, s are integers such that 0 ≤ p < 2, 0 ≤ q < 4, 0 ≤ r < 2 and 0 ≤ s < 2. By calculation, it can be shown that the equation

(z

^`₁¹

z

₂^`²

z

^`₃³

t

ⁿ₁¹

t

ⁿ₂²

t

ⁿ₃³

α

^p

β

^q

γ

^r

δ

^s

)

^d

= (0, 0, I)

has no integral solution for `

₁

, `

₂

, `

₃

, n

₁

, n

₂

, n

₃

, p, q, r, s with d > 0. This proves that Π is indeed torsion free, and thus is an AB-group.

6. Non-existence

Using notations from [1], we list all three-dimensional crystallographic groups with holonomy order greater than 16 below. Most of these groups are eliminated by the fact that they contain torsion subgroups which are not in the list 1, Z

₂

, Z

₃

, Z

₄

, Z

₆

or Z

₂

× Z

₂

. See Proposition 4.1.

The remaining 4 cases where this proposition does not apply are proved in Proposition 4.2. All the calculations were done using the program MATHEMATICA [7] and hand-checked.

• 48:

7/5/1/02: h(a, A), (b, B), (e, E)i ∼ = (Z

₂

)

³

.

7/5/1/03: h(a, A), (c, C)i non-commutative group of order 12.

7/5/1/04: h(a, A), (c, C)i non-commutative group of order 12.

7/5/2/02: h(a, A)(d, D), (c, C)i non-commutative group of order 6.

7/5/2/03: h(a, A), (c, C)i non-commutative group of order 12.

7/5/2/04: h(a, A), (c, C)i non-commutative group of order 12.

7/5/3/02: h(a, A)(d, D), (c, C)i non-commutative group of order 6.

(13)

• 24:

7/4/1/02: h(a, A), (c, C)i non-commutative group of order 12.

7/4/2/02: h(a, A), (c, C)i non-commutative group of order 12.

7/4/3/02: Proposition 4.2.

7/3/1/02: h(a, A)(d, D), (b, B)(c, C)i non-commutative group of order 6.

7/3/1/03: h(a, A)(b, B), (a, A)(c, C)i non-commutative group of order 12.

7/3/2/02: Proposition 4.2.

7/3/3/02: h(a, A)(d, D), (c, C)i non-commutative group of order 6.

7/2/1/02: h(a, A), (c, C)i non-commutative group of order 12.

7/2/1/03: Proposition 4.2.

7/2/2/02: h(a, A), (c, C)

²

i non-commutative group of order 12.

7/2/3/02: Proposition 4.2.

6/7/1/02: h(a, A), (c, C)i non-commutative group of order 6.

6/7/1/03: h(a, A), (c, C)i non-commutative group of order 6.

6/7/1/04: h(a, A), (c, C)i non-commutative group of order 6.

• 16:

4/7/1/10: CONSTRUCTION! (Theorem 5.1)

References

[1] H. Brown, R. B¨ ulow, J. Neub¨ user, H. Wondratschek, and H. Zassenhaus, Crys- tallographic Groups of Four-Dimensional Spaces, John Wiley & Sons, Inc., New York, 1978.

[2] K. Y. Ha, J. B. Lee, and K. B. Lee, Maximal holonomy of infra-nilmanifolds with 2-dimensional Quaternionic Heisenberg Geometry, Trans. Amer. Math. Soc. 357 (2005), no. 1, 355–383.

[3] I. Kim and J. R. Parker, Geometry of quaternionic hyperbolic manifolds, Math.

Proc. Cambridge Philos. Soc. 135 (2003), no. 2, 291–320.

[4] K. B. Lee and F. Raymond, Topological, affine and isometric actions on flat Riemannian manifolds, J. Differential Geom. 16 (1981), no. 2, 255–269.

[5] K. B. Lee, J. K. Shin, and S. Yokura, Free actions of finite abelian groups on the 3-torus, Topology Appl. 53 (1993), no. 2, 153–175.

[6] K. B. Lee and A. Szczepa´ nski, Maximal holonomy of almost Bieberbach groups for Heis

5

, Geom. Dedicata 87 (2001), no. 1-3, 167–180.

[7] S. Wolfram, Mathematica, Wolfram Research, 1993.

(14)

Kyung Bai Lee

Department of Mathematics University of Oklahoma Norman, OK 73019, U.S.A.

E-mail: kb lee@math.ou.edu Joonkook Shin

Department of Mathematics Chungnam National University Taejeon 305-764, Korea

E-mail: jkshin@cnu.ac.kr