THE BIERI–NEUMANN–STREBEL INVARIANTS VIA NEWTON
POLYTOPES

arXiv:1802.07049v2 [] 18 Mar 2018

DAWID KIELAK
Abstract. We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such
Newton polytopes are single polytopes (rather than formal differences of two
polytopes); this result can be seen as analogous to the fact that determinants
of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between
the Newton polytopes and invertibility of the matrices over Novikov rings,
thus establishing a connection with the invariants of Bieri–Neumann–Strebel
(BNS) via a theorem of Sikorav.
We offer several applications: we reprove Thurston’s theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group;
we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants
of Poincaré duality groups of type F in dimension 3 and groups of deficiency
one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result
partially confirms a conjecture of Friedl.
We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the L2 -torsion polytope of Friedl–Lück is
invariant under homotopy. We prove the vanishing of this polytope in the
presence of amenability, thus proving a conjecture of Friedl–Lück–Tillmann.

1. Introduction
In 1986, W. Thurston [Thu] proved a remarkable theorem about the ways in
which a compact oriented 3-manifold M can fibre over the circle. He introduced
a semi-norm x : H 1 (M ; R) → [0, ∞) (now called the Thurston norm and often
denoted by k · kT ), whose unit ball is a polytope Bx , such that if φ : π1 (M ) → Z is
a homomorphism induced by a fibration of M , then φ lies in the cone of an open
maximal face of Bx , and every other integral character π1 (M ) → Z lying in the
same cone also comes from a fibration (hence one can talk about fibred faces of
Bx ).
A year later, Bieri–Neumann–Strebel [BNS] introduced the geometric (or Σ, or
BNS) invariant of finitely generated groups. They observed that if G is the fundamental group of the manifold M above, then the Σ-invariant Σ(G) ⊆ H 1 (G; R)r{0}
coincides with the union of the cones of the fibred faces (with the origin removed).
Thus, Thurston’s theorem can be reinterpreted as a result about the structure of
Σ(G) – for 3-manifold groups, the BNS invariant is determined by the integral polytope Bx∗ ⊂ H1 (G; R) (which is the dual polytope of Bx ), where ‘integral’ means
that the vertices lie on the integer lattice, and ‘determined’ means that if a character lies in Σ(G), then any other character which attains its minimum on Bx∗ at
the same face as φ also lies in Σ(G). We can thus mark the vertices of Bx∗ dual to
the fibred faces, and say that Σ(G) is determined by a marked polytope.
Date: March 20, 2018.
1

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DAWID KIELAK

The BNS invariants where subsequently computed for many finitely generated
groups: finitely generated nilpotent or metabelian groups [BG, Theorem E], 2generator 1-relator groups and Houghton groups [Bro, Theorem 4.2 and Proposition
8.3], right-angled Artin groups (RAAGs) [MV, Theorem 5.1], groups with staggered
presentations and graph products (with underlying graphs without a central vertex)
[GM], pure symmetric automorphism groups of finitely generated free groups [OK],
limit groups [Koc2, Corollary 30], fundamental groups of compact Kähler manifolds
[Del], pure symmetric automorphism of RAAGs [KP, Theorem A], braided Thompson group F [Zar1, Theorem 3.4], some Artin groups [AK1, AK2, Alm], pure braid
groups [KMM], {finitely generated free}-by-Z groups with polynomially growing
monodromy [CL], and Lodha–Moore groups [Zar2]. In all of these examples, Σ(G)
is determined by an integral polytope.
The invariant has also been computed in [BNS, Theorem 8.1] for finitely generated groups of piecewise linear automorphisms of the interval [0, 1], which are
irreducible in the sense that there are no points in the interior of the interval fixed
by the entire group, and which have independent left and right derivatives in the
following sense: let λ and ρ be the left and right derivatives taken at the endpoints
of the interval [0, 1]; ‘independence’ means that λ(ker ρ) = λ(G) and vice versa. In
this case, Σ(G) = H 1 (G; R) r {κ log λ, κ log ρ | κ > 0}. Thus, Σ(G) is determined
by a polytope, but not necessarily integral – there is a specific example giving a
non-integral polytope. The example is shown by Stein [Ste] to be finitely presented
and of type FP∞ , but it is not of type F – this is relatively easy to see, since the
example contains Thompson’s group F .
It is worth noting that the vast majority of the papers above computes Σ(G), and
only then deduces that it is determined by a polytope. It is of course very valuable
to have a complete description of Σ(G), but such detailed knowledge cannot be
obtained even for all finitely presented groups – it was shown by Cavallo–Delgado–
Kahrobaei–Ventura [CDKV, Theorem 6.4] that computing Σ(G) is undecidable.
Thus, it is worthwhile to study the structure of Σ(G) from a more abstract viewpoint.
The original motivation behind this article was to study the BNS invariants of
free-by-cyclic groups. In particular, the author set out to prove that such BNS
invariants have only finitely many connected components, which was not known
prior to this work, but expected due to the analogies between free-by-cyclic groups
and fundamental groups of 3-manifolds. It turned out that the analogy goes much
further, and the methods developed found also other applications. Let us summarise
the results in this direction by the following statement.
Theorem. Let G belong to one of the following classes:
(1)
(2)
(3)
(4)

descending HNN extensions of finitely generated free groups; or
fundamental groups of compact connected oriented 3-manifolds; or
agrarian Poincaré duality groups of dimension 3 and type F; or
agrarian groups of deficiency one.

Then Σ(G) is determined by an integral polytope.
Here, ‘agrarian’ means that the integral group ring of the group in question
embeds into a skew-field (division algebra).
We can also draw a number of conclusions related to the L2 -torsion polytope
PL2 of Friedl–Lück [FL]. This polytope is associated to every finite L2 -acyclic
(i.e. with vanishing L2 -homology) classifying space of a group G satisfying the
Atiyah conjecture – the Atiyah conjecture is a statement about integrality of the L2 Betti numbers of CW-complexes on which G acts freely, properly and cocompactly.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

3

(In fact, the polytope PL2 can also be defined for other CW-complexes, not only
classifying spaces.)
The original definition of PL2 takes as input a classifying space rather then its
fundamental group, and if two different classifying spaces are taken, then the resulting polytopes differ by the Newton polytope of some element of the Whitehead
group of G. The Whitehead group is a (K-theoretic) group consisting of obstructions for G-homotopy equivalences of G-CW complexes being G-simple homotopy
equivalences.
Theorem. Let G be a finitely-generated torsion-free group satisfying the Atiyah
conjecture. The Newton polytope of any element of the Whitehead group Wh(G)
vanishes. Hence, if G is additionally L2 -acyclic and of type F, then the L2 -torsion
polytope PL2 (G) of G is well-defined.
Also, if G is amenable and G ∼
6 Z, then PL2 (G) is a singleton.
=
The methods used in the proofs of both theorems above are of an algebraic
character: we use Sikorav’s theorem (Theorem 5.5) to relate BNS invariants to
matrices over the group ring, and study these by extending scalars to a skewfield (given by the property of being agrarian – usually, we will use the skew-field
constructed by Linnell [Lin] for torsion-free groups satisfying the Atiyah conjecture).
Thus, the methods can be traced to the theory of L2 -invariants (already used for
the computation of the BNS invariants of some 2-generator 1-relator groups by
Friedl–Tillmann [FT]), but stripped of its analytic flavour. Let us also mention the
article of Friedl–Lück [FL], which is foundational for the point of view presented
here.
The paper is divided into four parts. Let us briefly discuss the contents of each
of these.
Twisted group rings, biorderable groups, and the Ore localisation. In
Section 2 we recall the three notions, which will be of central importance throughout
the paper.
Matrices and their polytopes. In Section 3 we prove the two main technical
results of the paper. We let H be a finitely generated free-abelian group, K be a
skew-field, and denote by KH some twisted group ring of H (this is a variation
on the usual group ring, which appears naturally when one considers extensions
of groups). The ring KH can be thought of as a group ring, but also as a ring of
twisted Laurent polynomials in several commuting variables over a skew-field.
We will embed KH into its skew-field of fractions D (formally, the Ore localisation), and study square matrices over KH which become invertible over D. For
any such matrix A, w have the Dieudonné determinant det A at our disposal; since
D is the skew-field of fractions, we can write det A as a fraction of two elements in
KH. Thinking of these elements as Laurent polynomials, we can take their Newton
polytopes (convex hulls of their supports), and thus associate to det A a formal
difference of two polytopes (one for the numerator, and one for the denominator).
The first result of significance is Theorem 3.14, which shows that in fact we
can represent det A by a single polytope, denoted by P (A). This rather innocuous
statement has interesting consequences (listed below). Theorem 3.14 is also interesting in its own right: were our coefficients K forming a commutative field, and
were the group ring KH not twisted, we would be dealing with matrices over an
abelian ring of classical interest, the ring of Laurent polynomials. It is clear that in
this case the determinant det A is a polynomial, and thus its Newton polytope is a
single polytope. In our setting, the Dieudonné determinant is not a polynomial (it

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DAWID KIELAK

is a rational function), but nevertheless we do obtain a single polytope, in analogy
with the classical case.
The second theorem we prove in Section 3 is Theorem 3.30. It says that the
shape of the Newton polytope P (A) determines the existence of right-inverses of A
φ
d . The superscript φ denotes a character φ : H → R, and
over the Novikov rings KH
φ

d is the ring of formal sums of elements in h with coefficients
the Novikov ring KH
in K (just like KH), with the caveat that sums with infinite support are allowed,
provided that the supports go to infinity ‘only in the direction of φ’. Theorem 3.30
φ
d if and only if φ attains its minimum
says that A admits a right-inverse over KH
on P (A) at a unique vertex.
The Novikov rings play a central role in the Σ-theory (theory of the BNS invariants), since Sikorav [Sik] proved that a character φ lies in Σ(G) if and only if
φ
d ) = 0 (see Theorem 5.5). Motivated by this result, we introduce the
H1 (G; ZG
notion of the Σ-invariant of the matrix A, denoted by Σ(A); Theorem 3.30 tells
us that Σ(A) is determined by P (A), and this will be the source of our polytopes
throughout the paper.

Agrarian groups. The focal point of the remainder of the paper will be the class
of agrarian groups, that is groups G whose group ring ZG embeds in a skew-field.
We introduce the class in Section 4, where we also introduce the potentially smaller
class of equivariantly agrarian groups.
It is clear that agrarian groups are torsion-free and satisfy the zero-divisor conjecture of Kaplansky; there are currently no other obstructions to being agrarian
known – there is not a single torsion-free example of a non-agrarian group. On
the other hand, there are many positive examples: torsion-free amenable groups
whose integral group rings have no zero-divisors, biorderable groups, and torsionfree groups satisfying the Atiyah conjecture are all examples of agrarian groups.
The class is also closed under subgroups, directed unions, and many extensions,
and being agrarian is a fully-residual property. Noteworthy from our point of view
is that the class is known to contain all descending HNN extensions of free groups,
and the fundamental groups of 3-manifolds which fibre over the circle.
Note that Sections 3 and 4 are completely independent.
Applications. In Section 5 we list the applications of Theorems 3.14 and 3.30.
Funke [Fun] introduced the notion of a group of polytope class. He defined it as
a subclass of the class of torsion-free groups satisfying the Atiyah conjecture and
with finitely generated abelianisations. We show (in Theorem 5.14) that in fact
every group in this (a priori) larger class is of polytope class. This allows us to use
the work of Funke in greater generality: we prove that the Newton polytopes of
elements of the Whitehead group Wh(G) of a finitely-generated torsion-free group
satisfying the Atiyah conjecture are trivial (Corollary 5.16). The Whitehead group
is a quotient of K1 (ZG), the first K-group, by the subgroup {±g | g ∈ G}. It
plays an important role in homotopy theory: Wh(G) is trivial if and only if every
homotopy equivalence between two CW-complexes with fundamental group G is a
simple-homotopy equivalence. The elements of Wh(G) can be though of as matrices
over ZG, and so it makes sense to talk about Newton polytopes of elements of
Wh(G). Our vanishing result can be seen as evidence towards the conjecture on
triviality of the Whitehead groups of torsion-free groups.
The vanishing of the Newton polytopes of the elements of Wh(G) implies that
the L2 -torsion polytope of Friedl–Lück [FL] is a homotopy invariant, and hence it
can be defined as a group invariant of L2 -acyclic groups of type F satisfying the
Atiyah conjecture. The L2 -torsion polytope is obtained as the Newton polytope of

THE BNS INVARIANTS VIA NEWTON POLYTOPES

5

(the determinant of) the universal L2 -torsion. This latter invariant can be thought
of as an L2 -version of the Whitehead torsion – we refer the reader to [FL], where
it is defined and discussed in more detail.
The next situation we look at occurs when the group G admits a finite subnormal
chain terminating in an amenable group N . We prove that if N is not abelian, or
G has trivial centre, then the L2 -torsion polytope PL2 (G) is a singleton. We use
this to prove a conjecture of Friedl–Lück–Tillmann [FLT], which states that if G
as above is amenable but not cyclic, then PL2 (G) is a singleton.
Afterwards, we look at an agrarian group G of deficiency one. For such a group
we show (Theorem 5.23) that Σ(G) is determined by an integral polytope. If G satisfies the Atiyah conjecture then we obtain a slightly stronger result (Theorem 5.25).
These results confirm a conjecture of Friedl (under the additional assumption of the
group being agrarian). They are also interesting in view of a conjecture of Bieri,
which states that any group of deficiency one with non-trivial Σ(G) is a descending
HNN extension of a free group. The BNS-invariants of descending HNN extensions
will be discussed below, but the structure we exhibit for the BNS-invariants for
agrarian groups of deficiency one is of the same kind as in the case of descending
HNN extensions.
We proceed to discuss precisely the descending HNN extensions of finitely generated free groups. Note that this class includes {finitely generated free}-by-Z
groups (usually referred to as free-by-cyclic groups). It was the author’s original
motivation to prove that for such a group G, the invariant Σ(G) has finitely many
connected components – this was not known before, except for free-by-cyclic groups
with polynomially growing monodromy, see the work of Cashen–Levitt [CL]. In fact
we show more; we prove a result fully analogous to the statement of Thurston for
3-manifolds: the L2 -torsion polytope PL2 (G) admits a marking of its vertices, and
a character φ lies in Σ(G) if and only if it attains its minimum on PL2 (G) uniquely
at a marked vertex (Theorem 5.29). Note that the polytope PL2 (G) is (for descending HNN extensions of free groups) very much analogous to the Thurston
polytope Bx∗ – when the group is a 3-manifold group (this happens for free-bycyclic groups with geometric monodromy), the L2 -torsion polytope coincides with
Bx∗ (see [FL]). Further analogies between PL2 (G) and Bx∗ for general descending
HNN extensions of free groups where studied by Funke and the author in [FK]. In
particular, Bx∗ contains enough information to recover the Thurston norm itself.
Using the same technique one can use PL2 (G) to induce a function H 1 (G; R) → R.
It was shown in [FK] that in fact this function is also a semi-norm, and given an
integral character it returns (up to a sign) the L2 -Euler characteristic of the kernel
of the character. In particular, when the kernel is finitely generated, it returns its
(usual) Euler characteristic, and so determines the rank of such a kernel (which is
necessarily a free group).
In the two final sections, we look at Poincaré duality groups of type F in dimension 3, and then at 3-manifolds themselves. In the first situation, again we prove
that Σ(G) is determined by an integral marked polytope (Theorem 5.32). When
M is a 3-manifold, we reprove Thurston’s theorem. Note that the proof we give is
completely independent, and of an algebraic character.

Acknowledgements. The author would like to thank Kai-Uwe Bux for comments
on an earlier version of the article, as well as Fabian Henneke and Stefan Witzel
for helpful discussions.
The author was supported by the Priority Programme 2026 ‘Geometry at infinity’
of the German Science Foundation (DFG).

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DAWID KIELAK

2. Twisted group rings, biorderable groups, and the Ore
localisation
We start by introducing three concepts that will be of the utmost importance.
2.1. Twisted group rings. Throughout the article, rings are unital and not necessarily commutative, modules are by default right-modules, and groups are discrete.
Definition 2.1. Let R be a ring. We denote its group of units by R× . An element
r is a zero-divisor if and only if r is non-zero, and there exist x ∈ R such that
xr = 0 or rx = 0.
Let R be a ring and G a group. We denote the set of functions G → R by
RG . Clearly, pointwise addition turns RG into an abelian group. Since R has a
distinguished 0, we can talk about supports.
Definition 2.2 (Support). For x ∈ RG , we define its support to be
supp x = {g ∈ G | x(g) 6= 0}
Note that for an element x ∈ RG we will use both the function notation, and
the sum notation
X
x=
x(g)g
g∈G

In the sum notation, we will often ignore the elements g with x(g) = 0, that is we
will focus on the values x attains on its support. The sum notation is particularly
common when talking about the subgroup RG of RG consisting of functions with
finite support. We now explain how to endow RG with a ring structure.
Definition 2.3 (Twisted group ring). Let φ : G → Aut(R) and µ : G × G → R×
be two functions satisfying

φ(g) ◦ φ(g ′ ) = c µ(g, g ′ ) ◦ φ(gg ′ )

µ(g, g ′ ) · µ(gg ′ , g ′′ ) = φ(g) µ(g ′ , g ′′ ) · µ(g, g ′ g ′′ )

where c : R× → Aut(R) takes r to the left conjugation by r. The functions φ and
µ are called structure functions, and turn RG into a twisted group ring by setting
(⋆)

rg · r′ g ′ = rφ(g)(r′ )µ(g, g ′ )gg ′

and extending to sums by linearity (here we are using the sum notation).
When φ and µ are trivial, we say that RG is untwisted (in this case we obtain the
usual group ring). We adopt the convention that the group ring with Z-coefficients
is always untwisted.
The following is the key example of a twisted group ring, and we will use it
repeatedly throughout the article.
Example 2.4. Let ZG be the (untwisted) integral group ring. Let α : G → H be
a quotient with kernel K, and let s : H → G be a set-theoretic section of α. Then
the map

−1 
α(g)
g 7→ g · s ◦ α(g)
induces an isomorphism of twisted group rings ZG ∼
= (ZK)H, where ZG and ZK
are untwisted, and the structure functions of (ZK)H are as follows: φ(h) is the
automorphism of ZK induced by the right conjugation by s(h), and µ(h, h′ ) =
−1
s(h)s(h′ ) s(hh′ )
.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

7

Note that in the generality that we introduced them, the rings RG are sometimes
called crossed products.
The structure functions do not appear in the notation, but they are (implicitly)
assumed to be specified. Note that the structure functions can be used to define
products on other subsets of RG as follows. Let x, y ∈ RG be two functions. For
every g ∈ G define Xg and Yg to be the smallest subsets of G such that if g = a · b
and (a, b) ∈ supp x × supp y, then (a, b) ∈ Xg × Yg . If Xg and Yg are finite for every
g, then (⋆) gives a method of calculating the product x · y by setting its value at g
to the value at g of
 X
 X

x(a)a ·
y(b)b
a∈Xg

b∈Yg

We will refer to this product as induced by the twisted convolution (⋆).
2.2. Biorderable groups. Biorderable groups form a class of groups whose (twisted)
group rings are well-behaved.
Definition 2.5. A group G is biorderable if and only if there exists a total ordering
6 on G which is invariant under right and left multiplication.
The class of biorderable groups is clearly closed under taking subgroups and
arbitrary products – for the latter, it is enough to order the factors of a product in
some way, and then use lexicographic order on the resulting group. Together these
two properties imply that residually biorderable groups are themselves biorderable.
The class is also closed under central extensions, and contains finitely generated freeabelian groups. Putting these properties together yields that residually torsion-free
nilpotent groups are biorderable: this is a rather large class, containing free groups
and surface groups (see e.g. [Bau]), as well as RAAGs (as shown by Droms [Dro]).
The class of biorderable groups is also closed under directed unions and free
products – see the book by Deroin–Navas–Rivas [DNR, Section 1.1.1 and Theorem
2.1.9].
The key property for us is that twisted group rings of biorderable groups with
skew-field coefficients embed in skew-fields.
Theorem 2.6 (Malcev [Mal]; Neumann [Neu]). Let G be a biorderable group, let
K be a skew-field, and let KG be a twisted group ring. The ring KG embeds into a
skew field F , its Malcev–Neumann completion.
Let us briefly describe the construction: fix a biordering 6 on G. We define
F ⊆ KG to be the subset of those functions G → K whose support is well-ordered
with respect to 6, that is such that any non-empty subset of the support has a
6-minimum. It is clear that F is an abelian group; it can be turned into a ring
using the twisted convolution (⋆) – one has to argue that when taking products,
it is always sufficient to take product of two finitely supported functions, but this
follows from the fact that functions in F have well-ordered supports. It turns out
that the ring F is actually a skew-field.
Lemma 2.7. Let G be a biorderable group, let K be a skew-field, and let KG denote
some twisted group ring of G. The units of KG have supports of cardinality 1.
Proof. Let 6 be a biordering of G. Take x ∈ KG× ; by definition, there exists
y ∈ KG such that xy = 1. Write
m
n
X
X
νj h j
λi gi , y =
x=
i=0

j=o

where the finite sequences (gi ) and (hj ) of elements of G are strictly 6-increasing,
and the coefficients λi and νj are never 0.

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DAWID KIELAK

We are now going to carry out the twisted multiplication and compute x · y. It
is immediate that the support of xy does not contain any group element 6-smaller
than g0 h0 . The coefficient at g0 h0 of xy is
λ0 φ(g0 )(ν0 )µ(g0 , h0 )
Crucially, none of the factors above are zero, and the multiplication is carried out
in a skew-field K. Therefore g0 h0 is the strictly 6-smallest element of the support
of xy.
Arguing analogously, we conclude that gn hm is the strictly 6-greatest element
of the support of xy. But xy = 1, and therefore 1 is the only element in supp xy.
We conclude that g0 h0 = gn hm , which is only possible when n = 0 = m, in which
case x has support of cardinality 1, as claimed.

2.3. Ore localisation. In this section we review the notion of Ore localisation,
and show how it comes into play when considering various twisted group rings of
amenable groups.
Definition 2.8. Let R be a ring. A subset S ⊆ R is said to satisfy the left Ore
condition if and only if for all (p, r) ∈ R × S there exists (q, s) ∈ R × S such that
sp = qr
(this can be interpreted as the existence of left common multiples). The right Ore
condition is defined analogously by the equation
ps = rq
The ring R is said to be an Ore ring if and only if the subset S of R consisting of
all non-trivial non-zero-divisors satisfies both the left and the right Ore condition.
If additionally R has no zero-divisors, then we say that it is an Ore domain.
Definition 2.9. Given a ring R and a subset S ⊆ R r {0} closed under multiplication and satisfying the left and right Ore conditions, we define the localisation of
R at S by
RS −1 = (R, S)/ ∼
where ∼ is the transitive closure of the relation identifying (p, r) with (px, rx) for
any x ∈ S.
When S is the subset of R consisting of all non-zero non-zero-divisors, we call
RS −1 the Ore localisation.
One should think of (p, r) as the right fraction p/r. The Ore conditions allow
us to change left fractions into right ones and vice versa; they also allow us to
find common denominators. Therefore the localisation is itself a ring, and the map
p 7→ p/1 is a ring morphism. The construction is explained in Cohn’s book [Coh,
Section 1.2], and in more details in Passman’s book [Pas, Section 4.4]
Remark 2.10. Clearly, the Ore condition can be used to find common denominators
for any finite number of elements in the localisation RS −1 .
The following facts may also be found in [Coh, Section 1.2].
Proposition 2.11. Let R be an Ore ring, and let S denote the set of non-zero
non-zero-divisors.
(1) R embeds into its Ore localisation RS −1 .
(2) Every automorphism of R extends to an automorphism of RS −1 .
(3) When R is an Ore domain, then any ring monomorphism R′ ֒→ R extends
to a monomorphism R′ S ′−1 ֒→ RS −1 , where S ′ is the group of units of R′ .
(4) When R is an Ore domain, then RS −1 is a skew-field.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

9

Let us now introduce amenability.
Definition 2.12 (Amenable groups). A countable group G is amenable if and only
if there exists a sequence (Fi ) of non-empty finite subsets of G (a Følner sequence),
such that for every g ∈ G we have
|Fi △ g.Fi |
−→ 0
|Fi |
as i −→ ∞ (here △ denotes the symmetric difference).
Note that in the definition above one can easily replace the left translates g.Fi
by the right translates Fi .g.
In the context of group rings, the Ore condition is (almost) equivalent to amenability of the underlying group, as shown by the author in an appendix to an article of
Bartholdi [Bar].
Theorem 2.13 ([Bar, Theorem A.1]). Let G be a group, and suppose that the
group ring ZG has no zero-divisors. Then ZG is an Ore domain if and only if G
is amenable.
One of the implications was shown by Tamari [Tam]. We will go back to his
proof, since we will require a version of his theorem for twisted group rings.
Theorem 2.14 (Tamari [Tam]). Let G be an amenable group, and let K be a skewfield. If a twisted group ring KG does not contain zero-divisors, then KG is an Ore
domain.
Proof. Since KG admits an anti-automorphism induced by g 7→ g −1 , it is enough
to check the Ore condition on one side. (Note that this anti-automorphism sends
λg to g −1 λ = φ(g −1 )(λ)g −1 for λ ∈ K.)
Let p, r ∈ KG with r 6= 0. Let F be a Følner set such that
1
|F △ F.g|
<
|F |
|U |
P
P
for every g ∈ U = supp p ∪ supp r. Consider q = f ∈F λf f and s = f ∈F λ′f f ;
we treat the elements λf , λ′f ∈ K as 2|F | variables. We now try to solve the linear
equation
qr(h) = sp(h)
for every h ∈ G. Note that this equation is trivial except possibly for
[
h∈F ∪
F.g
g∈U

that is except for fewer than 2|F | elements h. Hence we are solving a system of
fewer than 2|F | equations with 2|F | variables. Since we are working over a skewfield, a non-zero solution exists. Let us pick one such – this way we have defined q
and s such that qr = sp. If s = 0 then r is a zero-divisor, which is a contradiction.
Therefore the pair (q, s) is as required.

For the sake of completeness, let us also introduce the elementary amenable
groups.
Definition 2.15 (Elementary amenable groups). The class of elementary amenable
groups is the smallest class containing all finite groups and all abelian groups, and
closed under subgroups, quotients, direct unions, and extensions.
It is classical that all elementary amenable groups are amenable – the class
of amenable groups contains finite groups and Z, and is closed under subgroups,
colimits, quotients, and extensions.
In Section 5.5 we will need the following lemma.

10

DAWID KIELAK

Lemma 2.16. Let G be any group, and let N be a normal subgroup. Suppose that
S ⊆ ZN r {0} is closed under multiplication and satisfies the left Ore condition in
ZN . Then S G , the multiplicative closure of the G-conjugates of S, satisfies the left
Ore condition in ZG.
Proof. We first argue that S G satisfies the left Ore condition in ZN , and then we
argue that any G-invariant subset of ZN closed under multiplication and satisfying
the left Ore condition in ZN actually also satisfies the left Ore condition in ZG.
To prove the first claim, take p ∈ ZN and s = s1 . . . sn where each si lies in some
G-conjugate of S. We will find left common multiples of p and s. We argue by
induction on n. If n = 1 then the claim follows from the Ore condition for S (which
clearly holds for all conjugates of S as well). Otherwise, using the Ore condition,
there exist q ∈ ZN and tn in the same conjugate of S as sn , such that
tn p = qsn
Now by the inductive hypothesis there exist r ∈ ZN and u ∈ S G such that
uq = rs1 . . . sn−1
and therefore
utn p = rs
This proves the first claim.
To prove the second claim, take s ∈ S and p ∈ ZG, where we assume that
S is multiplicatively closed and G-invariant. We have ZG = (ZN ) G/N (as in
Example 2.4), and so we can write
n
X
pi
p=
i=1

where pi = κi gi , with gi ∈ G/N and κi ∈ ZN . Using the Ore condition in ZN
−1
we conclude that there exist κ′1 ∈ ZN and s1 ∈ S such that s1 κ1 = κ′1 sg1 ,
where exponentiation denotes right conjugation, and we are using the fact that S
is G-invariant.
We now argue by induction on n: if n = 1, then
s1 p = s1 κ1 g1 = κ′1 sg1

−1

g1 = κ′1 g1 s

and the Ore condition is verified. Otherwise, we have
n
X
s1 κi p i
s1 p − κ′1 g1 s = s1 p − s1 p1 =
i=2

and s1 κi ∈ ZN for every i. By the inductive hypothesis there exist q ∈ ZG and
t ∈ S such that
n
X
s1 κi pi ) = qs
t·(
i=2

and so
ts1 p = (tκ′1 g1 + q)s



3. Matrices and their polytopes
In this section, we consider a finitely generated free-abelian group H. With
the applications in mind, one will not err by thinking of H as the free part of
the abelianisation of a finitely generated group. On the other hand, the twisted
group ring KH (where K is a skew-field), which will be the main focal point of
the section, can be interpreted as a ring of twisted Laurent polynomials in finitely
many commuting variables, and the more algebraically-minded reader might prefer
this point of view.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

11

3.1. Polytopes. We start by looking at polytopes. Note that H1 (H; R) = H ⊗Z R
is a finite dimensional vector space over R.
Definition 3.1 (Polytopes). A polytope in H1 (H; R) is a compact subset which
is the intersection of finitely many affine halfspaces. Note that, in particular, the
empty set is a polytope.
Given a polytope P and a character φ ∈ H 1 (H; R), we define

Fφ (P ) = p ∈ P | φ(p) = min φ(q)
q∈P

(clearly, this is precisely the set of points in P on which φ attains its minimum).
The image Fφ (P ) is also a polytope. The collection {Fφ (P ) | φ ∈ H 1 (H; R)} is
the collection of faces of P . A face is called a vertex if and only if it has dimension
0. Note that P = F0 (P ) is also a face of P .
A polytope P is integral if and only if its vertices lie in H ⊆ H1 (H; R); a polytope
P is φ-flat if and only if P = Fφ (P ).
The Minkowski sum of two polytopes P and Q defined by
P + Q = {p + q | p ∈ P, q ∈ Q}

turns the set of all non-empty polytopes in H1 (H; R) into a cancellative abelian
monoid. It is clear that the Minkowski sum restricts to an operation on the set
of non-empty integral polytopes in H1 (H; R), turning this set into an abelian cancellative monoid as well.
We define P(H) to be the Grothendieck group of fractions of the monoid of
non-empty integral polytopes in H1 (H; R): the group P(H) is constructed from
the free-abelian group with basis equal to the set of non-empty integral polytopes
by factoring out relations given by the Minkowski sum. It is easy to see that every
element in P(H) is represented by a formal difference P − Q of polytopes. We say
that an element is a single polytope if and only if we can take Q to be a singleton.
In this case P is uniquely determined up to translation. Note that Fφ induces a
homomorphism Fφ : P(H) → P(H) given by Fφ (P − Q) = Fφ (P ) − Fφ (Q).
We also define PT (H) to be the quotient of P(H) by the subgroup consisting of
formal differences of singletons. It is immediate that this is equivalent to considering
pairs of polytopes P − Q up to translation of the first polytope. Note that a single
polytope in P(H) is represented by a unique single polytope in PT (H).
Let us now introduce duals, which establish a correspondence between polytopes
in H1 (H; R) and subsets of H 1 (H; R).
Definition 3.2 (Duals). Let P ⊂ H1 (H; R) be a polytope. Given a face Q of P ,
we define its duals to be the connected components of
{φ ∈ H 1 (H; R) r {0} | Fφ (P ) = Q}
Remark 3.3. If Q 6= P , then Q admits only a single dual. This is also true if Q = P
is of full dimension, or codimension at least 2. Hence the existence of non-unique
duals is rather pathological, but it does occur in situations of interest to us.
Remark 3.4. Observe also that duals are convex, and if P is not empty, then every
dual of a vertex of P is open, and the union of duals of vertices of P is dense in
H 1 (H; R).
Pn
and let Q be a
Lemma 3.5. Let P = i=1 Pi be a Minkowski sum of polytopes,
Pn
face of P . There exist unique faces Qi of Pi such that Q = i=1 Qi .

Proof. Let Q be a face of P . By definition, this means that there exists
P a character
ψ such that Q = Fψ (P ). Let Qi = Fψ (Pi ); we clearly have Q =
Qi . We now
argue that the definition of Qi is in fact independent of the choice of ψ.

12

DAWID KIELAK

Perturb ψ slightly inside of the dual of Q containing it; this way we can potentially decrease the dimensions of some of the faces Fψ (Pi ) without increasing the
dimension of any other. Since Q is well-defined, this proves that the faces Qi do
not depend on the choice of ψ locally. Now, the dual of a face is unique or there
are two duals, each consisting precisely of one character, the two characters being
antipodal. In the former case, the fact that Qi is well defined follows immediately
from connectivity; in the latter case, it is clear that if Q = Fψ (P ) = F−ψ (P ) then
Fψ (Pi ) = F−ψ (Pi ) for each i.

We now state a result of Funke which gives us a method of recognising single
polytopes among the elements of P(H).
Proposition 3.6 (Funke [Fun, Lemma 4.3]). Suppose that H is of rank at least 2.
For every P ∈ P(H), the element P is a single polytope if and only if Fφ (P ) is a
single polytope for each φ ∈ H 1 (H; Z) r {0}.
The following corollary occurs as [Fun, Lemma 4.7].
Corollary 3.7. Let P ∈ P(H). Suppose that for every φ ∈ H 1 (H; Z) r {0} there
exists a φ-flat polytope Xφ such that P + Xφ is a single polytope. Then P is a single
polytope.
Proof. We will argue by induction on the rank of H. When this rank is 0, every
element of P(H) is a single polytope. When the rank is 1, then the only φ-flat
polytopes are singletons, and so P is a single polytope.
Now suppose that the result holds for n − 1, and let H be of rank n (with
n > 2). We will argue that for every ψ ∈ H 1 (H; Z) r {0}, the face Fψ (P ) is a single
polytope; in view of Proposition 3.6, this suffices.
Note that Fψ (Xφ ) is a φ-flat polytope. Also, Fψ (P + Xφ ) = Fψ (P ) + Fψ (Xφ ) is
a single polytope, since P + Xφ is. But, up to translation, we have Fψ (P ) = Q − Q′
such that Q, Q′ and Fψ (Xφ ) lie in P(ker ψ), and the rank of ker ψ is lower than
that of H. Hence, by the inductive hypothesis, Fψ (P ) is a single polytope.

3.2. Dieudonné determinant. Let us start by introducing an important convention: whenever we talk about a matrix, we will explicitly state over which ring it
lies. In particular, properties like invertibility will always be taken in the ring over
which the matrix is defined. If we want to consider a matrix as a matrix over a
larger ring (via extension of scalars), we will use tensor notation. Unless specified
otherwise, we tensor over ZG. We will use Mn (R) to denote the ring of n × n
matrices over a ring R.
We are now going to introduce the Dieudonné determinant which can be computed for square matrices over a skew-field. We will later show how to associate a
polytope to such determinants.
Definition 3.8. Let A = (aij ) ∈ Mn (D). The canonical representative of the
Dieudonné determinant detc A ∈ D is defined inductively as follows:
(1) If n = 1 then detc A = a11 .
(2) If the last row of A consists solely of zeros, then detc A = 0.
(3) If ann 6= 0 then we form an (n − 1) × (n − 1) matrix A′ = (a′ij ) by setting
c
c ′
a′ij = aij − ain a−1
nn anj , and declare det A = det A · ann .
(4) If ann = 0 and not every entry in the least row of A is zero then let j be
maximal such that anj 6= 0. Let B be the symmetric matrix interchanging
j and n. We declare detc A = − detc (AB).
The Diedonné determinant det A is defined to be the image of detc A in
D× /[D× , D× ] ⊔ {0}

THE BNS INVARIANTS VIA NEWTON POLYTOPES

13

The procedure in step (3) corresponds to multiplying A on the right by elementary matrices until the last row has a single non-zero element. Thus, computing
the canonical representative of the Dieudonné determinant consists of putting the
matrix A into an upper-diagonal form, and then taking the product of the diagonal
entries. Hence it is clear that over commutative fields the Dieudonné determinant
coincides with the usual determinant.
Note that when writing det A we do not need to specify the skew-field, since we
have adopted the convention that every matrix comes with a specified ring over
which it lies.
Theorem 3.9 (Dieudonné [Die]). For any n, the Dieudonn’e determianant
det : Mn (D) → D× /[D× , D× ] ⊔ {0}
is multiplicative.
3.3. The Newton polytope of a matrix. Recall that K is a skew-field, and
KH is a twisted group ring of H, a finitely generated free-abelian group. We start
by introducing Newton polytopes of elements in KH. Note that KH has no zerodivisors – this follows directly from Theorem 2.6, since H is biorderable.
To study the Newton polytopes let us first introduce minima µφ , which are
algebraic counterparts to the face maps Fφ .
Definition 3.10 (µφ ). For any character φ ∈ H 1 (H; R) we define µφ : KH → KH
by setting

x(h) if φ(h) is minimal in φ(supp x)
µφ (x)(h) =
0
otherwise
(we are using the function notation here).
Note that supp µφ (x) consists of elements in H with the same value under φ.
Since KH has no zero-divisors, it is an easy exercise to see that µφ is multiplicative.
Definition 3.11. Let p ∈ KH be an element. The associated Newton polytope
P (p) is the convex hull of supp p taken in the R-vector space H1 (H; R).
Note that, for every φ ∈ H 1 (H; R) and every p ∈ KH, we have


Fφ P (p) = P µφ (p)

Lemma 3.12. The map P : KH r {0} → P(H) satisfies
P (pq) = P (p) + P (q)
for every p, q ∈ KH r {0}

Proof. The proof is an induction on the rank n of H. If n = 0, then both sides of
the desired equation are trivial. If n = 1, then KH is a twisted Laurent polynomial
ring in one variable, and the result is immediate.
Now suppose that n > 2. Take any character φ ∈ H 1 (H; R) r {0}. We have



Fφ P (pq) = P µφ (pq) = P µφ (p) · µφ (q)

Now we use the inductive
hypothesis (we might have to translate the polytopes

P µφ (p) and P µφ (q) first, so that they both lie in the same hyperspace of H).
We obtain






P µφ (p)·µφ (q) = P µφ (p) +P µφ (q) = Fφ P (p) +Fφ P (q) = Fφ P (p)+P (q)
Hence, for every non-trivial φ, we have


Fφ P (pq) − P (p) − P (q) = 0 = Fφ − P (pq) + P (p) + P (q)

14

DAWID KIELAK

But then
Proposition 3.6 tells us that P (pq) − P (p) − P (q) and − P (pq) − P (p) −

P (q) are single polytopes, which is only true when P (pq) − P (p) − P (q) is a
singleton. Using any of the maps Fφ we immediately see that this singleton is
precisely the origin of H1 (H; R), and therefore P (pq) = P (p) − P (q).

Since H is amenable and KH has no zero-divisors, KH is an Ore domain by
Theorem 2.14; let D denote the Ore localisation of KH – recall that D is a skewfield containing KH, and elements of D× are fractions of the form pq −1 with p, q ∈
KH r {0}.
Thanks to Lemma 3.12, we immediately see that the map P induces a homomorphism P : D× → P(H) defined by
P (pq −1 ) = P (p) − P (q)
Since P(H) is abelian, the homomorphism P gives a well-defined group homomorphism
D× /[D× , D× ] → P(H)
Hence we may talk about P (det A ⊗ D) ∈ P(H) ⊔ {∅} (setting P (0) = ∅).
Definition 3.13 (Newton polytope). Given a square matrix A over KH, we define
its Newton polytope to be P (A) = P (det A ⊗ D) ∈ P(H) ⊔ {∅}.
We are now ready for the first result of the article.
Theorem 3.14 (Single polytope). Let A be a square matrix over KH. Then P (A)
is empty or a single polytope.
Proof. Take φ ∈ H 1 (H; Z) r {0}, and let L = ker φ. Take z ∈ H with φ(z) =
1, the generator of im φ = Z. Let KL denote the subring of KH consisting of
elements of support lying in L. It is clear that KL is a twisted group ring of L
with coefficients K, and hence an Ore domain; let us denote the Ore localisation
of KL by L. The localisation L embeds into the Ore localisation D of KH by
Proposition 2.11(3), and the action by conjugation of z on KL extends to an action
on L by Proposition 2.11(2). Thus, we have an embedding of the twisted group
ring LZ (with Z generated by z) into D.
We will think of LZ as a twisted Laurent polynomial ring with variable z. We
apply Euclid’s algorithm over z to A ⊗ LZ: using only elementary matrices over LZ
we put
Pn A ⊗ LZ into an upper-triangular form. Thus det A ⊗ D can be represented
by i=−n κi z i , a Laurent polynomial in z with coefficients in L. Since L is the
Ore localisation of KL, there exist µi , ν ∈ KL such thatPκi = ν −1 µi for every i (see
Remark 2.10). Hence det A ⊗ D is represented by ν −1 ni=−n µi z i . If P (A) 6= ∅, it
immediately follows that
P (A) = P (det A ⊗ D) = P (

n
X

µi z i ) − P (ν)

i=−n

Pn
Crucially, P ( i=−n µi z i ) and P (ν) are single polytopes, and P (ν) is φ-flat. Thus
Corollary 3.7 tells us that P (A) is a single polytope.

3.4. Novikov rings. To see why one should be interested in Newton polytopes of
matrices over group rings, we need to introduce the Novikov rings.
Definition 3.15 (Truncated support). Let R be a ring and G a group. Let x ∈ RG
be a function. Given a character φ ∈ H 1 (G; R) and a constant κ ∈ R we define the
truncated support to be
suppφ,κ x = {g ∈ G | x(g) 6= 0 and φ(g) 6 κ}

THE BNS INVARIANTS VIA NEWTON POLYTOPES

15

Definition 3.16 (Novikov ring). Given a character φ ∈ H 1 (G; R) we define
φ

d = {x : G → R | suppφ,κ x is finite for every κ ∈ R}
RG
φ

d into an abelian group; to endow it with a ring
Pointwise addition turns RG
structure we use the twisted convolution (⋆) – this way the set-theoretic inclusion
d φ turns into an embedding of rings.
RG ⊆ RG
φ
S
d
d =T
For any S ⊆ H 1 (H; R) we define RG
φ∈S RG .

d φ as a left RG module, and a right RG
d φ module, so we will
We will treat RG
φ
d on the right.
tensor RG modules with RG
The Novikov rings play a crucial role in the Bieri–Neumann–Strebel invariants
via the theorem of Sikorav (Theorem 5.5), which we will discuss later. The key
technical feature of the current article is that we will discuss Σ-invariants of matrices.
3.5. Σ-invariants of matrices.

Definition 3.17 (Σ(A)). Let R be a ring, G a group, and RG a twisted group ring.
Let A be a (not necessarily square) matrix over RG. We define Σ(A) ⊆ H 1 (G; R),
φ
d admits a right
the Σ-invariant of A, by declaring φ ∈ Σ(A) if and only if A ⊗ RG
inverse.
φ

d admitting a right inverse means precisely that
Recall our convention: A ⊗ RG
φ
d as well.
the inverse lies over RG

Definition 3.18 (φ-identity). Let φ ∈ H 1 (G; R) be given. An n × m matrix A
φ
d is said to be a φ-identity if and only if for every entry x of A − I we have
over RG
φ(supp x) ⊆ (0, ∞). (Here, I denotes the identity matrix extended by a zero matrix
either on the right or at the bottom, so that I is an n × m matrix.)
φ

d
For the purpose of the above definition, we treat elements in RG
matrices.

as 1 × 1

Lemma 3.19. Suppose that a square matrix A over RG is a φ-identity for some
φ ∈ H 1 (G; R). There exists an open neighbourhood U ⊆ H 1 (G; R) of φ such that
U
d is right and left invertible.
A ⊗ RG

Proof. Since the supports of the entries of A − I are finite, there exist κ > 0 and an
open neighbourhood U of φ in H 1 (G; R) such that for every ψ ∈ U the supports of
the entries of A − I are sent by ψ to (κ, ∞). Thus
Y =

∞
X

(I − A)i

i=0

d U , as the supports of the entries of (I − A)i are mapped
defines a matrix over RG
by every ψ ∈ U to (iκ, ∞). Clearly AY = Y A = I.

Lemma 3.20. Let A be an n × m matrix over RG, and let φ ∈ Σ(A). There
exists an m × n matrix X over RG such that AX is a φ-identity. Moreover, there
U
d admits a right
exists an open neighbourhood U of φ in H 1 (G; R) such that A ⊗ RG
inverse.

16

DAWID KIELAK

Proof. We let supp A denote the union of the supports of the entries of A. Take
C ∈ N such that
C > − min φ(supp A)
φ

φ

d is right-invertible, there exists a matrix B = (bij ) over RG
d such
Since A ⊗ RG
P
that AB = I. Let a pair (i, j) be fixed for a moment. We have bij = λg g (where
the sum is typically infinite). We set
X
λg g ∈ RG
b0ij =
φ(g)<C

φ

+
0
d
and define b+
ij ∈ RG by the equation bij = bij + bij . We now apply the procedure
0
0
to every pair (i, j), and define matrices B = (bij ) and B + = (b+
ij ); observe that we
have B = B 0 + B + .
By the choice of C, the supports of the entries of AB + are mapped by φ to
(0, ∞). But I = AB = AB 0 + AB + , and so AB 0 is a φ-identity. We set X = B 0 .
The second assertion follows from Lemma 3.19, observing that AX is a square
matrix.


Note that, in particular, the above result implies that Σ(A) is an open subset of
H 1 (G; R), as one would expect.
As before, let K be a skew-field, KH a twisted group ring, and D the Ore
φ
d , we will use the
localisation of KH. In order to study the Novikov rings KH
minima µφ : recall that we introduced them (in Definition 3.10) as maps KH → KH.
It is however immediate that the definition extends and defines multiplicative maps
φ
d → KH.
µφ : KH
In what follows, all tensoring takes place over KH.
d
Lemma 3.21. For every φ ∈ H 1 (H; R) there exists a skew-field F such that KH
and D both embed into F in such a way that the embeddings agree on KH.

φ

Sketch proof. The skew-field F will be the Malcev–Neumann skew-field, which we
discussed in Theorem 2.6; a similar embedding was constructed in [FK, Lemma
5.5].
Let 6 be a biordering of H which makes φ : H → R into an order-preserving
homomorphism. Then F , the subset of KH consisting of functions with supports
φ
d is actually a subset
well-ordered with respect to 6, forms a skew-field. Now KH
of F . This also implies that KH is a subset of F ; let us denote the resulting
embedding by ι.
It is easy to see that the embedding ι : KH ֒→ F induces an embedding of the
Ore localisation of KH, that is of D – we simply set ι(pq −1 ) = ι(p)ι(q)−1 . The
injectivity of this extended ι is immediate.

We obtain the following, immediate corollary.
φ

d has no zero-divisors.
Corollary 3.22. The ring KH

Recall that we have implicit structural functions in the definition of KH. The
φ
d ; in fact
same functions (via formula (⋆)) are used to define multiplication in KH
they can be also used to define the multiplications
KH × KH → KH and KH × KH → KH

Definition 3.23. We say that an element x ∈ D is represented by y ∈ KH if and
only if x = pq −1 with p, q ∈ KH and p = yq.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

17

Note that x can be represented by different elements in KH , since KH contains
zero-divisors with respect to the twisted convolution. The situation is however
φ
d .
different if x is represented by an element of KH
d
Lemma 3.24. Let x ∈ D be an element represented by y and y ′ , both lying in KH
1
′
for some φ ∈ H (H; R). Then y = y .

φ

Proof. By definition, we have x = pr−1 = p′ r′−1 with p, r, p′ , r′ ∈ KH and r, r′ 6= 0.
We also have p = yr and p′ = y ′ r′ . Now, using Ore condition, there exist q, s ∈ KH
such that s 6= 0 and sp = qr. Now pr−1 = p′ r′−1 means precisely that sp′ = qr′ as
well. Thus
qr′ = sp′ = sy ′ r′
d φ contains no zero-divisors, we conclude
and so (q − sy ′ )r′ = 0. As r′ 6= 0 and KH
that q = sy ′ . Therefore
syr = sp = qr = sy ′ r
and arguing as above (using s, r 6= 0) we see that y − y ′ = 0.


φ

d if and
Because of the above lemma, we will say that x ∈ D is equal to y ∈ KH
only if it is represented by y.
Lemma 3.25. Let A be a square matrix over KH, such that A ⊗ D is invertible.
C
d admits a right-inverse, then
Let C ⊆ H 1 (H; R) be a non-empty subset. If A ⊗ KH
this inverse is unique.
Proof. Take φ ∈ C. Let F denote the skew-field from Lemma 3.21, which contains
d φ and D as a subring. We will carry all matrix multiplication out in F .
KH
Suppose that we have two right-inverses, X and X ′ . Then
A · (X − X ′ ) = 0

But we have assumed that A ⊗ D is invertible, and so there exists a matrix Y over
F such that Y · A = I. Hence
X − X ′ = Y · A · (X − X ′ ) = Y · 0 = 0
and so X − X ′ = 0.



Corollary 3.26. Let A be a square matrix over KH, such that A ⊗ D is invertible.
′
d C and X ′ be a right inverse of A ⊗ KH
d C for
Let X be a right inverse of A ⊗ KH
some C, C ′ ⊆ H 1 (H; R) with C ∩ C ′ 6= ∅. Then X = X ′ , and X is a right inverse
′
d C∪C .
of A ⊗ KH

Proof. Let φ ∈ C ∩ C ′ . By assumption, there exist right inverses X and X ′ of,
C
C′
d and A ⊗ KH
d . In particular, both X and X ′ can be viewed
respectively, A ⊗ KH
φ
d , and so X = X ′ by Lemma 3.25. Now X is a matrix over
as matrices over KH
C

d ∩ KH
d
KH

C′

d
= KH

C∪C ′

.



Lemma 3.27. Let S ⊆ H 1 (H; R) be any subset, and let C denote its convex hull.
Then
S
C
d = KH
d
KH
S

d be any element. Take φ1 , . . . , φk ∈ S, and t1 , . . . , tk ∈ [0, 1]
Proof. Let x ∈ KH
Pk
Pk
d ψ.
with i=1 ti = 1. Let ψ = i=1 ti φi . It is enough to show that x ∈ KH

18

DAWID KIELAK

Pick κ ∈ R and for each ρ ∈ H 1 (H; R) consider the truncation xρ,κ defined by

x(h) if ρ(h) 6 κ
xρ,κ (h) =
0
otherwise
We need to show that supp xψ,κ is finite for every κ. To this end, take h ∈ supp xψ,κ .
If φi (h) > κ for all i, then the same is true for ψ(h), which is a contradiction. So
there exists i such that φ(h) 6 κ. Thus h lies in the support of xφi ,κ . So supp xψ,κ
is contained in the union of the supports of the elements xφi ,κ . But this union is
finite.

Let us summarise the results obtained so far.
Lemma 3.28. Let A be a square matrix over KH, such that A ⊗ D is invertible.
C
d is rightEvery connected component C of Σ(A) is open and convex, and A ⊗ KH
invertible.
Proof. For every φ ∈ C, by Lemma 3.20, there exists an open neighbourhood Uφ
U
d φ admits a right-inverse Xφ . We may assume Uφ
of φ in Σ(A) such that A ⊗ KH
to be connected. Since C is a connected component, it is immediate that Uφ ⊆ C
for every φ.
Consider the transitive closure ∼ of the relation on U = {Uφ | φ ∈ C} given by
having non-empty intersection. The union of all sets Uφ belonging to an equivalence class of ∼ is open, and two such unions for two distinct equivalence classes
are disjoint. Since C is connected, ∼ admits only one equivalence class. Therefore for any two sets Uφ and Uψ in U there exists a finite sequence of sets Uφ =
U0 , U1 , . . . , Un = Uψ in U such that Ui ∩ Ui+1 6= ∅ for every i. Let Xi denote the
matrix corresponding to Ui . Corollary 3.26 implies that Xi = Xi+1 for every i, and
therefore Xφ = XSψ . Thus, letting X = Xφ for any φ ∈ C, we see that X lies over
U
C
T d Uφ
d φ φ = KH
d .
= KH
φ KH
The connected component C is convex by Lemma 3.27 and open by the discussion
above (or by Lemma 3.20).

Before stating the main theorem of this section let us look at the following lemma,
whose central purpose is to elucidate the statement of the theorem, as well as some
arguments used in its proof.
Lemma 3.29. Let x ∈ KH be any element, let φ ∈ H 1 (H; R) be any character, and
let C the unique dual of a face of P (x) containing φ. The following are equivalent.
C

d .
(1) The element x is invertible in KH
φ
d .
(2) The element x is invertible in KH
(3) The support supp µφ (x) is a singleton.
(4) The dual C is a dual of a vertex of P (x).
d
Proof. (1)⇒(2) This is immediate, since KH
φ

C

φ

d by definition.
6 KH

d such that xz = 1. Using the minima we see that
(2)⇒(3) There exists z ∈ KH
µφ (x)µφ (z) = 1

as well. But the minima lie in KH, and the units in KH have support of size 1 by
Lemma 2.7. Thus µφ (x) is supported on a singleton.


(3)⇒(4) We have Fφ P (x) = P µφ (x) , which is a singleton. Thus, φ lies in a
dual of a vertex of P (x), by the very definition of dual. But this dual is C, and so
C is as claimed.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

19


(4)⇒(1) For every ψ ∈ C, we have Fψ P (x) equal to a singleton, say h ∈ H.
Thus µψ (x) = λh with λ ∈ K r {0}. Therefore µψ (x) is invertible in KH, and
y = µψ (x)−1 x is an element of KH which is a ψ-identity. Hence y, and therefore
d Uψ , where Uψ is some open neighbourhood of ψ, by
also x, is invertible over KH
Lemma 3.19. Since the open sets Uψ cover C, we have C ⊆ Σ(x). Therefore x is
C
d by Lemma 3.28 (applied to the 1 × 1 matrix x).
invertible in KH

We may view x above as a 1×1 matrix; in this case we have P (x) = P (det x⊗D),
d φ is equivalent to φ ∈ Σ(x). The following result
and the invertibility of x over KH
generalises this to square matrices of arbitrary size, and constitutes the second main
result of the article.
Theorem 3.30 (Σ-invariants of matrices). Let A be a square matrix over KH with
A ⊗ D invertible. Let φ ∈ H 1 (H; R) be any character, and let C denote the unique
dual of a face of P (A) containing φ. The following are equivalent.
C

d is right-invertible.
(1) The matrix A ⊗ KH
(2) We have φ ∈ Σ(A).
(3) The dual C is a dual of a vertex of P (A).
C

φ

d 6 KH
d by definition, and φ ∈ Σ(A)
Proof. (1)⇒(2) This is immediate, since KH
φ
d is right-invertible.
means precisely that A ⊗ KH

(2)⇒(3) Set P = P (A) = P (det A ⊗ D). Recall that this is a single polytope by
Theorem 3.14 – it cannot be empty, since A ⊗ D is invertible and so its Dieudonné
determinant is not zero. We start by taking φ ∈ Σ(A), and claiming that C is a
dual of a vertex of P .
By Lemma 3.20, there exists a matrix X over KH such that AX is a φ-identity.
Set AX = (xij ). Consider the last row of AX. Its last entry xnn is invertible over
φ
d by Lemma 3.19, since it is itself a φ-identity. Thus, we may multiple AX
KH
on the right by a number of elementary matrices in such a way that the resulting
matrix A′ has zeroes in the last row, with the exception of the last entry (which
remains unchanged). Observe that we may use elementary matrices whose non-zero
off-diagonal entries are precisely the elements −x−1
nn xni with i ∈ {1, . . . , n − 1}.
Each of these off-diagonal entries is a product of an element of KH and an
φ
d . Thus, we may see each
inverse of such an element, the inverse taken in KH
of these entries as lying over D, since D contains all inverses of elements in KH.
Moreover, the support of such an off-diagonal entry is taken by φ into (0, ∞), since
AX is a φ-identity, and therefore φ(supp xnn ) ⊂ [0, ∞) and φ(supp xin ) ⊂ (0, ∞),
implying that φ(supp x−1
nn xin ) ⊂ (0, ∞) as well. It is easy to see that multiplying a
φ-identity by elementary matrices whose non-zero off-diagonal entries are as above
yields another φ-identity. Therefore A′ is a φ-identity.
Repeating the process for other rows, we conclude the existence of a matrix Y
over D, with det Y = 1, and such that AXY is upper triangular, and the diagonal
entries are φ-identities. Hence det AX ⊗ D = det AXY ⊗ D can be represented by
φ
d which is also a φ-identity.
an element x ∈ KH
We can also view x as an element in D, that is as a fraction of elements p, q ∈ KH
(since D is the Ore localisation of KH). Thus we have x = pq −1 , and so
xq = p
Since p and q have finite support in H, we can carry out the multiplication above
d φ . Since x is a φ-identity, we have µφ (x) = 1, and therefore µφ (q) = µφ (p).
in KH

20

DAWID KIELAK

Thus we have
Fφ (P (p)) = Fφ (P (q))
But we know that P (det AX ⊗ D) is a single polytope by Theorem 3.14, and so
Fφ P (det AX ⊗ D) = Fφ (P (p)) − Fφ (P (q)) is a singleton. We have
P (AX) = P (A) + P (X)

×

since P : D → P(H) is a homomorphism. Both P = P (A) and P (X) are single

polytopes, again using Theorem 3.14. Therefore so are Fφ P (A) and Fφ P(X) ,
and their sum is a singleton. This is only possible
 when both Fφ P (A) and
Fφ P (X) are singletons. In particular, Fφ P (A) is a singleton, which means
precisely that C is a dual of a vertex of P (A) (as φ ∈ C). This proves the claim.
dC
(3)⇒(1) Now let C be a dual of a vertex of P . We aim at showing that A ⊗ KH
admits a right-inverse.
Since C is open, there exists φ ∈ C ∩ H 1 (H; Z). Let us take such a φ. We will
proceed in 3 steps.
Step 1: We claim that there exists a φ-flat polytope Q, such that for every intersecd C0 is right-invertible.
tion C0 of a dual of a vertex of Q with C, the matrix A ⊗ KH
To prove the claim we start by introducing some extra notation (which is exactly
the same as the one used in the proof of Theorem 3.14). Let L = ker φ. The ring
KH embeds into LZ, where L is the Ore localisation of KL, and the embedding is
induced by φ : H → Z. We now use Euclid’s algorithm (treating LZ as a Laurent
polynomial ring in one variable): it gives us a matrix X over LZ (a product of
elementary matrices) such that A ⊗ LZ · X = Y and Y is an upper-triangular
matrix over LZ. Also, det X ⊗ D = 1.
Let us focus first on X. It is a matrix over LZ, and L is the Ore localisation
of KL. Thus, there exists an element p ∈ KL such that for every entry x of X we
have px ∈ KL (see Remark 2.10). This implies that for every ζ ∈ H 1 (H; R), if p is
ζ
ζ
d then every entry of X can be represented by an element in KH
d
invertible in KH
ζ

d ). In fact more is true: if C ′
(in this case, we will simply say that X lies over KH
C′
d , as Lemma 3.29 tells
denotes a dual of some vertex of P (p), then X lies over KH
C′

d .
us that p is invertible in KH
Now let us look
Q at Y = (yij ). Since Y is upper-triangular, we have det Y ⊗ D
represented by i yii . Therefore
X
P (yii )
P = P (det A ⊗ D) = P (det Y ⊗ D) =
i

Take ψ ∈ C. Since φ, ψ ∈ C, we have Fφ (P ) = Fψ (P ), and so
X
X


Fψ P (yii )
Fφ P (yii ) = Fφ (P ) = Fψ (P ) =
i

i

Now Lemma 3.5 implies that
Fψ P (yii ) = Fφ P (yii )
for every i, since the decomposition of a face of P into a sum of faces of the polytopes
P (yii ) is unique.
Recall that yii lies in LZ for every i. Let us now argue as for the matrix X
before: we see that for every i there exists qi ∈ KL such that
qi yii ∈ (KL)Z
For notational convenience, let us fix i for the moment. We then have
qi yii = (r0 + r1 z + · · · + rm z m )z n

THE BNS INVARIANTS VIA NEWTON POLYTOPES

21

for some m ∈ N and n ∈ Z, where z stands for a generator of im φ = Z, the
coefficients ri lie in KL, and r0 6= 0 6= rm . We have
µφ (qi yii ) = r0 z n
Since qi ∈ KL, we also have µφ (qi ) = qi , and so
qi · µφ (yii ) = r0 z n
Applying the polytope map and using that Fψ P (yii ) = Fφ P (yii ) we obtain

P (qi ) + Fψ P (yii ) = P (r0 z n )

It is now clear that replacing P (qi ) by Fψ P (qi ) on the left-hand side results in
obtaining a polytope on the right-hand side which is a subset of P (r0 z n ). But we
also have



Fψ P (qi ) + Fψ P (yii ) = Fψ P ((r0 + r1 z + · · · + rm z m )z n )

We conclude that Fψ P ((r0 + r1 z + · · · + rm z m )z n ) is a subset of P (r0 z n ), which
implies that µψ ((r0 + r1 z + · · · + rm z m )z n ) = µψ (r0 z n ).
Let C ′′ denote the non-empty intersection of C with a dual of some specified
vertex of P (r0 ), and take ρ ∈ C ′′ . Since ρ lies in a dual of a vertex of P (r0 ),
and hence also its translate P (r0 z n ) = P (r0 ) + P (z n ), Lemma 3.29 tells us that
µρ (r0 z n ) is supported on a singleton. Taking ψ = ρ above, we see that

µρ (r0 + r1 z + · · · + rm z m )z n = µρ (r0 z n )

is also supported on a singleton. Therefore, using Lemma 3.29 again, we see that
C ′′′
d
qi yii = (r0 + r1 z + · · · + rm z m )z n is invertible in KH
, where C ′′′ is a dual of
a vertex of P (qi yii ) containing ρ. Now it is clear that C ′′ ⊆ C ′′′ since for every
ξ ∈ C ′′ we have
µξ (qi yii ) = µξ (r0 z n ) = µρ (r0 z n ) = µρ (qi yii )
C ′′

d . We set si = r0 , and again
Hence qi yii , and therefore also yii , isP
invertible in KH
allow i to vary. We set Q = P (p) + P (si ).
Let C0 be the intersection of C with some dual of a vertex of Q. A dual of
a vertex of Q is a subset of duals of vertices of the summands P (p) and P (si ).
d C0 is right-invertible, as
Therefore, by the discussion above, the matrix A ⊗ KH
claimed.
Step 2: We now claim that C r Σ(A) has empty interior and is φ-convex, that is
for any ψ ∈ C r Σ(A) we have sψ + tφ ∈ C r Σ(A) for every (s, t) ∈ (0, ∞) × [0, ∞).
The first property is immediate, as C ∩ Σ(A) contains the intersection of C with
the union of all duals of vertices of Q (by Step 1). The union of all duals of vertices
of any polytope is dense in H 1 (H; R); this in particular applies to Q. Since C is
open, its intersection with a dense subset of H 1 (H; R) is dense in C.
Now let us look at the second property. Since Q is φ-flat, every dual of a face
of Q is φ-convex (this is easy to verify). Also, the intersection of such a dual with
C is still φ-convex, as C is φ-convex (since it is convex and contains φ), and the
intersection of two φ-convex sets is φ-convex (this is immediate). Therefore, it is
enough to show that C ∩ Σ(A), and therefore also C r Σ(A), is the intersection of
C with some union of duals of faces of Q.
Let ψ ∈ C ∩ Σ(A) be any character, and let F denote a face of Q in whose dual
ψ lies. Let C1 denote the connected component of ψ in C ∩ Σ(A). By Lemma 3.20,
there exists an open neighbourhood U of φ in Σ(A). It is easy to see that for every
vertex v of F , the neighbourhood U intersects some dual Cv of v in a non-trivial
manner. Lemma 3.28 implies that C1 contains C ∩Cv for every v, as the sets Cv are
contained in C ∩Σ(A) by Step 1, and they are convex and thus connected; moreover,

22

DAWID KIELAK

Lemma 3.28 says that C1 is convex. Thus, C1 contains the entire intersection of C
with the dual of F containing ψ. This shows that every C1 is the intersection of C
with with some union of duals of faces of Q, as claimed.
Step 3: We now claim that C ⊆ Σ(A), which will finish the proof thanks to
Lemma 3.28, since C is connected, and therefore is a subset of a connected component of Σ(A).
Suppose for a contradiction that C 6⊆ Σ(A): let ρ ∈ C r Σ(A) be a character. Since C is a dual of a vertex, it contains an open neighbourhood U of ρ; we
find integral characters φ1 , . . . , φk ∈ U which form a basis of the R-vector space
H 1 (H; R), and such that ρ cannot be generated (over R) using fewer than k of
these characters. We span a non-degenerate k-simplex in H1 (H; R) with vertices
ρ, φ1 , . . . , φk , which necessarily has non-empty interior. Now we use the fact that
C r Σ(A) is φi -convex for every i (shown in Step 2, as φ was an arbitrary element
of C ∩ H 1 (H; Z)), and conclude that the interior of the simplex lies in C r Σ(A).
But this contradicts the fact that C r Σ(A) has empty interior.

4. Agrarian groups
In this section we introduce agrarian groups, that is groups whose (untwisted)
integral group rings embed in skew-fields (division algebras). Such groups will be
of central importance for us – the embedding into a skew-field allows us to take
determinants of square matrices over the group ring; the determinants in turn lead
to their Newton polytopes which in many cases control the Bieri–Neumann–Strebel
invariants.
To prove the inheritance properties and to list known classes of examples of
agrarian groups we need to first cover the Atiyah conjecture.
4.1. The Atiyah conjecture. Recall our important convention: whenever we talk
about a matrix, we will explicitly state over which ring it lies.
Let A be a matrix over the integral group ring ZG of a group G. Naturally, A
represents a linear map between two finitely generated free right ZG modules, say
A : ZGn → ZGm
Tensoring with L2 (G) we obtain
A ⊗ L2 (G) : L2 (G)n → L2 (G)m
Let N (G) denote the von Neumann algebra of G, that is the algebra of bounded
G-equivariant operators on L2 (G). The kernel of A ⊗ L2 (G) is a Hilbert N (G)module, which essentially means that it is a topological space with a continuous
module structure over N (G).
The von Neumann algebra has two properties of interest here. Firstly, the nonzero-divisors in N (G) satisfy the Ore condition, and so N (G) embeds into its Ore
localisation. Secondly, Hilbert N (G)-modules have a von Neumann dimension,
which is an element of [0, ∞] – for details see the book of Lück [Lüc, Section 1.1]).
Definition 4.1. Let G be a torsion free group. We say that G satisfies the Atiyah
conjecture if and only if for every matrix A over ZG, the von Neumann dimension
of ker A ⊗ L2 (G) is an integer.
The Atiyah conjecture can also be formulated for groups with torsion – for such
a formulation, as well as a discussion of counterexamples, see [Lüc, Chapter 10].
There are however no torsion-free groups known which do not satisfy the Atiyah
conjecture.
Also, one can define the Atiyah conjecture over group rings of G with other
coefficients. This is however not relevant to our discussion here.

THE BNS INVARIANTS VIA NEWTON POLYTOPES

23

Theorem 4.2 (Linnell [Lin], Schick [Sch1]; Linnell–Okun–Schick [LOS]; Schreve
[Sch2]). Let C1 be the smallest class of groups containing all free groups which is
closed under directed unions and extension by elementary amenable groups.
Let C2 be the smallest class of groups containing the trivial group, closed under
subgroups, colimits and inverse limits over directed systems, and such that if we have
a quotient q : G → H with G torsion-free, H elementary amenable and q −1 (F ) ∈ C2
for every finite subgroup F of H, then G ∈ C2 .
Let C3 denote the union of the class of {right-angled Artin}-by-{elementary
amenable} groups and the class of {right-angled Coxeter}-by-{elementary amenable}
groups where in the latter case in addition the finite subgroups of the elementary
amenable quotients are 2-groups.
Let C4 denote the class of virtually cocompact special groups.
If G is a torsion-free group lying in C1 ∪ C2 ∪ C3 ∪ C4 , then G satisfies the Atiyah
conjecture.
Note that C2 contains all residually {torsion-free solvable} groups, which is a
very rich class. Also, let us elaborate on the class C4 : it contains random groups
(in the density model) with density less than 1/6 – this follows from the result
of Olliver–Wise [OW], who have shown that such random groups are cocompactly
cubulated, combined with the fact that they are hyperbolic, and cocompactly cubulated hyperbolic groups are virtually cocompact special by the work of Agol [Ago,
Theorem 1.1].
Lemma 4.3 ([Lüc, Lemma 10.4]). The class of torsion-free groups satisfying the
Atiyah conjecture is closed under taking subgroups and directed unions.
In fact one can also give statements for groups with torsion, but they will not
be relevant in our context.
We did not give a proper definition of the von Neumann dimension appearing
in the Atiyah conjecture, since we will be using a reformulation due to Linnell. To
state it we need one more definition.
Definition 4.4 (Division closure). Let R be a subring of a ring S. The division
closure of R in S is the smallest subring of S containing R such that if an element
of this subring is invertible in S, then it is also invertible in the subring.
Theorem 4.5 (Linnell [Lin]). Let G be a torsion-free group, and let D(G) denote
the division closure of ZG in the Ore localisation of the von Neumann algebra N (G).
The group G satisfies the Atiyah conjecture if and only if D(G) is a skew-field.
Note that in Linnell’s paper, D(G) denotes the division closure of CG, rather
than ZG. The proofs however work verbatim for ZG.
Proposition 4.6. Let G be a torsion-free group satisfying the Atiyah conjecture.
Every automorphism of G extends to an automorphism of D(G). Also, if K is
a subgroup of G then the natural embedding ZK ֒→ ZG extends to an embedding
D(K) → D(G).
Proof. Let x be an automorphism of G. It is easy to see that x extends to an
automorphism of L2 (G), hence of N (G), and therefore of the Ore localisation of
N (G).
The second statement follows immediately from observing that the von Neumann
algebra of a subgroup embeds into the von Neumann algebra of a supergroup in
such a way that non-zero-divisors stay non-zero-divisors.

Since we are talking about the Atiyah conjecture, let us introduce the L2 homology.

24

DAWID KIELAK

Definition 4.7. The nth L2 -homology group of a group G is defined to be Hn (G; L2 (G)),
with the caveat that one computes the homology topologically, that is one divides
the cycles by the closure of the boundaries.
A group G is L2 -acyclic if and only if its L2 -homology vanishes for every n. In
this context one may in fact take the usual notion of homology, without taking
closures.
Proposition 4.8 (Lück [Lüc, Lemma 10.28(3)]). Let G be a torsion-free group
satisfying the Atiyah conjecture. Then
Hn (G; L2 (G)) = Hn (G; D(G))
for every n.
Again, Lück deals with the division closure of CG, but the proof carries over to
the case of D(G) being the division closure of ZG.
4.2. Agrarian and equivariantly agrarian groups. We now introduce the class
of groups of main interest to us.
Definition 4.9. A group G is agrarian if and only if the integral group ring ZG
embeds into a skew-field. If the embedding can be made equivariant with respect
to Aut(G) (which also acts on ZG), then G is said to be equivariantly agrarian.
If ZG embeds in a skew-field D, then we treat D as a left ZG module, and a
right D module. Thus we will tensor ZG modules on the right with D, and obtain
right D modules.
Let us list some properties of agrarian groups.
Proposition 4.10.
(1) Every equivariantly agrarian group is agrarian.
(2) Subgroups of agrarian groups are agrarian.
(3) Countable directed unions of agrarian groups are agrarian.
(4) Agrarian groups are torsion free and satisfy the zero-divisor conjecture of
Kaplansky, that is their integral group rings have no zero-divisors.
(5) Torsion-free groups satisfying the Atiyah conjecture are equivariantly agrarian.
(6) Amenable groups whose integral group rings have no zero-divisors are equivariantly agrarian.
(7) Biorderable groups are agrarian.
(8) {Equivariantly agrarian}-by-H groups are agrarian, provided that H is biorderable or H is amenable and its twisted group rings with skew-field coefficients
have no zero-divisors.
(9) Countable fully-residually agrarian groups are agrarian.
Proof.
(1) Obvious.
(2) Obvious.
(3) Let (Gi ) be a sequence of agrarian groups, and suppose that ji : ZGi → Di is
an embedding into a skew-field for every i. Let D denote the ultraproduct
of the skew-fields Di ; it is a standard
exercise to show that D is itself a
S
skew-field. SNow we define j : Z( Gi ) → D by putting j(g) = (ji (g)) for
every g ∈ Gi (where
S we put ji (g) = 0 if g 6∈ Gi ), and then extending
linearly. Take x ∈ Z( Gi ): it is clear that j(x) = 0 implies that ji (x) = 0
for infinitely many values i, and hence that x = 0. Thus j is injective.
(4) Skew-fields have no zero-divisors, and groups with torsion have zero-divisors
in their integral group rings.
(5) Follows from Theorem 4.5.
(6) Follows from Theorem 2.14 and Proposition 2.11(2).

THE BNS INVARIANTS VIA NEWTON POLYTOPES

25

(7) Follows from Theorem 2.6.
(8) Let K → G → H be an extension, with K equivariantly agrarian. The
very definition tells us that the untwisted group ring ZK can be embedded
into a skew-field K in an Aut(K)-equivariant fashion. In particular, the
embedding is equivariant with respect to the conjugation action of G on K,
and so we have
ZG = (ZK)H ֒→ KH
Now the statement follows from Theorem 2.6 when H is biorderable and
from Theorem 2.14 when H is amenable and KH has no zero-divisors.
(9) This is similar to Item 3: if G is fully-residually agrarian, then we obtain
a sequence qi of epimorphisms qi : G → Hi such that every Hi is agrarian,
and for every x ∈ ZG there exists j0 such that qj (x) 6= 0 for every j > j0 ,
where qj now denotes the induced map on group rings. We embed ZHi into
a skew-field Di , and then take D to be the ultraproduct of the skew-fields.
The maps qj define an embedding ZG ֒→ D.

We see that agrarian groups form a large class – they contain torsion-free groups
satisfying the Atiyah conjecture (including all elementary amenable groups and
free groups), but they also contain extensions of such groups by free groups (as free
groups are biorderable). Thus, for example, free-by-free groups are agrarian, but
are not all known to satisfy the Atiyah conjecture; a related class of hyperbolic extensions of free groups has been studied by Dowdall–Taylor [DT]. Another example
of agrarian groups which are not known to satisfy the Atiyah conjecture is the class
of 4-manifolds which up to homotopy are surface-by-surface bundles (studied e.g.
by Hillman [Hil]) – again, these are agrarian since they are Atiyah-by-biorderable.
There is no torsion-free example of a non-agrarian group known.
Lemma 4.11. Let G be an agrarian group, and suppose that K → G → H is an
exact sequence of groups. Then ZK embeds into a skew-field K in a G-equivariant
way, and so we have an embedding
ZG ∼
= (ZK)H ֒→ KH
Proof. Let K be the skew-field coming from the definition of agrarianism of G.
Since G embeds therein, we have an action of G on K by conjugation. It is clear
that this action preserves ZK.

5. Applications
Throughout the applications, we will look at agrarian groups G with finitely
generated abelianisations. We will always set H = Gfab , the free part of the
abelianisation of G, K = ker(G → Gfab ), and identify the untwisted group ring
ZG with the twisted group ring (ZK)H, as in Example 2.4.
Since G is agrarian, we will embed ZK into a skew-field K, as explained in
Lemma 4.11, and hence obtain an embedding
ZG ∼
= (ZK) ֒→ KH
We will use D to denote the Ore localisation of KH.
Tensoring in this section is always taken over ZG.
5.1. Bieri–Neumann–Strebel invariants. In [BNS] Bieri–Neumann–Strebel introduced the geometric invariant Σ(G) (also known as the Σ-invariant or the BNSinvariant), where G is any finitely-generated group. Later, Bieri–Renz [BR] extended the definition to a series of invariants Σm (G; Z), with Σ(G) = Σ1 (G; Z). We
recall the definition below. (Note that for us Z is a right module, and so we avoid
the minus sign which usually appears in the equality above.)

26

DAWID KIELAK

Definition 5.1. For every m ∈ N ⊔ {∞}, we define
Σm (G; Z) ⊆ H 1 (G; R) r {0}
by declaring φ ∈ Σm (G; Z) if and only if the monoid {g ∈ G | φ(g) > 0} is of type
FPm . We will write Σ1 (G) for Σ1 (G; Z).
Recall that a monoid M is of type FPm if and only if the trivial ZM -module Z
admits a resolution by projective ZM -modules such that the first m terms of the
resolution are finitely generated (here ZM is the untwisted monoid ring). When
m = ∞ we require all the terms of the resolution to be finitely generated.
Note that T
M is of type FP∞ if and only if it is of type FPm for all m ∈ N, and so
Σ∞ (G; Z) = m∈N Σm (G; Z).
The definition of type FPm applies of course also to groups. In the context of
groups let us introduce one more notion.
Definition 5.2. A group G is of type F if and only if it admits a finite CW-model
of its classifying space, and of type F∞ if and only if it admits a CW-model of its
classifying space with finite skeleton in every dimension.
Remark 5.3. Note that Bieri–Renz talk about Σm (G; Z) as a subset of the character
sphere obtained by identifying classes in H 1 (G; R)r{0} under positive homotheties.
We prefer to define Σm (G; Z) as a subset of the whole cohomology – in our setup
Σm (G; Z) is of course stable under multiplication by positive scalars.
Theorem 5.4 (Bieri–Renz [BR, Theorem A]). Let G be a finitely generated group,
and let m ∈ N. The sets Σm (G; Z) are open subsets of H 1 (G; R).
5.2. Sikorav’s theorem. Fundamental to the way we will treat BNS-invariants is
the following theorem of Sikorav [Sik], which gives an interpretation of Σm (G; Z) in
φ
d . (We give
terms of group homology of G with coefficients in the Novikov ring ZG
a reference to an easily available paper of Bieri, rather than to the original thesis
of Sikorav.)
Theorem 5.5 (Sikorav [Bie, Theorem 2]). Let G be a group of type FPm . For every
φ
d ) = 0 for every
φ ∈ H 1 (G; R) r {0} we have φ ∈ Σm (G; Z) if and only if Hi (G; ZG
i 6 m.
φ

d ) using the cellular chain complex of
In practice, we will compute Hi (G; ZG
the universal covering of some CW-model for the classifying space of G; the chain
complex is naturally a free resolution of Z by ZG-modules. The group homology
φ
with coefficients in the Novikov completion d
ZG is obtained by tensoring the chain
φ

d and then computing the homology.
complex with ZG
When the group G is of type F∞ , then we can arrange for the above complex
to be a complex of free finitely-generated ZG-modules; hence the boundary maps
are naturally matrices over ZG, and the homology computation is related to their
invertibility upon extension of scalars. This is the reason for our interest in Σinvariants of matrices.
5.3. Σ-invariants of matrices revisited. Let us record the following, which highlights some similarities between BNS-invariants for groups and for matrices over a
group ring.
Lemma 5.6. Let G be a group, and let A be a square matrix over ZG. Then Σ(A)
is stable under positive homotheties, and is an open subset of H 1 (G; R).

THE BNS INVARIANTS VIA NEWTON POLYTOPES

27

Proof. The first assertion is obvious, since multiplying φ by a positive scalar does
φ
not change the ring d
ZG .
The latter assertion follows immediately from Lemma 3.20.

To discuss the Σ-invariants of matrices further, we introduce the notion of a
marked polytope.
Definition 5.7. A function m : H 1 (G; R) r {0} → {0, 1} (where {0, 1} is endowed
with the discrete topology) is a marking function if and only if m−1 (0) is closed.
A character mapped to 1 will be called marked.
Let P be a polytope. We say that m is a marking of P if and only if m is
constant on duals of faces of P . A dual mapped to 1 will be called marked. A
polytope endowed with a marking will be called a marked polytope. When m−1 (1)
consists solely of duals of vertices, we say that P is a polytope with marked vertices.
Note that the definition above leads to statements which may seem a little awkward at first, since we will often assume the existence of a marked polytope, and
then immediately treat being marked as a property of a character.
Remark 5.8. When every face of P admits a single dual, the definition of a marked
polytope above gives the more natural notion of a marking, where it is faces of the
polytope which are marked. The fact that m−1 (0) is closed guarantees that faces
of marked faces are also marked.
Theorem 5.9. Let A be a square matrix over ZG with A ⊗ D invertible. The polytope P (A) admits a marking of its vertices, such that for every φ ∈ H 1 (H; R) r {0}
we have φ ∈ Σ(A) if and only if φ is marked.
Proof. This is a fairly straightforward corollary of Theorem 3.30: we know that
φ
d admits a right-inverse if and only if φ lies in a dual of a vertex of P (A).
A ⊗ KH
d φ is certainly a necessary condition for being
Now being right-invertible over KH

dφ , as ZG
dφ is a subring of KH
d φ . Thus characters lying in
right-invertible over ZG
duals of faces of dimension at least 1 do not lie in Σ(A).
We now define the marking on P (A). A dual C of a vertex is marked if and only
if C intersects Σ(A) non-trivially. We need only show that if C intersects Σ(A)
non-trivially, then in fact C ⊆ Σ(A). Take φ, ψ ∈ C, with φ ∈ Σ(A). Theorem 3.30
C
d is right-invertible; let X denote its inverse. We also know
tells us that A ⊗ KH
φ

φ

d . Right-inverses over KH
d are unique by
that A admits a right inverse Y over ZG
C
φ
d ∩ ZG
d =
Lemma 3.25, and so X = Y , which in turn implies that X lies over KH
C

d
ZG , and we are done.



Let us now discuss ways of combining various marked polytopes into a single
one.
Definition 5.10. An atlas of markings consists of a finite open cover U1 , . . . , Un of
H 1 (G; R) r {0}, and a finite collection (P1 , m1 ), . . . , (Pn , mn ) of marked polytopes
in H1 (G; R), such that for every pair (i, j) we have mi |Ui ∩Uj = mj |Ui ∩Uj .
It is clear that an atlas of markings gives a well defined marking function
m : H 1 (G; R)r{0} → {0, 1}. Hence, we will often specify m instead of the markings
mi . It is clear that m−1 (1) is open, since it is open inside every chart Ui . Thus
m−1 (0) is closed.
Lemma 5.11. Let U1 , . . . , Un and P1 , . . . , Pn bePan atlas of markings. The induced
n
marking function m defines a marking of P = i=1 Pi .

28

DAWID KIELAK

Proof. Recall
P Lemma 3.5: for any face Q of P , we have unique faces Qi of Pi such
that Q = ni=1 Qi .
Let C denote a dual of a face Q of P . We argue that either all characters
in C are marked, or none is. To this end, let φ ∈ C. There exists i such that
φ ∈ Ui , and φ is marked if and only if the dual of Fφ (Pi ) containing φ is marked.
Now, for every ψ in the connected component of C ∩ Ui containing φ we have
Fφ (Pi ) = Qi = FψS
(Pi ), and so ψ is marked if and only if φ is. This shows that the
marking map m :
Ui → {0, 1} is continuous on C, which is connected. Since the
target of the marking map is discrete, the map must be constant on C.

Lemma 5.12. Let S = {s1 , . . . , sn } be a finite generating set of G, let P be a
polytope in H1 (G; R), and let ki ∈ N∪{0} be given for every i ∈ {1, . . . , n}. Suppose
that for every i there exists a marking of vertices of Pi = P + ki · P (1 − si ) such
that the collections U1 , . . . , Un and P1 , . . . , Pn form an atlas of markings, where
Ui = {φ ∈ H 1 (G; R) | φ(si ) 6= 0}
Then the induced marking function m defines a marking of vertices of P .
Proof. The proof is identical to the previous one, with only one exception: the
proof that Fφ (Pi ) = Fψ (Pi ) holds for every ψ in the connected component of C ∩ Ui
containing φ, where C is a dual, is different. To show the desired statement, take
such C, Ui , φ and ψ. Since φ and ψ lie in a connected component of Ui , the real
numbers φ(si ) and ψ(si ) have the same sign. Therefore


Fφ ki P (1 − si ) = Fψ ki P (1 − si )
Also, φ, ψ ∈ C implies that Fφ (P ) = Fψ (P ). Thus Fφ (Pi ) = Fψ (Pi ) as required.


5.4. Polytope class. In [Fun] Funke introduced the following notion.
Definition 5.13. Let G be a torsion-free group satisfying the Atiyah conjecture
and such that the abelianisation of G is finitely generated. We say that G is of
polytope class if and only if for every square matrix A over ZG the following holds:
if A ⊗ D(G) is invertible then the Newton polytope P (det A ⊗ D(G)) is a single
polytope.
Recall that D(G) is the skew-field of Linnell – see Theorem 4.5.
Funke proved in [Fun, Theorem 4.1] that torsion-free amenable groups G satisfying the Atiyah conjecture and with finitely generated abelianisation are of polytope
class. We show that one does not need to assume amenability.
Theorem 5.14. Every torsion free group G with finitely generated abelianisation,
and with G satisfying the Atiyah conjecture, is of polytope class.
Proof. This is an immediate corollary of Theorem 3.14, taking D = D(G).



This fact has interesting consequences for the Whitehead group.
Definition 5.15. Let G be a group. Let GL(ZG) denote the directed union over n
of the groups of invertible n × n matrices over ZG, where the embeddings are given
by introducing the identity matrix in the bottom-right corner. The group K1 (G)
is the abelianisation of GL(ZG). The Whitehead group Wh(G) is the quotient of
K1 (G) by the subgroup generated by 1 × 1 matrices of the form (±g), with g ∈ G.
It is not hard to see that the map A 7→ det A ⊗ D is well-defined on GL(ZG).
Hence, one can talk about the Newton polytopes of elements of GL(ZG). It is even
easier to see that if we quotient the polytope group P(Gfab ) by the translations (and

THE BNS INVARIANTS VIA NEWTON POLYTOPES

29

obtain PT (Gfab )), then in fact we obtain a well defined map Wh(G) → PT (Gfab ).
Thus, we may talk about the Newton polytopes of elements of the Whitehead group.
It is an open question whether there exists a torsion-free group G with non-trivial
Wh(G). We show that in our setting at least the Newton polytopes vanish.
Corollary 5.16. Let G be an agrarian group with finitely generated abelianisation.
The Newton polytope of every element of Wh(G) is a singleton.
Proof. Since G is agrarian, fix a skew-field D into which ZG embeds.
Let A ∈ Wh(G) be given. We can think of A as a square matrix over ZG, which
is invertible. Recall that P (A) denotes P (det A ⊗ D).
We have
P (A) + P (A−1 ) = P (AA−1 ) = P (det I ⊗ D) = 0
and so P (A) = −P (A−1 ). But both are single polytopes, and so P (A) = P (A−1 ) =
0 in PT (G).

Note that a version of the above corollary for torsion-free groups satisfying the
Atiyah conjecture is proven in [Fun, Lemma 3.4] under the extra assumption that
G is of P ≤ 0-class, which is implied by being of polytope class.
The corollary has a direct application to the study of the L2 -torsion polytopes.
These polytopes are defined for any free finite G-CW-complex X which is L2 -acyclic,
provided that G satisfies the Atiyah conjecture. The construction is due to Friedl–
Lück [FL], and is a little complicated – roughly speaking, to X one associates an
L2 -version of the Whitehead torsion, called the universal L2 -torsion. This torsion is
naturally an element of D× /[D× , D× ], and hence one can take its Newton polytope
up to translation. This element of PT (Gfab ) is precisely the L2 -torsion polytope
PL2 (X, G).
The universal L2 -torsion is a simple homotopy invariant, but not a homotopy
invariant – when one passes to a complex G-homotopy equivalent to X, the universal
L2 -torsion gets multiplied by an element of Wh(G). When passing to the L2 -torsion
polytope, the situation was a priori the same. However, the corollary above tells us
that the Newton polytopes of elements in Wh(G) are trivial, and so the L2 -torsion
polytope is actually a homotopy invariant. Let us summarise this discussion.
Theorem 5.17. Let G be an L2 -acyclic group of type F, and suppose that G satisfies
the Atiyah conjecture. For any two finite models X and Y for the classifying space
of G, we have
PL2 (X, G) = PL2 (Y, G)
Therefore, we may purge X from the notation and talk about PL2 (G), the L2 -torsion
polytope of G.
5.5. Vanishing of the L2 -torsion polytope in the presence of amenability.
We now use [Fun, Lemma 5.2] and the proof of [Fun, Theorem 5.3] of Funke to
prove the following.
Theorem 5.18. Let G be a group of type F satisfying the Atiyah conjecture. Suppose that G admits a finite chain of subnormal subgroups with the last term being a
non-trivial amenable group N . Suppose further that N is not abelian or that G has
trivial centre. Then G is L2 -acyclic and the L2 -torsion polytope of G is a singleton.
Proof. Note that N is amenable and infinite, since G is of type F, and thus torsion
free. Therefore the L2 -acyclicity of G follows from [Lüc, Theorem 1.44] of Lück.
Since G is of type F, it is finitely generated, and hence so is its abelianisation.
Thus G is of polytope class by Theorem 5.14
Now, to fix notation, we denote the elements of the subnormal chain by Ni , with
N0 = G and Nk = N . Let M = N ∩ K (recall that K = ker G → Gfab ).

30

DAWID KIELAK

Case 1: Suppose that M is not trivial, and hence M is an infinite amenable group.
Let S be the set of non-zero elements of ZM . Note that the untwisted group ring
ZG has no zero-divisors, since it embeds into the skew-field D(G). Thus ZM has
no zero-divisors either. Therefore the subset S of ZM satisfies the Ore condition
inside of ZM . Note that M is a normal subgroup of Nk−1 , and that S is Nk−1 invariant. Now Lemma 2.16 tells us that S satisfies the left Ore condition in ZNk−1 .
Set Sk−1 = S. Replacing Sk−1 by Sk−2 , the multiplicative closure of the Nk−2 conjugates of Sk−1 , and using Lemma 2.16 again, we see that Sk−2 satisfies the left
Ore condition in ZNk−2 . We repeat the procedure until we arrive at the closure
S0 which satisfies the left Ore condition in ZG. The same argument shows that S0
satisfies the right Ore condition, and hence the Ore conditions on both sides.
Note that the Newton polytope of every element in S0 is a singleton, since this
is true for elements in S by construction (as these lie in the group ring of the kernel
K) and is clearly preserved under conjugation and multiplication.
We are now going to use [Fun, Lemma 5.2] of Funke. Considering only a classifying space for the group in question, and using Theorem 5.14 to remove the
assumption of P ≥ 0-class (which is weaker than being of polytope class), we may
state it as follows:
Let G be a group of type F satisfying the Atiyah conjecture. Let
T ⊆ ZG be a multiplicative subset satisfying the left and right Ore
conditions, and such that P (t) is a singleton for every t ∈ T . If
Hi (G; T 1− ZG) = 0 for every i, then PL2 (G) is a singleton.
The assumptions of [Fun, Lemma 5.2] are satisfied (taking T = S0 ) – we need
only that
Hn (G; S0−1 ZG) = 0
for all n. To prove it, note that extending scalars to a localisation is exact, and
so the only potential problem occurs for H0 (G; S0−1 ZG), since Hi (G; ZG) = 0 for
all i > 0, and H0 (G; ZG) = Z. But S0 contains an element of the form 1 − g with
g ∈ M r {1} ⊆ G r {1} by assumption, and so H0 (G; S0−1 ZG) = 0 (as we could
compute it from a presentation complex coming from a presentation of G containing
g as a generator, and then the first boundary map is clearly onto over S0−1 ZG).
Case 2: Suppose that M is trivial. Then N embeds into H = Gfab , and so is
abelian. Thus G has trivial centre by hypothesis.
Since N is normal in Nk−1 , and embeds into H = Gfab , the action of Nk−1 on
N must be trivial, and so N lies in the centre of Nk−1 . We replace N by the centre
of Nk−1 : if this new N does not embed into H, we are back in case 1. Otherwise,
note that N is in fact characteristic in Nk−1 , and so normal in Nk−2 . We repeat
the argument, and replace N by the centre of Nk−2 . We continue doing so until
either the new N intersects K non-trivially, or until we have produced a non-trivial
centre of G, which is impossible.

The theorem above allows us to confirm a conjecture of Friedl–Lück–Tilmann.
Conjecture 5.19 (Friedl–Lück–Tillmann [FLT, Conjecture 6.4]). Let G be an L2 acyclic group of type F satisfying the Atiyah conjecture and with Wh(G) = 0. If G
is amenable but not virtually Z, then PL2 (G) is a singleton.
We prove the following, stronger version.
Corollary 5.20. Let G be an L2 -acyclic group of type F satisfying the Atiyah
conjecture. If G is amenable but not isomorphic to Z, then PL2 (G) is a singleton.
Proof. If G is non-abelian, then the result follows directly from Theorem 5.18 by
taking N = G. If G is abelian, then type F tells us that G is a finitely generated

THE BNS INVARIANTS VIA NEWTON POLYTOPES

31

free abelian group, which is not Z. Therefore the universal L2 -torsion of G vanishes

by [FL, Theorem 2.5(5)]. Hence PL2 (G) = 0.
Note also that G = Z fails the assertion of the conjecture: we have
PL2 (Z) = −P (1 − z)
where z is a generator of Z.
Remark 5.21. Theorem 5.18 corresponds nicely with [Weg, Theorem 4.10] of Wegner: we are assuming that G satisfies the Atiyah conjecture, whereas Wegner assumed that G has semi-integral determinant, which is a related condition. Also,
in his theorem one requires G to posses an elementary amenable subnormal subgroup, whereas for our purposes amenable subnormal subgroups suffice. Wegner
proved the vanishing of the L2 -torsion, whereas we prove the vanishing of the L2 torsion polytope. Both invariants are determined by the universal L2 -torsion of
Friedl–Lück [FL].
5.6. Agrarian groups of deficiency 1. Throughout this section, G is going to
be an agrarian group of deficiency d > 1, that is G will admit a presentation with
the number of generators exceeding the number of relators by d (formally speaking,
it also means that there is no presentation with an even greater discrepancy). We
will fix a presentation of G with a finite generating set S = {s1 , . . . , sk } realising
the deficiency. We will also build a CW-classifying space for G whose 2-skeleton
coincides with the presentation complex, and let (C∗ , ∂∗ ) denote the cellular chain
complex of the universal covering of the classifying space, with ∂n : Cn → Cn−1 .
We will choose representatives for the G-orbits of cells, and hence identify C∗ with
a chain complex of free ZG-modules. Note that, by our convention, C∗ is a chain
complex of right modules, and so the boundary maps are matrices acting on the
left.
We will look at two boundary maps more closely. The first one, ∂1 , is the row
vector (1 − si )i . The second one, ∂2 , is a matrix with d more rows than columns,
and we will denote it by A. We let Ai denote A with the ith row removed.
Theorem 5.22 (Bieri–Neumann–Strebel [BNS, Theorem 7.2]). For any finitely
generated group G, if d > 1 then Σ1 (G) = ∅.
From now on we will assume that d = 1. Crucially for us, in this case the
matrices Ai are square, since C2 has rank precisely one less than C1 .
Friedl conjectured that a group G of deficiency one which admits an aspherical
presentation complex realising the deficiency has Σ1 (G) determined by a polytope.
We prove this conjecture under the assumption of G being agrarian, but we will
not need the assumption of asphericity of the presentation complex.
Theorem 5.23. Let G be an agrarian group of deficiency 1. Then there exists a
marked integral polytope P in H1 (G; R), such that φ ∈ Σ1 (G) if and only if φ is
marked.
Proof. In view of Sikorav’s Theorem (Theorem 5.5), for every character
φ ∈ H 1 (G; R) r {0}
we have φ ∈ Σ1 (G) if and only if
φ
φ
H1 (C∗ ⊗ d
ZG ) = H0 (C∗ ⊗ d
ZG ) = 0

φ

d without changing
Let si ∈ S. If φ(si ) 6= 0, we can modify the complex C∗ ⊗ ZG
its homology as follows: let v denote the ith basis vector of C1 . Since φ(si ) 6= 0, the
φ
dφ , and so the boundary map ∂1 ⊗ d
element 1 − si is invertible in ZG
ZG restricts to

32

DAWID KIELAK
φ

φ

d . This immediately implies
an isomorphism from the d
ZG -span of v onto C0 ⊗ ZG
φ
φ
d and
that H0 (C∗ ⊗ d
ZG ) = 0. Also, we may split off the span of v from C1 ⊗ ZG
write
φ
C1 ⊗ d
ZG = hvi ⊕ C1′
where C1′ is spanned by the remaining basis vectors of C1 . We will use this procedure
repeatedly, whenever we are working over a ring R in which 1 − si is invertible. As
a shorthand, we will say that we ‘replace C1 ⊗ R by the R-module C1′ ’.
φ
d ) = 0 if and only if the composition
It is immediate that H1 (C∗ ⊗ ZG
φ

φ

d → C1 ⊗ d
C2 ⊗ ZG
ZG → C1′

φ

of ∂2 and the projection is onto. But this map is precisely Ai ⊗ d
ZG , and it is onto
if and only if φ ∈ Σ(Ai ). This in turn is equivalent to φ being marked under mi ,
where the marking mi comes from Theorem 5.9 and is a marking of P (Ai ).
Now the collections U1 , . . . , Uk and P (A1 ), . . . , P (Ak ) form an atlas of markings,
where
Ui = {ψ ∈ H 1 (G; R) r {0} | ψ(si ) 6= 0}
An application of Lemma 5.11 allows us to combine the marked polytopes P (A1 ), . . . , P (Ak )
into a single marked polytope P which satisfies the assertion of our theorem.

Remark 5.24. The bulk of the above proof will be used repeatedly in the sequel.
Under stronger hypotheses we can in fact prove a somewhat stronger theorem:
we will assume now that G satisfies the Atiyah conjecture, and use the L2 -torsion
polytope PL2 (G).
Theorem 5.25. Let G be a torsion-free group of deficiency 1, which satisfies
the Atiyah conjecture and has trivial first L2 -Betti number. We have Σ1 (G) =
Σ∞ (G; Z), and there exists an integral polytope P in H1 (G; R) with marked vertices, such that φ ∈ Σ1 (G) if and only if φ is marked.
Moreover, the Cayley 2-complex of any presentation of G realising the deficiency
is contractible.
Proof. We continue with the notation of the proof of Theorem 5.23. We first claim
that the boundary map ∂3 : C3 → C2 is trivial. Since G satisfies the Atiyah conjecture and is torsion free, we have the skew-field D(G) of Linnell (introduced in
Theorem 4.5) at our disposal. Since we also know that the first L2 -Betti number
of G vanishes, we have H1 (G; D(G)) = 0 by Proposition 4.8.
Take v to be the first basis vector of C1 . Since (1 − s1 ) is invertible over D(G),
we argue precisely as in the proof of Theorem 5.23, and replace C1 ⊗ D(G) by C1′ ,
a free module over D(G) of rank one less than C1 . The vanishing of H1 (G; D(G))
tells us that the matrix
A1 ⊗ D(G) : C2 ⊗ D(G) → C1′
is onto. But the ranks of C2 ⊗ D(G) and C1′ are equal, and so A1 ⊗ D(G) is also
injective. Thus ∂2 is injective, and so ∂3 = 0. This implies that the Cayley 2complex of G is contractible, and so we may take X to be the presentation complex
(which is of dimension 2).
Now we argue precisely as in the proof of Theorem 5.23: letting A = ∂2 , and
setting Ai as before to be a square matrix over ZG, we conclude that for every
φ ∈ H 1 (G; R) r {0} we have φ ∈ Σ1 (G) if and only if φ lies in the dual of a marked
vertex of P (Ai ) for every i with φ(si ) 6= 0. Now, instead of combining the polytopes
P (Ai ) into a single polytope by taking the sum, we will use the fact that
P (Ai ) − P (1 − si ) = PL2 (G)

THE BNS INVARIANTS VIA NEWTON POLYTOPES

33

for every i – this follows from the additivity of the universal L2 -torsion that underpins the L2 -torsion polytope, see [FL, Lemma 1.9]. (For the special case of
descending HNN extensions of free groups, a more direct reference is [FK, Theorem
3.2(1)]; the computation given there applies to the current setting as well.)
If the rank of H = Gfab is 0 then there is nothing to prove. Suppose that the
rank of H is equal to 1. We take P = P (1 − si ), where si is not trivial in H. We
mark the vertices of P in the unique way making the assertion about Σ1 (G) true.
Now suppose that the rank of H is at least 2. We claim that PL2 (G) is a single
polytope. To this end, let φ ∈ H 1 (G; Z) be non-trivial. There exists si ∈ S such
that φ(si ) 6= 0, and so


Fφ PL2 (G) = Fφ P (Ai )


up to translation, since Fφ P (1 − si ) is a singleton. But P (Ai ), and so Fφ P (Ai ) ,
is a single polytope. Now the claim follows from Proposition 3.6.
We conclude the existence of the desired marked polytope by applying Lemma 5.12
with P = PL2 (G) and Pi = P (Ai ).
We are left with the claim about Σi (G; Z) for i > 0. Let φ ∈ H 1 (G; R) r {0},
and take i such that φ(si ) 6= 0. We have shown above that φ ∈ Σ1 (G) if and only
φ
d . But Bieri [Bie, Theorem 3] (building on the
if Ai admits a right inverse over ZG
φ

d is von Neumann finite, which means
work of Kochloukova [Koc1]) showed that ZG
φ
d admits a right-inverse if and only if it admits a twothat a square matrix over ZG
φ

sided inverse. Thus φ ∈ Σ1 (G) if and only if Ai ⊗ d
ZG is an isomorphism. Also,
2
1
dφ is injective, by Theorem 5.5.
φ ∈ Σ (G; Z) if and only if φ ∈ Σ (G) and Ai ⊗ ZG
φ
d )=0
Thus we have established Σ1 (G) = Σ2 (G; Z). For i > 2 we have Hi (G; ZG
since X if of dimension 2, and so another application of Theorem 5.5 completes the
proof.


5.7. Descending HNN extensions of free groups. We are now going to focus
on a special kind of groups of deficiency 1, namely descending HNN extensions of
finitely generated free groups.
Definition 5.26. Let Fn = ha1 , . . . , an i denote the free group of rank n, and let
g : Fn → Fn be a monomorphism. The descending HNN extension Fn ∗g of Fn by
g is defined by the presentation
Fn ∗g = ha1 , . . . , an , t | t−1 a1 tg(a1 )−1 , . . . , t−1 an tg(an )−1 i
In particular, {finitely generated free}-by-Z groups are descending HNN extensions of free groups, where g is an automorphism.
Note that changing the sign of the stable letter t in the presentation gives the
definition of an ascending HNN extension.
Let us mention the following conjecture of Bieri:
Conjecture 5.27 (Bieri [Bie, Conjecture]). Let G be a group of deficiency 1 with
Σ1 (G) 6= ∅. Then G is a descending HNN extenson of a finitely generated free
group.
The conjecture is known to hold (by [Bie, Corollary B]) if there exists a character
φ ∈ H 1 (G; R) r {0} such that {φ, −φ} ⊆ Σ1 (G).
Before proceeding to the main statement of this section, let us discuss semi-norms
induced by polytopes, which will be important in the proof.
Definition 5.28 (Semi-norms). Let P be a polytope in H1 (G; R), where G is a
group with finitely generated abelianisation. We define k · kP : H 1 (G; R) → [0, ∞)

34

DAWID KIELAK

by
kφkP = max |φ(x) − φ(y)|
x,y∈P

It is easy to see that k · kP is a semi-norm.
Let P −Q represent an element R of PT (G), where P and Q are single polytopes.
We define
k · kR = k · kP − k · kQ : H 1 (G; R) → R
Again, one readily sees that this function is independent of the choice of P and Q.
When P = PL2 (G) is defined, it is interesting to study k · kP . This has in particular been done for descending HNN extensions of non-trivial free groups in [FK],
where it was shown that the function is indeed a semi-norm (see [FK, Corollary
3.5]); this semi-norm was christened the Thurston norm (and denoted k · kT ), due
to analogies with the 3-manifold case (which we will discuss later).
For every character φ with im φ = Z, the number kφkT is equal to minus the
L2 -Euler characteristic of ker φ. When ker φ is finitely generated (which happen
precisely when {φ, −φ} ⊂ Σ1 (G)), then it is in fact a free group (as shown by
Geoghegan–Mihalik–Sapir–Wise [GMSW, Theorem 2.6 and Remark 2.7]). Thus,
for such a φ, the number kφkT is equal to minus the usual Euler characteristic of
ker φ, that is we have
G∼
= F1+kφkT ⋊ Z
with φ being equal to the projection map to Z.
Theorem 5.29. Let G be a descending HNN extension of a finitely generated nontrivial free group Fn . We have Σ1 (G) = Σ∞ (G; Z), and there exists a marking of
the vertices of the L2 -torsion polytope PL2 (G) such that φ ∈ Σ1 (G) if and only if φ
is marked.
Proof. The group G satisfies the Atiyah conjecture: it fits into a short exact sequence
hhFn ii → G → Z
with a locally-free kernel. Hence, G belongs to the class C1 of Theorem 4.2. It is
clear that G is also torsion-free.
The group G admits an obvious 2-dimensional classifying space – see [FK, Lemma
3.1]. It is L2 -acyclic since it is a mapping torus, and mapping tori are L2 -acyclic
by [Lüc, Theorem 1.39]. Hence the assumptions of Theorem 5.25 are satisfied.
We now argue as in the proof of Theorem 5.25. If the rank of Gfab is at least 2,
then PL2 (G) is the polytope constructed in the proof of Theorem 5.25.
Suppose that the rank of Gfab is 1 (which the smallest possible value). [FK,
Corollary 3.5] tells us that PL2 (G) is a single polytope – it is a difference of two
segments, inducing a semi-norm, and so in particular a non-negative function. Thus
we may argue as in the proof of Theorem 5.25 even when the rank of Gfab is 1. 
Let us note that understanding the BNS invariants of free-by-cyclic groups is
related to a large body of current research, see [DKL1, DKL2, DKL4] by Dowdall–
I. Kapovich–Leininger and [AKHR] by Algom-Kfir–Hironaka–Rafi. In particular,
Dowdall–I. Kapovich–Leininger [DKL3, Theorem 1.2] showed that if G = Fn ⋊g Z
is hyperbolic, φ denotes the induced character, and g is fully irreducible, then every
other integral character in the same component of Σ1 (G) as φ also comes from a freeby-cyclic decomposition of G with a fully-irreducible monodromy. In our language,
this implies that if G is hyperbolic then we can talk about fully-irreducible vertices
of PL2 (G).

THE BNS INVARIANTS VIA NEWTON POLYTOPES

35

5.8. Poincaré duality groups. We are moving towards (aspherical) manifolds in
dimension 3. First, however, we will discuss their homological cousins.
Definition 5.30. A Poincaré duality group in dimension n is a group G such
that there exists a G-module O (the orientation module) isomorphic to Z as an
abelian group, and a class f ∈ Hn (G; O) (the fundamental class), such that for any
G-module M the cup product with f gives an isomorphism
H i (G; M ) ∼
= Hn−i (G; M ⊗ O)
When O is the trivial G-module, we say that G is orientable.
Note that the notation above requires a G-bimodule structure on M and O. This
is readily available for any right ZG module – the left multiplication by g is equal
to the right multiplication by g −1 .
A natural example of a Poincaré duality group is the fundamental group of a
closed aspherical manifold. Davis [Dav] constructed other examples, which however
are not finitely presented. It is open whether there exist finitely presented Poincaré
duality groups which are not fundamental groups of closed aspherical manifolds.
Note that the fundamental class f is not trivial, since H0 (G; Z) is not trivial.
The author suspects that the following is well-known, but was unable to find a
reference in the literature.
Proposition 5.31. Let G be a Poincaré duality group in dimension n (with n > 3)
which is of type F. There exists a free resolution of length n of the trivial module
Z by finitely generated ZG-modules. Moreover, we can arrange for the 0th term of
the resolution to be of rank 1, and the boundary map ∂1 can be taken to be a row
vector (1 − si )i , where S = {s1 , . . . , sk } is some generating set of G.
Proof. Let C = (C∗ , ∂∗ ) be the cellular chain complex of the universal cover of some
finite CW-classifying space X of G. We denote the cochain complex by (C ∗ , ∂ ∗ )
with ∂ i : C i−1 → C i . By collapsing a maximal tree in the 1-skeleton of X, we easily
arrange for C0 to be of rank 1 as a ZG-module, and for ∂1 to be as desired.
Let us represent the fundamental class f by a cycle cf in Cn . Wall in the proof
of [Wal, Lemma 1.1] explains how to make sense of taking the cup product with a
cycle; thus the cup product with cf gives us a chain map ξ : C ∗ → Cn−∗ . The map
ξ induces the isomorphism Hi (C ∗ ⊗ M ) ∼
= Hn−i (C∗ ⊗ M ⊗ O) for every ZG-module
M . In particular, this holds for M = ZG; observe that Hi (C∗ ⊗ O) = Hi (C ∗ )
for every i, as both coincide with the homology of the universal cover of X. Since
the modules Cn and C n are free (and hence projective) and finitely generated,
Whitehead’s theorem tells us that ξ is a chain homotopy equivalence. This means
that there exist chain maps ξ ′ : C∗ → C n−∗ , p : C∗ → C∗+1 and p′ : C ∗ → C ∗−1
such that ξξ ′ = id −∂p − p∂ and ξ ′ ξ = id −∂p′ − p′ ∂.
···

/ Ci+1
O [

∂i+1

ξn−i−1

···

/ C n−i−1
O [
/ Ci+1
O

∂

/ C n−i−1

n−i

pi

∂

n−i

∂

/ C n−i+1
O

∂i

/ Ci
O
ξn−i

/ C n−i

n−i+1

pi−1

ξi′

p′n−i

/ ···

ξn−i+1

/ C n−i
O [

∂i+1

ξn−i−1

···

/ Ci−1
O

ξn−i

′
ξi+1

···

∂i

/ C0
O [

∂

′
ξi−1

/ Ci−1
O

p′n−i+1
n−i+1

/ ···

/ ···

′
ξi−1

/ C n−i+1

/ ···

36

DAWID KIELAK

Let m be maximal such that Cm 6= 0. Note that m > n, since Hn (G; Z) 6= 0, as
it contains f . We claim that we can modify the chain complex C so that m = n,
and the new complex is still a free resolution of Z by finitely generated ZG-modules.
If m = n then we are done. Suppose that m > n. We modify C to form
a new chain complex, D, as follows: D is identical to C except Dm = 0, and
Dm−2 = Cm−2 ⊕ Cm ; the boundary map Dm−1 → Dm−2 is equal to the transpose
(∂m−1 , pm−1 )T , and the boundary map Dm−2 → Dm−3 is equal to (∂m−2 , 0).
C=

Cm e

∂m

/ Cm−1

∂m−1

/ Cm−2

∂m−2

/ Cm−3

/ ···

/ Cm−3

/ ···

pm−1
T

D=

0

(∂m−1 ,pm−1 )
/ Cm−1
/ Cm−2 ⊕ Cm

(∂m−2 ,0)

It is immediate that D is a chain complex of finitely generated free modules; it is
also immediate that it is exact everywhere, except possibly at Dm−1 and Dm−2 .
Crucially, since C n−m = 0, we have ξn−m = 0 and so pm−1 ∂m = id. Let x
be a cycle in Dm−1 = Cm−1 . By exactness of C, we see that x ∈ im ∂m ; let y
denote a preimage of x in Cm . Since x is a cycle, we have pm−1 (x) = 0. But
pm−1 (x) = pm−1 ∂m (y) = y, and so y = 0 and thus x = 0.
Now let x ∈ Dm−2 be a cycle. We write x = (x0 , x1 )T ∈ Cm−2 ⊕ Cm ; being a
cycle means that x0 is a cycle in Cm−2 , and so there exists y ∈ Cm−1 such that
x0 = ∂m−1 (y). We can pick y so that pm−1 (y) = 0, using exactness of C and the
fact that pm−1 ∂m = id. Now
(∂m−1 , pm−1 )T (y + ∂m (x1 )) = (x0 , pm−1 (y) + pm−1 ∂m (x1 ))T = (x0 , x1 )T = x
since pm−1 (y) = 0 (by construction) and pm−1 ∂m = id.
We replace C by D; note that we can still define the chain homotopy equivalence
ξ between D and its dual cochain complex. Alternatively, we can realise D as the
cellular chain complex of the universal cover of a finite CW-classifying space of G:
we can remove each m-cell from X and attach an (m − 2) sphere instead of it. Then
we modify the attaching maps of the (m − 1) cells so that they coincide (in the
universal cover) with the boundary map in D. The resulting space is aspherical,
since D is exact away from degree 0. Since m > n > 3, the fundamental group did
not change under these modifications.
Note that when passing from C to D we did not alter C1 nor the boundary map
∂1 : C1 → C0 . We now repeat the argument until the length of the resolution is
n.

Theorem 5.32. Let G be a Poincaré duality group in dimension 3. Suppose further
that G is agrarian and of type F. We have Σ∞ (G; Z) = Σ1 (G), and there exists a
marked polytope P such that for every φ ∈ H 1 (G; R) r {0} we have φ ∈ Σ1 (G) if
and only if φ is marked.
Proof. Proposition 5.31 tells us that there exists a free resolution C = (C∗ , ∂∗ ) of
length 3 of the trivial ZG-module Z by finitely generated ZG-modules. We also
know that C0 has rank 1, and ∂1 = (1 − s)i , where S = {s1 , . . . , sk } is a finite
generating set of G.
Take φ ∈ H 1 (G; R) r {0}, and take s ∈ S with φ(s) 6= 0. Let Uφ denote a
closed neighbourhood of φ such that for every ψ ∈ Uφ we have ψ(s) 6= 0. Assume
further that Uφ is the convex hull of finitely many characters. It is immediate that
U
U
U
d φ is onto (since 1 − s is invertible in ZG
d φ ), and so H0 (C ⊗ ZG
d φ ) = 0.
∂1 ⊗ ZG
Uφ

Poincaré duality tells us that the transpose of ∂3 ⊗ d
ZG is onto as well. Thus,
Uφ
d is free, we see that it is a summand of C2 ⊗ ZG
dUφ . Therefore we
since C3 ⊗ ZG

THE BNS INVARIANTS VIA NEWTON POLYTOPES

have

Uφ

C2 ⊗ d
ZG

Uφ

Uφ

= C3 ⊗ d
ZG

37

⊕ C2′

d -module C ′ .
for some (stably free) ZG
2
We will modify the complex C by taking the direct sum with the complex
C3

id

/ C3

/0

(where the 0 module lies in degree 0). It is clear that the new chain complex
D = (D∗ , δ∗ ) is a free resolution of Z by finitely generated ZG-modules, and thus
it computes the homology of g in the same way as C did.
Let ki denote the rank of the free module Di ; we have already mentioned that
k0 = 1. Consider the homology of D ⊗ D. Since D is a skew-field, we see that
the Euler characteristic of D ⊗ D is k0 − k1 + k2 − k3 . But, since G is a Poincaré
duality group of odd dimension, it is immediate that this Euler characteristic must
be equal to 0. So
k1 + k3 = k2 + 1
We have
Uφ
Uφ
dUφ = C2 ⊗ d
ZG ⊕ C3 ⊗ d
ZG
D2 ⊗ ZG

Uφ
Uφ
ZG
= C3 ⊗ d
ZG ⊕ C2′ ⊕ C3 ⊗ d
Uφ
Uφ
∼
ZG ⊕ C2 ⊗ d
ZG
= C3 ⊗ d

dUφ
The last isomorphism implies the existence of an invertible matrix M over ZG
Uφ
d ), such that M δ3 is the identity k3 × k3
(the change of basis matrix of D2 ⊗ ZG
matrix extended at the bottom by the (k2 − k3 ) × k3 zero matrix.
Now we will replace the matrix M by a matrix M0 over ZG by truncating the
entries of M (the argument is very similar to the one used in Lemma 3.20). Recall
that Uφ is the convex hull of finitely many characters, say ρ1 , . . . , ρl . Pick C ∈ R
such that C + ρi (g) > 0 for every i and every g ∈ G appearing in the support of
some entry of δ3 (which is a matrix over ZG). Now, let M0 denoteSa matrix over
(0, C)
ZG obtained from M by truncating the entries of M to the subset i ρ−1
i
of G – formally speaking, we
treat
the
entries
of
M
as
functions
G
→
Z,
we
first

S
restrict these functions to i ρi−1 (0, C) and then extend back to functions from

S
G by setting their values outside of i ρi−1 (0, C) to be 0. By the choice of C,
we see that M0 δ3 is a ρi -identity for every i. Now, a simple convexity argument
immediately shows that in fact M0 δ3 is a ψ-identity for every ψ ∈ Uφ .
The fact that M0 δ3 is a ψ-identity implies that the span of the last (k2 − k3 )
U
d φ is a complement to the image of the (injective) map
basis vectors of D2 ⊗ ZG
U

U

φ
d φ . Therefore, when computing the homology of
M 0 δ3 : D 3 ⊗ d
ZG → D2 ⊗ ZG
Uφ
d , we can disregard D3 and replace D2 by this complement D′ . This
D ⊗ ZG
2
ψ
d ) = 0 for every i > 2 and every ψ ∈ Uφ .
already tells us that Hi (G; ZG
Crucially, the differential D2′ → D1 is a matrix defined over ZG, say A. The
matrix A is a k1 × (k2 − k3 ) matrix. But k2 − k3 = k1 − 1, and so A has one
row more than column. We now form Aφ by removing the row of A corresponding
to s, that is, if s = si then we remove the ith row. Again, Aφ is defined over
ZG, and φ ∈ Σ1 (G) if and only if Aφ is right-invertible. But, by definition, Aφ is
right-invertible if and only if φ ∈ Σ(Aφ ). Moreover, φ ∈ Σ2 (G; Z) if and only if
φ ∈ Σ1 (G) and Aφ is injective. We have already seen that a result of Bieri [Bie,
Theorem 3] tells us that if Aφ is right-invertible then it is also injective. Hence
we obtain: φ ∈ Σ1 (G) if and only if φ ∈ Σ2 (G; Z) if and only if φ ∈ Σ(Aφ ). We

38

DAWID KIELAK
φ

have also already shown that higher homology groups of G with d
ZG coefficients
are trivial, and so Σ1 (G) = Σ∞ (G).
By Theorem 5.9, φ ∈ Σ(Aφ ) is equivalent to φ being marked by a marking of
P (Aφ ). Now, we observe that there are finitely many characters φ1 , . . . , φn such that
every character in H 1 (G; R)r{0} is contained in at least one of the neighbourhoods
Uφi up to positive homothety – this follows immediately from the compactness of
the unit sphere in H 1 (G; R). Therefore, Uφ1 , . . . , Uφn and P (Aφ1 ), . . . , P (Aφn ) form
an atlas of markings, and we are done after an application of Lemma 5.11, which

combines the marked polytopes P (Aφi ) into a single marked polytope P .
Note that if we were to strengthen the hypothesis by assuming G to satisfy the
Atiyah conjecture, it is not clear whether we would be able to extract a stronger
result – it is not even clear that the L2 -torsion polytope (even though it does
exist) would be a single polytope; more importantly for us, it is not clear what the
contribution to the polytope coming from the top-degree differential is. In the case
of an honest 3-manifold, it is the same as the contribution of the differential of the
lowest degree, but this does not seem to follow from Poincaré duality alone.
5.9. 3-manifolds. We will now give a new proof of the following theorem of Thurston.
Theorem 5.33 (Thurston [Thu, Theorem 5]). Let M be a compact, oriented 3manifold. The set F of cohomology classes in H 1 (M ) representable by non-singular
closed 1-forms is some union of the cones on open faces of Bx , minus the origin.
The set of elements in H 1 (M ; Z) whose Lefschetz dual is represented by a fibre of
a fibration consists of the set of all lattice points in F .
The object Bx appearing above is the unit ball of the Thurston semi-norm
x : H 1 (M ; R) → [0, ∞). The semi-norm is first defined on the integral characters,
then defined on positive multiples of such characters by the formula x(λφ) = λx(φ),
and lastly it is extended by continuity to the whole of H 1 (M ; R). Given an integral
character φ (that is a character with im φ = Z, the standard copy of Z in R), we
define
x(φ) = min −χ− (Σ)
Σ

where Σ runs through all embedded (not necessarily connected) oriented surfaces
in M dual to φ via Poincaré–Lefschetz duality, and χ− (Σ) denotes the sum of the
Euler characteristics of the connected components of Σ which are not spheres.
Thurston also defined the polytope Bx∗ ⊆ H1 (M ; R) dual to Bx ; he proved
in [Thu, Theorem 2] that Bx∗ is an integral polytope. He also proved in [Thu,
Theorem 3] that the open faces of Bx forming the union of cones F are all faces of
maximal dimension (called fibred faces) – on the level of the dual polytope Bx∗ this
translates to the statement that F is the union of some duals of vertices of Bx∗ .
The identification of the set of elements in H 1 (M ; Z) whose Lefschetz dual is
represented by a fibre of a fibration with the set of all (integral) lattice points in
F follows immediately from the work of Tischler [Tis]: one has to observe that
(topological) 3-manifolds can always be taken to be smooth – it was shown by
Moise [Moi, Theorems 1 and 3] that 3-manifolds can be triangulated in such a way
that closed stars of vertices are piecewise-linearly homeomorphic to a simplex in
R3 . This is enough to construct a smooth structure.
Now let us argue that without loss of generality one may take M to be connected
in the statement of Thurston’s theorem: suppose that we have shown the result
for connected manifolds, and let M = M1 ⊔ · · ·Q⊔ Mm be a decomposition into
connected components. Note that H 1 (M ; R) = i H 1 (Mi ; R), and the Thurston
norm x restricted to H 1 (Mi ; R) is equal to the Thurston norm xi on Mi . Thus, Bx∗
is simply the product of the polytopes Bx∗i , and duals of faces of Bx∗ are products

THE BNS INVARIANTS VIA NEWTON POLYTOPES

39

of duals of corresponding faces of Bx∗i . Now, a cohomology class in H 1 (M ; R) is
representable by a non-singular closed 1-form if and only if its component in every
H 1 (Mi ; R) is.
When M is connected, Bieri–Neumann–Strebel observed in [BNS, Corollary F]
that F = Σ1 (π1 (M )), thus connecting Thurston’s result to the BNS invariants.
Let us summarise this discussion in a statement that is equivalent to Thurston’s
theorem above.
Theorem 5.34 (Reformulation of Thurston’s theorem). Let M be a compact, connected, oriented 3-manifold, and let G = π1 (M ). There exists a marking of vertices
of Bx∗ such that for every φ ∈ H 1 (G; R) r {0} we have φ ∈ Σ1 (G) if and only if φ
is marked.
Proof. If Σ1 (G) = ∅, then the statement holds vacuously. Let us assume that
Σ1 (G) 6= ∅.
We will first argue that G satisfies the Atiyah conjecture. Since Σ1 (G) is open (by
Theorem 5.4), it contains an integral surjective character φ : G → Z, and so M fibres
smoothly over the circle (by [Tis]). The fibre must be a compact orientable surface.
Irrespectively of whether it is closed or not, its fundamental group is residually
{torsion-free nilpotent} (see e.g. [Bau, Corollary 2]). Note that residual nilpotence
is equivalent to saying that the intersection of all the terms in the lower central series
is trivial. But the lower central series consists of characteristic subgroups, and so
any extension of a residually nilpotent group by Z is residually solvable. Moreover,
every extension of a residually {torsion-free nilpotent} group by Z is residually
{torsion-free solvable}. Such groups satisfy the Atiyah conjecture – they are torsion
free elements of the class C2 of Theorem 4.2. In particular, our fundamental group
G satisfies the Atiyah conjecture.
By a result of Lück [Lüc, Theorem 1.39], M is L2 -acyclic, as it is a mapping
torus over a finite CW complex. Also, since M fibres with fibre an oriented surface,
it is immediate that either M is homeomorphic to S2 × S1 (as there are no nontrivial fibrations over S1 with fibre S2 ), or M is aspherical, since the circle and any
oriented surface which is not S2 are aspherical. In the latter case it is also clear
that the boundary of M is either empty or a collection of tori.
When M ∼
= S1 × S2 , then Σ1 (G) = H 1 (G; R) r {0}, and we declare all faces of
Bx∗ marked. One can compute Bx∗ directly, and it turns out to be a single point,
the origin of H 1 (G; R). Thus Bx∗ is a polytope with marked vertices.
We are now ready for the main case: M is an L2 -acyclic aspherical manifold
with empty or toroidal boundary whose fundamental group satisfies the Atiyah
conjecture. At this point we can already use Theorem 5.32, but since we have the
Atiyah conjecture at our disposal and the manifold structure, we will actually learn
more about the marked polytope occurring in Theorem 5.32.
Pick a finite CW-structure for M , and let C denote the cellular chain complex
of the universal cover of M . The complex C is a free ZG-resolution of the trivial
module Z, since M is aspherical.
Suppose first that M is closed. McMullen in the proof of [McM, Theorem 5.1]
shows that we may take C as follows:
C=

C3

∂3

/ C2

∂2

/ C1

∂1

/ C0

where C0 and C3 are of rank one, C1 and C2 are of the same rank, ∂3 is the transpose
of ∂1 , which in turn is the vector (1 − s)s∈S with S being a finite generating set
of G as usual. For any φ ∈ H 1 (G; R) r {0} there exists si ∈ S with φ(si ) 6= 0.
Now let Ai denote the matrix obtained from ∂2 by removing the ith row and the
ith column. The matrix Ai is a square matrix over ZG, and the structure of ∂1

40

DAWID KIELAK

and ∂3 , together with the invertibility of 1 − si over ZGφ , immediately imply that
dφ ) = 0 if and only if φ ∈ Σ(Ai ). Thus Σ(Ai ) induces a marking of the
H1 (G; ZG
vertices of P (Ai ), using Theorem 5.9 as before.
Since G is L2 -acyclic, the L2 -torsion polytope PL2 (G) is defined, and we have
PL2 (G) = P (Ai ) − 2P (1 − si ) – this follows from additivity of the universal L2 torsion, see [FL, Lemma 1.9]. In fact, PL2 (G) is a single polytope: if the rank of Gfab
is at least 2, it follows from Corollary 3.7; otherwise it follows from [FL, Theorem
2.39], since the induced function k · kPL2 (G) can be a semi-norm only when PL2 (G)
is a single polytope (in the case of Gfab of rank 1). Now we need to define a marking
on the vertices of PL2 (G), using the given markings on the polytopes P (Ai ). This
is achieved by Lemma 5.12, taking Ui = {ψ ∈ H 1 (G; R) r {0} | ψ(si ) 6= 0}.
Now suppose that M has boundary. Again, using the proof of [McM, Theorem
5.1], we see that
∂2
/ C1 ∂1 / C0
C=
C2
where C1 is of rank one greater than C2 . We argue precisely as before, reducing
the problem of whether φ ∈ Σ1 (G) to whether the matrix Ai obtained by removing
dφ (as usual, we
the column of ∂2 corresponding to si admits a right-inverse over ZG
have φ(si ) 6= 0).
In this case we have PL2 (G) = P (Ai ) − P (1 − si ), and the argument proceeds
exactly as before.
It was shown by Friedl–Lück [FL, Theorem 3.37] that PL2 (G) coincides with

Bx∗ . This concludes the proof.

It is immediate from Thurston’s construction that the equality of semi-norms
x = k · kBx∗ is satisfied. Friedl–Lück proved that Bx∗ = PL2 (G) when M is
aspherical, not homeomorphic to S1 ×S2 , and G satisfies the Atiyah conjecture. This
equality yields a new point of view on the value x(φ) for φ with im φ = Z: when ker φ
is finitely generated, it was known that x(φ) gives minus the Euler characteristic
of the surface, which is the fibre of the corresponding fibration. When ker φ is not
finitely generated, we see now that x(φ) gives minus the L2 -Euler characteristic of
the kernel.
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Dawid Kielak
dkielak@math.uni-bielefeld.de
Fakultät für Mathematik
Universität Bielefeld
Postfach 100131
D-33501 Bielefeld
Germany

