A finite presentation for a group 𝐺 is balanced if it has an equal number of generators and relations.
This notion has been studied in connection with finite groups and is central to the Andrews–Curtis conjecture on presentations of the trivial group, but here, we shall consider finitely generated infinite nilpotent groups.
It is well known that finitely generated torsion-free nilpotent groups of Hirsch length at most 3 are either free abelian or are central extensions of
Z
2
by ℤ and have balanced presentations.
Work by Lubotzky and by Freedman, Hain and Teichner relating to the Golod–Shafarevich Theorem shows that if a finitely generated nilpotent group 𝐺 has a balanced presentation, then
β
1
(
G
;
Q
)
⩽
3
(see [6, 12]).
Freedman, Hain and Teichner conjecture that if 𝐺 is such a group and
β
1
(
G
)
=
3
, then 𝐺 is virtually
Z
3
.
We shall confirm this conjecture and show that there is one torsion-free nilpotent group with Hirsch length
h
(
G
)
=
4
and
β
2
(
G
;
Q
)
=
β
1
(
G
;
Q
)
=
2
, none with
h
(
G
)
=
5
and several with
h
(
G
)
=
6
.
Theorem 10 and Corollary 14 below were originally part of [10], in which it is observed that if the fundamental groups
π
X
and
π
Y
of the complementary regions of a closed hypersurface in
S
4
are both solvable (or even elementary amenable), then they satisfy the condition
β
2
(
G
;
F
)
⩽
β
1
(
G
;
F
)
for
G
=
π
X
or
π
Y
and all fields 𝐹.
Abelian groups satisfying this condition are easily listed.
Nilpotent groups are particularly amenable to homological arguments (cf. [5]), and so it is natural to consider them next.
(In the topological setting, other issues become intractable if there is non-trivial torsion. Our arguments below allow us to work with torsion-free groups, for the most part.)
In § 1, we set out our notation and prove five lemmas.
(These are mostly well known, but are included for convenience, as their proofs are short.)
Our main result is in § 2, where Theorem 7 proves that if
β
2
(
G
;
k
)
=
β
1
(
G
;
k
)
=
3
for all prime fields 𝑘, then
G
≅
Z
3
. In particular, if
β
1
(
G
;
Q
)
=
3
and
h
(
G
)
>
3
, then
β
2
(
G
;
Q
)
>
β
1
(
G
;
Q
)
.
Thereafter, we assume that
β
1
(
G
;
Q
)
=
2
.
In § 3, we show that there is just one finitely generated torsion-free nilpotent group 𝐺 with
β
2
(
G
;
Q
)
=
β
1
(
G
;
Q
)
and
h
(
G
)
=
4
(Theorem 10), and none with
h
(
G
)
=
5
(Corollary 14).
In § 4, we show that there are no other finitely generated, metabelian nilpotent groups 𝐺 with
h
(
G
)
>
4
and
β
2
(
G
;
Q
)
=
β
1
(
G
;
Q
)
(Theorem 15).
In § 5, we construct a torsion-free nilpotent group 𝐺 with Hirsch length 6 and
β
2
(
G
;
F
)
=
β
1
(
G
;
F
)
=
2
for all fields 𝐹.
In the final section, we consider briefly the related work in Lie theory [3, 15, 16, 18] which prompted us to look beyond the cases with
h
⩽
4
.
We outline why there are other nilpotent groups with Hirsch length 6 and
β
2
(
G
;
Q
)
=
β
1
(
G
;
Q
)
=
2
, but we do not attempt to construct further explicit examples.
1 Preliminaries
If 𝐺 is a group,
ζ
G
,
G
′
,
G
τ
and
γ
n
G
shall denote the centre, the commutator subgroup, the preimage in 𝐺 of the torsion subgroup of
G
/
G
′
and the 𝑛th stage of the lower central series, respectively.
Our convention for commutators is
[
x
,
y
]
=
x
y
x
-
1
y
-
1
.
The nilpotency class of 𝐺 is 𝑛 if
γ
n
G
≠
1
and
γ
n
+
1
G
=
1
.
If 𝐺 is a finitely generated nilpotent group, then it has a finite composition series with cyclic factors, and the Hirsch length
h
(
G
)
is the number of infinite factors in such a series.
(If 𝐴 is an abelian group, then
h
(
A
)
=
dim
Q
Q
⊗
A
is the rank of 𝐴.)
If 𝐺 is torsion-free and not cyclic, then 𝐺 has nilpotency class at most
h
(
G
)
+
1
-
h
(
G
/
G
′
)
, which is strictly less than
h
(
G
)
.
As most of our results involve comparing
β
2
(
G
;
Q
)
with
β
1
(
G
;
Q
)
, for various groups 𝐺, we shall simplify the notation by dropping the coefficients, except where they seem needed for clarity.
Thus
H
i
(
G
)
,
H
i
(
G
)
and
β
i
(
G
)
shall denote homology, cohomology and Betti numbers with coefficients ℚ, respectively.
We shall often simplify the notation further by setting
β
=
β
1
(
G
)
.
We use also without further comment the facts that if 𝐺 is a finitely generated nilpotent group with
h
(
G
)
=
h
, then 𝐺 is an orientable
PD
h
-group (Poincaré duality group of formal dimension ℎ) over ℚ and that
χ
(
G
)
=
0
.
See [2, Theorem 9.10].
A group 𝐺 is homologically balanced if 𝐺 is finitely generated and
β
2
(
G
;
R
)
⩽
β
1
(
G
;
R
)
for all fields 𝑅.
If 𝐺 has a balanced presentation, then it is homologically balanced.
However, there are groups 𝐺 such that
H
1
(
G
;
Z
)
=
H
2
(
G
;
Z
)
=
0
and which have no presentation of deficiency 0 (see [7]).
Such groups are homologically balanced but have no balanced presentation.
The examples constructed in [7] include both finite groups and torsion-free groups.
Lemma 1
A finitely presentable group 𝐺 is homologically balanced if and only if
β
2
(
G
)
=
β
1
(
G
)
and
H
2
(
G
;
Z
)
is torsion-free.
Proof
We may assume that
H
1
(
G
;
Z
)
=
G
/
G
′
≅
Z
β
⊕
T
and
H
2
(
G
;
Z
)
≅
Z
β
⊕
U
,
where
β
=
β
1
(
G
)
and 𝑇 and 𝑈 are finite.
If 𝐴 is an abelian group and 𝑝 is a prime, let
r
p
(
A
)
=
dim
F
p
A
/
p
A
.
Then
β
1
(
G
;
F
p
)
=
β
+
r
p
(
T
)
and
β
2
(
G
;
F
p
)
=
β
+
r
p
(
U
)
+
r
p
(
Tor
(
T
,
Z
)
)
,
by the Universal Coefficient Theorem for homology.
Since
Tor
(
T
,
Z
)
≅
T
(non-canonically),
β
2
(
G
;
F
p
)
=
β
1
(
G
;
F
p
)
for all primes 𝑝 if and only if
U
=
0
.
∎
Let
F
(
r
)
be the free group of rank 𝑟.
If
G
=
F
(
β
)
/
γ
k
F
(
β
)
, then
H
2
(
G
;
Z
)
≅
γ
k
F
(
β
)
/
γ
k
+
1
F
(
β
)
by the five-term exact sequence of low degree for the homology of 𝐺 as a quotient of
F
(
β
)
.
This abelian group has rank
1
k
∑
d
|
k
μ
(
d
)
β
k
d
, where 𝜇 is the Möbius function, by the Witt formulae [13, Theorems 5.11 and 5.12].
Hence
H
2
(
G
;
Z
)
has rank greater than 𝛽 unless
β
=
1
or
β
=
2
and
k
⩽
3
or
β
=
3
and
k
=
2
.
Thus the only relatively free nilpotent groups with
β
2
(
G
)
⩽
β
1
(
G
)
are
G
≅
Z
β
with
β
⩽
3
or
G
≅
F
(
2
)
/
γ
3
F
(
2
)
.
These groups have balanced presentations and Hirsch length at most 3.
Applying the Witt formulae inductively, we see that
h
(
F
(
2
)
/
γ
4
F
(
2
)
)
=
5
and
h
(
F
(
2
)
/
γ
5
F
(
2
)
)
=
8
.
A finitely generated nilpotent group has a maximal finite normal subgroup, with torsion-free quotient [19, 5.2.7].
We may use the following lemma to extend arguments based on the torsion-free cases.
Lemma 2
Let 𝐺 be a finitely generated nilpotent group, and let 𝑇 be its maximal finite normal subgroup.
Then the natural epimorphism
p
:
G
→
G
/
T
induces isomorphisms
H
i
(
p
)
=
H
i
(
p
;
Q
)
for all
i
⩾
0
.
Proof
Since 𝑇 is finite,
H
i
(
T
)
=
0
for
i
>
0
.
Hence the LHS spectral sequence for the homology of 𝐺 as an extension of
G
/
T
by 𝑇 collapses, and the edge homomorphisms
H
i
(
p
)
are isomorphisms for all
i
⩾
0
.
∎
We may also arrange that the abelianization is torsion-free.
Lemma 3
Suppose that 𝐺 is a finitely generated torsion-free nilpotent group, and let
β
=
β
1
(
G
)
.
Then 𝐺 has a subgroup 𝐽 of finite index which can be generated by 𝛽 elements and such that
H
i
(
J
)
≅
H
i
(
G
)
for all
i
⩾
0
.
Proof
Let
f
:
F
(
β
)
→
G
be a homomorphism such that
H
1
(
f
)
is an isomorphism, and let
J
=
Im
(
f
)
.
Then
[
G
:
J
]
is finite, and 𝐽 is subnormal in 𝐺 [19, 5.2.4].
The inclusion
j
:
J
→
G
induces a map between the LHS spectral sequences for 𝐽 and 𝐺 as central extensions of
J
/
J
∩
ζ
G
and
G
/
ζ
G
, respectively.
Induction on the nilpotency class of 𝐺 now shows that
H
i
(
j
)
is an isomorphism for all
i
⩾
0
.
∎
We may also use induction on the nilpotency class to show that 𝐽 and 𝐺 have isomorphic Mal’cev completions.
Lemma 4
[12]
Let 𝐺 be a finitely generated nilpotent group.
Then 𝐺 can be generated by 𝑑 elements, where
d
=
max
{
β
1
(
G
;
F
p
)
∣
p
prime
}
.
Proof
If 𝐺 is abelian, this is an easy consequence of the structure theorem for finitely generated abelian groups.
In general, if 𝐺 is nilpotent and the image in
G
/
G
′
of a subset
X
⊂
G
generates
G
/
G
′
, then 𝑋 generates 𝐺 [13, Lemma 5.9].
∎
The expression for 𝑑 is clearly best possible.
The following lemma was prompted by [3, Lemma 2].
Lemma 5
Let 𝐺 be a group with a subgroup
Z
⩽
ζ
G
∩
G
′
such that
z
=
h
(
Z
)
⩾
1
,
and let
G
¯
=
G
/
Z
and 𝑅 be ℤ or a field.
Let
β
¯
i
=
β
i
(
G
¯
;
R
)
for
i
⩾
0
.
Then
β
¯
2
⩾
z
and
β
¯
2
-
z
+
max
{
β
¯
1
z
-
β
¯
3
,
0
}
⩽
β
2
(
G
;
R
)
⩽
β
¯
2
-
z
+
β
¯
1
z
+
(
z
2
)
.
Proof
The quotient
G
¯
acts trivially on the cohomology of 𝑍 since 𝑍 is central in 𝐺.
Hence the rank of the
E
1
,
1
2
term of the LHS spectral sequence
H
p
(
G
¯
;
H
q
(
Z
;
R
)
)
⇒
H
p
+
q
(
G
;
R
)
for the homology of 𝐺 as a central extension of
G
¯
is
β
¯
1
z
.
The
E
2
,
0
2
term has rank
β
¯
2
, the
E
0
,
1
2
term has rank 𝑧, the
E
3
,
0
2
term has rank
β
¯
3
, and the
E
0
,
2
2
term has rank
(
z
2
)
.
The differential
d
2
,
0
2
must be surjective since
Z
⩽
G
′
, and so
β
¯
2
⩾
z
.
The lemma follows easily.
∎
When
Z
≅
Z
, the spectral sequence reduces to the Gysin sequence associated to 𝐺 as an extension of
G
¯
by ℤ, and the bounds given by Lemma 5 simplify to
β
2
(
G
¯
;
R
)
-
1
⩽
β
2
(
G
;
R
)
⩽
β
2
(
G
¯
;
R
)
-
1
+
β
1
(
G
¯
;
R
)
.
Note also that if
h
=
h
(
G
¯
)
⩽
6
(and
R
=
Q
), then
β
3
(
G
¯
)
is determined by
β
1
(
G
¯
)
and
β
2
(
G
¯
)
via Poincaré duality and the condition
χ
(
G
¯
)
=
0
.
From another point of view, if
G
≅
F
(
β
)
/
γ
k
F
(
β
)
, then rearranging the right-hand inequality gives the bound
1
-
β
+
1
k
∑
d
|
k
μ
(
d
)
β
k
d
⩽
β
2
(
G
¯
)
.
2 The case
β
=
3
Lubotzky has shown that if a nilpotent group 𝐺 can be generated by 𝑑 elements and 𝑝 is a prime such that
d
=
β
1
(
G
;
F
p
)
, then either
β
2
(
G
;
F
p
)
>
d
2
4
or
G
=
1
, ℤ or
Z
2
.
Hence homologically balanced nilpotent groups can be generated by 3 elements [12].
There is a similar inequality for the rational Betti numbers.
Lemma 6
Let 𝐺 be a finitely generated nilpotent group such that
h
(
G
)
>
2
.
Then
β
2
(
G
)
>
1
4
β
1
(
G
)
2
.
Proof
We may assume that 𝐺 can be generated by
β
=
β
1
(
G
)
elements, by Lemma 3.
Then
β
1
(
G
;
F
p
)
=
β
, so
β
2
(
G
;
F
p
)
>
β
2
4
for all primes 𝑝 (see [12]).
Since 𝐺 is a finitely generated nilpotent group,
H
2
(
G
;
Z
)
is finitely generated, and so the torsion subgroup of
H
2
(
G
;
Z
)
is finite.
If 𝑝 does not divide the order of this torsion subgroup, then
β
2
(
G
)
=
β
2
(
G
;
F
p
)
.
The lemma follows.
∎
An immediate consequence is that if 𝐺 is a finitely generated nilpotent group such that
β
2
(
G
)
⩽
β
1
(
G
)
, then
β
1
(
G
)
⩽
3
.
We shall use the following result from commutative algebra to settle the case
β
1
(
G
)
=
3
.
Lemma
Lemma ([4, Lemma 1])
Let 𝑅 be a Cohen–Macaulay local ring of Krull dimension 𝑑 and with residue field 𝑘.
Let 𝑀 be a non-zero 𝑅-module of finite length.
Then
dim
k
Tor
1
R
(
k
,
M
)
⩾
dim
k
k
⊗
R
M
+
d
-
1
.∎
We apply this result to modules over formal power series rings
R
=
k
[
[
x
,
y
,
z
]
]
, where 𝑘 is a prime field.
(Thus
k
=
Q
or
F
p
for some prime 𝑝.)
All such rings are regular and hence are Cohen–Macaulay [11, page 187].
Theorem 7
Let 𝐺 be a finitely generated nilpotent group such that
G
/
G
′
≅
Z
3
.
If
G
′
≠
1
, then
β
2
(
G
;
k
)
>
3
for some prime field 𝑘.
Proof
Let
{
x
,
y
,
z
}
be a fixed basis for
G
/
G
′
.
If
G
′
≠
1
, then
G
′
/
G
′′
is non-zero and is finitely generated.
Hence
A
=
H
1
(
G
′
;
k
)
≠
0
for some prime field 𝑘.
Let
k
Λ
=
k
[
G
/
G
′
]
, and let
X
=
1
-
x
,
Y
=
1
-
y
,
Z
=
1
-
z
.
Then
I
=
(
X
,
Y
,
Z
)
is the augmentation ideal in
k
Λ
.
Since 𝐺 is nilpotent, the
k
Λ
-module 𝐴 is annihilated by a power of ℐ, and
r
=
dim
k
A
is finite.
Hence 𝐴 is a module of finite length over the ℐ-adic completion
R
=
k
Λ
^
, which is a regular local ring of Krull dimension 3.
Moreover,
A
≅
R
⊗
k
Λ
A
.
Since completion is faithfully flat,
H
i
(
G
/
G
′
;
A
)
=
Tor
i
k
Λ
(
k
,
A
)
≅
Tor
i
R
(
k
,
A
)
for all 𝑖.
Let
b
i
=
dim
k
Tor
i
R
(
k
,
A
)
.
Then
b
0
=
dim
k
k
⊗
R
A
>
0
since
A
≠
0
, and
b
1
⩾
b
0
+
2
by [4, Lemma 1] since 𝑅 has Krull dimension
d
=
3
.
We may bound
β
2
(
G
;
k
)
below by means of the LHS spectral sequence for 𝐺 as an extension of
G
/
G
′
by
G
′
.
Since
dim
Q
H
i
(
G
/
G
′
;
A
)
=
b
i
for all 𝑖, the spectral sequence gives
β
2
(
G
;
k
)
⩾
max
{
3
-
b
0
,
0
}
+
max
{
b
1
-
1
,
0
}
⩾
max
{
3
-
b
0
,
0
}
+
b
0
+
1
⩾
4
.
This proves the theorem.
∎
We note the following corollaries.
Corollary 8
Let 𝐺 be a finitely generated nilpotent group with torsion subgroup 𝑇 and such that
β
1
(
G
)
=
3
.
Then the following are equivalent:
β
2
(
G
)
=
β
1
(
G
)
=
3
.
Proof
Clearly, (1) and (2) are equivalent and imply (3), while if (3) holds, then
G
′
is finite, and so (1) holds by the theorem.
∎
Corollary 9
If 𝐺 is a nilpotent group with a balanced presentation and such that
β
1
(
G
)
=
3
, then
G
≅
Z
3
.∎
Freedman, Hain and Teichner have also shown that
β
2
(
G
)
>
1
4
β
1
(
G
)
2
(for nilpotent groups 𝐺 such that
h
(
G
)
>
2
) and have used the Dwyer filtration [5] to refine this estimate for certain groups [6].
The following assertion is a paraphrase of one of their main results.
Theorem
Theorem ([6, Theorem 2])
If 𝐺 is a finitely generated nilpotent group such that
h
(
G
)
>
2
and
G
/
γ
r
G
≅
F
(
β
)
/
γ
r
F
(
β
)
for some
r
⩾
2
, then
β
2
(
G
)
>
(
r
-
1
)
r
-
1
r
r
β
1
(
G
)
r
.
∎
Setting
r
=
2
recovers the inequality
β
2
(
G
)
>
1
4
β
1
(
G
)
2
since we may assume that
G
/
G
′
≅
Z
β
by Lemma 3.
In seeking examples of nilpotent groups 𝐺 with
β
2
(
G
)
=
β
1
(
G
)
=
2
, we may focus on groups that can be generated by 2 elements, by Lemma 3.
If 𝐺 is such a group, then
G
/
γ
2
G
=
F
(
2
)
/
γ
2
F
(
2
)
, and
G
/
γ
r
G
is a quotient of
F
(
2
)
/
γ
r
F
(
2
)
for all
r
>
2
.
If
G
/
γ
3
G
is a proper quotient of
F
(
2
)
/
γ
3
F
(
2
)
, then
γ
2
G
/
γ
3
G
is finite since
γ
2
F
(
2
)
/
γ
3
F
(
2
)
≅
Z
.
Hence
G
′
=
γ
2
G
is finite, and so
β
2
(
G
)
=
1
.
Thus we may assume that
G
/
γ
3
G
≅
F
(
2
)
/
γ
3
F
(
2
)
.
On the other hand, if
G
/
γ
5
G
≅
F
(
2
)
/
γ
5
F
(
2
)
, then setting
β
=
2
and
r
=
5
in the inequality of [6, Theorem 2] gives
β
2
(
G
)
>
β
1
(
G
)
.
Thus we may assume that
γ
4
G
/
γ
5
G
has rank at most 2.
(The argument for [6, Theorem 2] may be tweaked to show that
γ
4
G
/
γ
5
G
must have rank 1.)
The authors of [6] conjecture that there is no nilpotent group 𝐺 such that
G
/
γ
4
G
≅
F
(
2
)
/
γ
4
F
(
2
)
and
β
2
(
G
)
=
β
1
(
G
)
.
3 Hirsch length at most 5
As our primary interest is in the torsion-free case and the cases with
β
⩾
3
or
h
⩽
3
have been settled, we shall assume henceforth that
β
=
2
and
h
⩾
4
.
The next theorem was the origin of this paper.
Theorem 10
There is just one torsion-free nilpotent group of Hirsch length 4 which is homologically balanced.
Proof
Let 𝑁 be a torsion-free nilpotent group of Hirsch length 4.
If 𝐹 is a field, then
β
1
(
N
;
F
)
⩾
2
since 𝑁 is not virtually cyclic, and
β
2
(
N
;
F
)
=
2
(
β
1
(
N
;
F
)
-
1
)
as 𝑁 is an orientable
PD
4
-group and
χ
(
N
)
=
0
.
Hence if
β
2
(
N
;
F
)
⩽
β
1
(
N
;
F
)
, then
β
1
(
N
;
F
)
=
2
.
If 𝑁 is homologically balanced, this holds for all fields 𝐹, and so
N
/
N
′
≅
Z
2
.
Therefore, 𝑁 can be generated by 2 elements.
Since 𝑁 is a quotient of
F
(
2
)
, the quotient
γ
2
N
/
γ
3
N
is cyclic.
If
γ
2
N
/
γ
3
N
were finite, then all subsequent factors of the lower central series would be finite [19, 5.2.5], and so
h
(
N
)
=
2
, contradicting the hypothesis
h
(
N
)
=
4
.
Therefore,
γ
2
N
/
γ
3
N
≅
Z
, and so
N
/
γ
3
N
≅
F
(
2
)
/
γ
3
F
(
2
)
.
Hence
γ
3
N
≅
Z
.
Since 𝑁 is nilpotent, it follows that
γ
3
N
is central in 𝑁.
Hence 𝑁 has a presentation
⟨
x
,
y
,
u
,
z
∣
u
=
[
x
,
y
]
,
[
x
,
u
]
=
z
a
,
[
y
,
u
]
=
z
b
,
x
z
=
z
x
,
y
z
=
z
y
⟩
,
in which
x
,
y
represent a basis for
N
/
γ
3
N
and 𝑧 represents a generator for
γ
3
N
.
Since
N
/
N
′
≅
Z
2
, we must have
(
a
,
b
)
=
1
, and after a change of basis for
F
(
2
)
, we may assume that
a
=
1
and
b
=
0
.
The relation
y
z
=
z
y
is then a consequence of the others, and so the presentation simplifies to
⟨
x
,
y
∣
[
x
,
[
x
,
[
x
,
y
]
]
]
=
[
y
,
[
x
,
y
]
]
=
1
⟩
.
∎
An alternative argument uses the observations that
N
′
≅
Z
2
(since it is torsion-free nilpotent and
h
(
N
′
)
=
2
) and that the image of 𝑁 in
Aut
(
N
′
)
preserves the flag
1
<
γ
3
N
<
N
′
.
Since 𝑁 is nilpotent and
N
/
N
′
≅
Z
2
, this image is infinite cyclic, and so
N
≅
C
⋊
Z
, where 𝐶 is the centralizer of
N
′
in 𝑁.
Moreover,
C
≅
Z
3
since
C
/
N
′
≅
Z
.
Hence
N
≅
Z
3
⋊
A
Z
, where
A
∈
SL
(
3
,
Z
)
is a triangular matrix.
It is not hard to see that, since
N
/
N
′
is torsion-free of rank 2, there is a basis for 𝐶 such that
A
=
(
1
0
0
1
1
0
0
1
1
)
∈
SL
(
3
,
Z
)
.
Hence 𝑁 has the 2-generator balanced presentation
⟨
t
,
u
∣
[
t
,
[
t
,
[
t
,
u
]
]
]
=
[
u
,
[
t
,
u
]
]
=
1
⟩
,
in which 𝑢 corresponds to the column vector
(
1
,
0
,
0
)
tr
in
Z
3
.
Corollary 11
The group 𝑁 described in Theorem 10 is the only torsion-free nilpotent group of Hirsch length 4 which can be generated by 2 elements.
Proof
If 𝐺 is a nilpotent group which can be generated by 2 elements, then either
G
/
G
′
≅
Z
2
or
h
(
G
)
⩽
1
.
Hence, if
h
(
G
)
=
4
, then
β
1
(
F
)
=
2
for all fields 𝐹.
Since 𝐺 is an orientable
PD
4
-group and
χ
(
G
)
=
0
, it follows that 𝐺 is homologically balanced, and so
G
≅
N
.
∎
This group is a quotient of
F
~
=
F
(
2
)
/
γ
4
F
(
2
)
by a maximal infinite cyclic subgroup of
ζ
F
~
.
The quotient of
F
~
by the subgroup generated by the 𝑝th power of a non-trivial central element is nilpotent of Hirsch length 4 and has a finite normal subgroup of order 𝑝.
Thus the condition on torsion is necessary for this corollary.
We do not know whether every homologically balanced nilpotent group of Hirsch length 4 is torsion-free.
We remark here that there is just one indecomposable nilpotent Lie algebra of dimension 4.
This derives from the Lie group
G
4
of [15].
(In the context of Thurston geometries, this is also known as
Nil
4
[9, Chapter 7].)
Theorem 12
Let 𝐺 be a finitely generated nilpotent group such that
β
1
(
G
)
=
2
and
h
(
G
)
=
5
.
Then
β
2
(
G
)
=
3
.
Proof
We may assume that 𝐺 is torsion-free, by Lemma 2.
The intersection
G
′
∩
ζ
G
is non-trivial [19, 5.2.1], and so we may choose a maximal infinite cyclic subgroup
A
⩽
G
′
∩
ζ
G
.
Let
G
¯
=
G
/
A
.
Then
h
(
G
¯
)
=
4
, and
G
¯
is also torsion-free since the preimage of any finite subgroup in 𝐺 is torsion-free and virtually ℤ.
Hence
G
¯
is an orientable
PD
4
-group.
Moreover, we have
β
1
(
G
¯
)
=
2
since
A
⩽
G
′
, and so
G
¯
τ
/
[
G
¯
,
G
¯
τ
]
≠
1
.
The group 𝐺 is a central extension of
G
¯
by ℤ.
The LHS cohomology spectral sequence associated to this extension reduces to a “Gysin” exact sequence [14, Example 5C]
0
→
Q
e
→
H
2
(
G
¯
)
→
H
2
(
G
)
→
H
1
(
G
¯
)
→
∪
e
H
3
(
G
¯
)
→
⋯
,
where
e
∈
H
2
(
G
¯
;
Z
)
classifies the extension.
If 𝐽 is any finitely generated group, then the kernel of the homomorphism
ψ
J
:
∧
2
H
1
(
J
)
→
H
2
(
J
)
induced by cup product is isomorphic to
Hom
(
J
τ
/
[
J
,
J
τ
]
,
Q
)
(see [8]).
In our case,
∧
2
H
1
(
G
)
→
H
2
(
G
)
has rank 1 since
β
1
(
G
¯
)
=
2
.
Hence
ψ
G
¯
=
0
since
G
¯
τ
/
[
G
¯
,
G
¯
τ
]
≠
1
.
Since the cup product of
H
3
(
G
¯
)
with
H
1
(
G
¯
)
is a non-singular pairing, it follows that
α
∪
e
=
0
for all
α
∈
H
1
(
G
¯
)
.
Hence
β
2
(
G
)
=
2
-
1
+
2
=
3
.
∎
We shall use the following extension of this result at the end of § 5.
Corollary 13
Let 𝐺 be a finitely generated torsion-free nilpotent group such that
G
/
G
′
≅
Z
2
and
h
(
G
)
=
5
.
Then
β
2
(
G
;
F
p
)
=
3
for all primes 𝑝.
Proof
The argument is essentially as in the theorem, but the appeal to [8] deserves comment.
We replace
J
τ
by the subgroup of 𝐽 generated by
J
′
and all 𝑝th powers; if
p
=
2
, we note also that if
z
∈
H
1
(
G
¯
;
F
2
)
, then 𝑧 is the reduction of an integral class, and so
z
2
=
0
.
∎
Theorem 12 implies that there are no homologically balanced nilpotent groups of Hirsch length 5.
Corollary 14
Let 𝑁 be a finitely generated nilpotent group of Hirsch length
h
(
N
)
=
5
.
Then
β
2
(
N
)
>
β
1
(
N
)
.
Proof
Since we may assume that
β
1
(
N
)
⩽
3
, the statement follows immediately from Theorems 7 and 12.
∎
4 Metabelian nilpotent groups
We shall settle the case of metabelian nilpotent groups with the next theorem.
Theorem 15
Suppose that 𝐺 is a finitely generated metabelian nilpotent group.
If
h
(
G
)
>
4
, then
β
2
(
G
)
>
β
1
(
G
)
.
Proof
We may assume that 𝐺 is torsion-free and can be generated by 2 elements, by Lemmas 2 and 3, and Theorem 7.
Hence
G
/
G
′
≅
Z
2
and
G
′
≅
Z
r
with
r
=
h
(
G
)
-
2
>
2
.
We shall again bound
β
2
(
G
)
below by means of the LHS spectral sequence for 𝐺 as an extension of
G
/
G
′
.
Let
A
=
H
1
(
G
′
)
=
Q
⊗
G
′
≅
Q
r
, and let
b
i
=
dim
Q
H
i
(
G
/
G
′
;
A
)
for
i
⩾
0
.
The homology groups
H
i
(
G
/
G
′
;
A
)
may be computed from the complex
0
→
A
→
∂
2
A
2
→
∂
1
A
→
0
arising from the standard resolution of the augmentation
Q
[
G
/
G
′
]
-module.
Since this complex has Euler characteristic 0,
b
0
-
b
1
+
b
2
=
0
.
Since 𝐺 is nilpotent and
G
′
is infinite,
h
(
G
/
γ
3
G
)
=
3
, and so
b
0
=
1
.
Hence
b
1
=
b
2
+
1
.
Let
ρ
=
dim
Q
H
0
(
G
/
G
′
;
∧
2
A
)
.
As
G
′
is abelian,
H
2
(
G
′
)
≅
∧
2
A
and so has rank
(
r
2
)
.
It is not hard to see that
ρ
⩾
2
if
r
>
3
.
There are no non-zero differentials in the spectral sequence which begin or end at the
(
1
,
1
)
position, and so
β
2
(
G
)
⩾
b
1
+
max
{
ρ
-
b
2
,
0
}
⩾
ρ
+
1
>
2
.
If
r
=
3
, then
dim
Q
H
0
(
G
/
G
′
;
∧
2
A
)
=
1
, but the conclusion still holds by Theorem 14.
(In this case, the differential
d
2
,
1
2
must be 0.)
∎
The theorem is not true if
h
(
G
)
⩽
4
, as is shown by the groups
Z
2
,
F
(
2
)
/
γ
3
F
(
2
)
and the group of Theorem 10.
If 𝐺 is any group, then
G
′′
=
[
G
′
,
G
′
]
⩽
γ
4
G
, and so all nilpotent groups of class 3 are metabelian.
This can be improved slightly for 2-generator groups since
F
(
2
)
/
γ
5
F
(
2
)
is metabelian.
Corollary 16
Suppose that 𝐺 is a nilpotent group of class at most 4 and such that
β
1
(
G
)
=
2
.
If
h
(
G
)
>
4
, then
β
2
(
G
)
>
β
1
(
G
)
.
Proof
We may again assume that 𝐺 can be generated by 2 elements, by Lemma 3.
As it is then a quotient of
F
(
2
)
/
γ
5
F
(
2
)
, it is metabelian, and so the theorem applies.
∎
5 Hirsch length 6
In this section, we shall give an example of a torsion-free nilpotent group of Hirsch length 6 which is homologically balanced.
The construction was inspired by some of the features of the examples of nilpotent Lie algebras reported to us by Grant Cairns, but stays within the realm of group theory.
The example is a central extension of a group of Hirsch length 5 by ℤ, and the claim is verified by means of the Gysin sequences (with coefficients in a field) associated to the extension.
Let 𝐾 be the central extension of
A
=
Z
4
by ℤ, with presentation
⟨
y
,
c
,
d
,
e
,
f
∣
[
y
,
d
]
=
[
y
,
e
]
=
f
,
[
c
,
d
]
=
f
-
1
,
[
y
,
c
]
=
[
c
,
e
]
=
[
d
,
e
]
=
1
,
f
central
⟩
.
We may define an automorphism 𝜃 of 𝐾 by setting
θ
(
y
)
=
c
y
,
θ
(
c
)
=
d
c
,
θ
(
d
)
=
e
d
,
θ
(
e
)
=
e
and
θ
(
f
)
=
f
.
(It is straightforward to check that this definition is compatible with the relations for 𝐾.)
The semidirect product
G
=
K
⋊
θ
Z
has the presentation
⟨
x
,
y
,
c
,
d
,
e
,
f
∣
c
=
[
x
,
y
]
,
d
=
[
x
,
c
]
,
e
=
[
x
,
d
]
,
[
y
,
d
]
=
[
y
,
e
]
=
f
,
[
c
,
d
]
=
f
-
1
,
[
x
,
e
]
=
[
y
,
c
]
=
[
c
,
e
]
=
[
d
,
e
]
=
[
x
,
f
]
=
[
y
,
f
]
=
1
⟩
.
It is clearly nilpotent, and torsion-free of Hirsch length 6, and has abelianization
G
/
G
′
≅
Z
2
and centre
ζ
G
=
⟨
f
⟩
≅
Z
.
Note also that
γ
4
G
≅
Z
2
, and so
G
/
γ
4
G
is a proper quotient of
F
(
2
)
/
γ
4
F
(
2
)
.
Let
θ
¯
be the automorphism of 𝐴 induced by 𝜃, and let
G
¯
=
G
/
ζ
G
≅
A
⋊
θ
¯
Z
.
Then
G
¯
is a torsion-free nilpotent group of Hirsch length 5, and so
β
2
(
G
¯
)
=
3
by Theorem 12.
The group 𝐺 is a central extension of
G
¯
by ℤ.
Let
η
∈
H
2
(
G
¯
;
Z
)
classify this extension.
Let
{
x
*
,
y
*
}
be a basis for
H
1
(
G
¯
;
Z
)
=
Hom
(
G
¯
,
Z
)
which is Kronecker dual to the basis for
G
/
G
′
represented by
{
x
,
y
}
.
Then
x
*
|
A
=
0
, while
y
*
|
A
is trivial on
{
c
,
d
,
e
}
.
Let
{
y
*
|
A
,
c
*
,
d
*
,
e
*
}
be the basis for
H
1
(
A
;
Z
)
=
Hom
(
A
,
Z
)
which is Kronecker dual to
{
y
,
c
,
d
,
e
}
.
The matrix of
M
*
=
H
1
(
M
)
with respect to the basis
{
y
*
,
c
*
,
d
*
,
e
*
}
is the transpose of the matrix for 𝑀 with respect to
{
y
,
c
,
d
,
e
}
, and so
M
*
(
y
*
)
=
y
*
,
M
*
(
c
*
)
=
c
*
+
y
*
,
M
*
(
d
*
)
=
d
*
+
c
*
and
M
*
(
e
*
)
=
e
*
+
d
*
.
We shall henceforth write
y
*
instead of
y
*
|
A
, where appropriate, and drop the
∪
sign for cup products.
Then
H
2
(
A
;
Z
)
has basis
{
y
*
c
*
,
y
*
d
*
,
y
*
e
*
,
c
*
d
*
,
c
*
e
*
,
d
*
e
*
}
.
We may identify the restriction
η
|
A
by considering the induced extensions of 2-generator subgroups of 𝐴, and we see that
η
|
A
=
y
*
d
*
+
y
*
e
*
-
c
*
d
*
.
Then we have
η
2
≠
0
since
η
2
|
A
=
-
2
y
*
c
*
d
*
e
*
generates
H
4
(
A
)
.
Similarly, we have
y
*
η
|
A
=
-
y
*
c
*
d
*
≠
0
.
We claim that
x
*
η
2
≠
0
.
(This is most easily seen topologically.
The group 𝐺 is the fundamental group of the mapping torus of a homeomorphism of the 4-torus, with fundamental group 𝐴.
The 4-torus intersects a loop representing 𝑥 transversely in one point and so represents the Poincaré dual to
x
*
.
Thus
x
*
η
2
∩
[
G
]
=
η
2
∩
(
x
*
∩
[
G
]
)
=
η
2
|
A
∩
[
A
]
=
±
1
.
There is a more algebraic argument based on [1].)
Since
η
2
|
A
=
2
y
*
η
|
A
e
*
and
x
*
η
2
≠
0
, it follows that
x
*
y
*
η
≠
0
.
On the other hand,
x
*
2
η
=
y
*
2
η
=
0
, and so
x
*
η
and
y
*
η
are linearly independent.
Hence cup product with 𝜂 is injective on
H
1
(
G
¯
)
.
It now follows from the exactness of the Gysin sequence that
β
2
(
G
)
=
2
.
The same argument with coefficients
F
=
F
p
shows that
β
2
(
G
;
F
p
)
=
2
=
β
1
(
G
;
F
p
)
for all primes 𝑝, and so 𝐺 is homologically balanced.
(It is at this point that we need Corollary 13.)
It can be shown that the relations
[
c
,
e
]
=
[
d
,
e
]
=
1
,
[
c
,
d
]
=
f
-
1
,
[
y
,
d
]
=
f
in the above presentation for 𝐺 are consequences of the other relations.
Using four of the remaining relations to eliminate the generators
c
,
d
,
e
,
f
leads to a presentation with 2 generators and 4 relations.
However, we do not know whether 𝐺 has a balanced presentation.
6 Nilpotent Lie algebras and Lie groups
In the parallel world of nilpotent real Lie algebras 𝔫, there is a similar issue, of whether there are such algebras 𝔫 with
β
2
(
n
)
=
β
1
(
n
)
and of large dimension.
There are just 3 non-abelian nilpotent Lie algebras 𝔫 of dimension at most 7 with
β
2
(
n
)
=
β
1
(
n
)
, and for these, the common value is 2 (see [3]).
One has dimension 4 and the other two have dimension 6.
The two 6-dimensional nilpotent Lie algebras with
β
2
=
β
1
each have 1-dimensional centre, and they may be distinguished by the dimensions of the maximal abelian Lie ideals of the quotients by the centre, which are 4 and 3, respectively.
These Lie algebras correspond to the Lie groups
G
4
,
G
6
,
12
and
G
6
,
14
of [15].
There are direct connections between these aspects of nilpotency.
A finitely generated torsion-free nilpotent group 𝐺 has a Mal’cev completion
G
R
, which is a 1-connected nilpotent real Lie group of dimension
h
(
G
)
in which 𝐺 is a lattice (i.e., a discrete subgroup such that the coset space
G
R
/
G
is compact).
Conversely, every nilpotent real Lie algebra of dimension at most 6 is isomorphic to one with rational structure coefficients, and so the corresponding 1-connected Lie group admits lattices by Mal’cev’s criterion [18, Chapter 2].
The coset space
M
=
G
R
/
G
is aspherical, and so
H
q
(
G
;
R
)
≅
H
DR
q
(
M
)
, the de Rham cohomology of 𝑀.
This is in turn the cohomology of the associated Lie algebra 𝔤 (see [16]).
The paper [17] shows directly that the rational cohomology rings of a finitely generated nilpotent group and the associated Lie algebra are isomorphic.
We may find further examples of torsion-free nilpotent groups 𝐺 such that
h
(
G
)
=
6
and
β
2
(
G
)
=
β
1
(
G
)
=
2
among lattices in
G
6
,
12
and
G
6
,
14
.
Each such lattice Γ is nilpotent and of Hirsch length 6, and we may invoke [16] to conclude that
β
2
(
Γ
)
=
β
1
(
Γ
)
=
2
.
(However, we do not know whether such lattices must be homologically balanced.
We remark that, when 𝐺 is the group constructed in § 5, then
G
R
≅
G
6
,
12
since
G
¯
=
G
/
ζ
G
has an abelian normal subgroup of rank 4.)
Some information is lost in passing from groups to Lie algebras since commensurable nilpotent groups have isomorphic completions.
Moreover, from dimension 7 onwards, there are uncountably many isomorphism classes of nilpotent Lie algebras, and so most do not derive from discrete groups.
The main result of [3] is that there are no nilpotent Lie algebras 𝔫 of dimension 7 with
β
2
(
n
)
=
β
2
(
n
)
=
2
, and so there are no such groups of Hirsch length 7.
It remains an open question whether there are any examples at all of homologically balanced nilpotent groups of Hirsch length greater than 7.
Acknowledgements
I would like to thank Grant Cairns for providing me with a copy of [3], and for subsequent advice.
I would also like to thank the referees for their scrutiny and constructive suggestions towards improving the exposition.
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