# Notes on nilspaces: algebraic aspects

Pablo Candela

title = Notes on nilspaces – algebraic aspects, author = Pablo Candela,
plaintextauthor = Pablo Candela, runningtitle = Notes on nilspaces: algebraic
aspects, runningauthor = P. Candela, year=2017, number=15, received=31 May
2016, revised=25 May 2017, published=7 September 2017, doi=10.19086/da.2105,

[classification=]

###### Contents

  1. 0 Introduction
    1. 1 Discrete cubes
    2. 2 Definition of a nilspace
  2. 1 Main examples of nilspaces
    1. 1 Standard cubes on abelian groups
    2. 2 Filtered groups of degree $k$ as $k$-step nilspaces
      1. 1 Cubes on a general filtered group
      2. 2 Nilspace morphisms between filtered groups: polynomial maps
      3. 3 Defining cubes on a filtered group in terms of equations
      4. 4 Cubes of degree $k$ on abelian groups
    3. 3 Quotients of filtered groups of degree $k$ as $k$-step nilspaces
    4. 4 Motivation for a general algebraic characterization
  3. 2 Algebraic characterization of nilspaces
    1. 1 Some basic notions
      1. 1 Cubespaces, simplicial completion, and concatenation
      2. 2 Nilspaces and parallelepiped structures
      3. 3 Tricubes and the tricube composition
      4. 4 Arrow spaces
    2. 2 $k$-step nilspaces as abelian bundles of degree $k$ acted upon by filtered groups
      1. 1 Characteristic factors
      2. 2 $k$-fold ergodic nilspaces as degree-$k$ abelian torsors
      3. 3 Abelian bundles, and the bundle decomposition
      4. 4 Translations on a $k$-step nilspace form a filtered group of degree $k$
    3. 3 Additional tools
      1. 1 Nilspace morphisms as bundle morphisms
      2. 2 Fibre-surjective morphisms and restricted morphisms
      3. 3 Extensions and cocycles
      4. 4 Translation bundles

## Chapter 0 Introduction

These notes constitute the first part of a detailed exposition of the theory
of nilspaces developed by Camarena and Szegedy in the paper [1]. We treat what
can be called the algebraic part of the theory, in which nilspaces are studied
without any topological assumption. We elaborate on some of the central
concepts and arguments from the first two chapters of [1], present more
detailed proofs, and illustrate some of the key notions with concrete
examples.

The main objects studied in [1] are _nilspaces_. These objects are defined in
terms of some axioms, which rely on the basic notions of discrete cubes and
cube morphisms. We shall begin in the next section by recalling these notions
and their elementary properties.
As stated in [1], important examples of nilspaces are provided by filtered
groups and filtered nilmanifolds. In Chapter 1, we elaborate on this by
treating the theory of cubes on filtered groups, which was pioneered by Host
and Kra [11, 12] and developed in several other works including [6, 7, 14]. We
treat this theory from the viewpoint of general nilspaces. This provides
natural motivation for various important objects and results related to this
theory, notably polynomial maps and Leibman’s theorem, and it also provides
illustrations of several tools and ideas used by Camarena and Szegedy.
One of the central results in the algebraic part of [1] is the description of
a general nilspace as an iterated _abelian bundle_. In Chapter 2, after having
gathered some basic tools in Section 1, we present a detailed proof of this
result in Section 2, following the arguments of Camarena and Szegedy. The
final section of the chapter concerns additional algebraic tools introduced in
[1]. These tools are of inherent interest but are also important for the
topological part of the theory of nilspaces.
We treat the topological part of the theory separately, in [2]. That part
studies nilspaces equipped with a compact topology that is compatible with the
cube structure, called _compact nilspaces_. This study is motivated
principally by the analysis of _uniformity norms_ in arithmetic combinatorics,
and of the analogous _uniformity seminorms_ in ergodic theory, leading in
particular to the theory of _higher order Fourier analysis_ ; see [3] for an
introduction to this theory from the viewpoint of arithmetic combinatorics. An
alternative treatment of compact nilspaces is given by Gutman, Manners, and
Varjú in [8, 9, 10].

Acknowledgements. I am very grateful to Balázs Szegedy for discussions that
were crucial for my understanding of [1], and to anonymous referees for advice
that helped to improve this paper. I also thank Yonatan Gutman, Frederick
Manners and Péter Varjú for informing me of their work prior to publication,
and for useful comments. The present work was supported by the ERC
Consolidator Grant No. 617747.

###  1 Discrete cubes

Recall that an _affine homomorphism_ from an abelian group
$\operatorname{Z}_{1}$ to an abelian group $\operatorname{Z}_{2}$ is a map of
the form $x\mapsto\phi(x)+t$, where
$\phi:\operatorname{Z}_{1}\to\operatorname{Z}_{2}$ is a homomorphism and $t$
is some fixed element of $\operatorname{Z}_{2}$.

######  Definition 1.1.

Let $n$ be a non-negative integer. The _discrete $n$-cube_ is the set
$\\{0,1\\}^{n}$. We denote the elements $(0,\ldots,0),(1,\ldots,1)$ of
$\\{0,1\\}^{n}$ by $0^{n},1^{n}$ respectively. A _discrete-cube morphism_ is a
map $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ that is the restriction of an affine
homomorphism $\mathbb{Z}^{m}\to\mathbb{Z}^{n}$. An _automorphism_ of the
discrete $n$-cube is a bijective morphism $\\{0,1\\}^{n}\to\\{0,1\\}^{n}$.
These automorphisms form a group, denoted by
$\operatorname{Aut}(\\{0,1\\}^{n})$.

We recall that $\operatorname{Aut}(\\{0,1\\}^{n})$ is generated by the group
$S_{n}$ of permutations of $[n]=\\{1,2,\ldots,n\\}$ together with the
reflections $v\scalebox{0.8}{$(i)$}\mapsto 1-v\scalebox{0.8}{$(i)$}$,
$i\in[n]$, more precisely we have $\operatorname{Aut}(\\{0,1\\}^{n})\cong
S_{n}\ltimes(\mathbb{Z}/2\mathbb{Z})^{n}$.
For $v\in\\{0,1\\}^{n}$, we shall write $\operatorname{Supp}(v)$ for the
_support_ of $v$, i.e.

$\operatorname{Supp}(v)=\\{i\in[n]:v\scalebox{0.8}{$(i)$}=1\\},$

and we shall denote by $|v|$ the cardinality of $\operatorname{Supp}(v)$.
Morphisms of discrete cubes can be described more explicitly in several ways.
Below we give a description that will be quite useful later on. Let us first
denote the four maps $\\{0,1\\}\to\\{0,1\\}$ as follows: we denote by
$\mathbf{0}$ and $\mathbf{1}$ the maps with constant value 0 and 1
respectively, we denote the identity map by $\tau_{0}$, and we denote by
$\tau_{1}$ the reflection map (the bijection sending 0 to 1).

######  Lemma 1.2.

A map $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ is a discrete-cube morphism if and
only if it is of the form
$\phi\big{(}v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(m)$}\big{)}=\big{(}w\scalebox{0.8}{$(1)$},\ldots,w\scalebox{0.8}{$(n)$}\big{)}$
where, for each $j\in[n]$, either $w\scalebox{0.8}{$(j)$}$ is a constant
function of $v$ or there is a unique $i=i(j)\in[m]$ and $k=k(j)\in\\{0,1\\}$
such that $w\scalebox{0.8}{$(j)$}=\tau_{k}(v\scalebox{0.8}{$(i)$})$.

######  Example 1.3.

To illustrate the lemma, we can consider a morphism
$\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$. We could have for instance
$\phi(v)=(v\scalebox{0.8}{$(1)$},1-v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(2)$}\big{)}$,
where no coordinate of $\phi(v)$ is a constant function of $v$. In this case,
embedding $\\{0,1\\}^{3}$ in $\mathbb{R}^{3}$ as the set of vertices of the
unit cube, the image of $\phi$ consists of the vertices of a square going
diagonally through the cube, namely $\\{(0,1,0),(1,0,0),(0,1,1),(1,0,1)\\}$.
We could also have that some coordinate of $\phi(v)$ is constant, for instance
$\phi(v)=(v\scalebox{0.8}{$(1)$},0,1-v\scalebox{0.8}{$(2)$}\big{)}$. In this
case, in the embedding just mentioned, the image of $\phi$ consists of the
vertices of a face of the unit cube, namely
$\\{(0,0,1),(1,0,1),(0,0,0),(1,0,0)\\}$.

###### Proof of Lemma 1.2.

Note that, firstly, from the definition of a morphism we can deduce that
$\phi(v)=Mv+\phi(0^{m})$, for some matrix $M\in\mathbb{Z}^{n\times m}$. Then,
since $\phi$ takes values in $\\{0,1\\}^{n}$, for each $j\in[n]$ the $j$-th
row of $M$ has at most one non-zero entry, with index $i=i(j)$. This entry
must then equal 1 or $-1$, which then determines the $i$-th entry of
$\phi(0^{m})$ to be 0 or 1 respectively, and the result follows. ∎

We shall now define an important family of subsets of a discrete cube, namely
its _faces_. For this we use the following notation. Given a morphism
$\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ and $i\in[m]$, we denote by $J_{\phi}(i)$
the set of indices of the coordinates of $\phi(v)$ that are not constant
functions of $v\scalebox{0.8}{$(i)$}$ (in other words, the coordinates of
$\phi(v)$ which vary as $v\scalebox{0.8}{$(i)$}$ varies), that is

$J_{\phi}(i):=\\{j\in[n]:\phi(v)\scalebox{0.8}{$(j)$}=\tau_{0}(v\scalebox{0.8}{$(i)$})\textrm{
or }\tau_{1}(v\scalebox{0.8}{$(i)$})\\}.$ (1)

In the example $\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},1-v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(2)$}\big{)}$
seen above, we would have $J_{\phi}(1)=\\{1,2\\}$ and $J_{\phi}(2)=\\{3\\}$.

Note that a morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ is injective if and
only if $J_{\phi}(i)$ is non-empty for every $i\in[m]$. We define

$J_{\phi}:=\bigsqcup_{i\in[m]}J_{\phi}(i).$ (2)

This is the set of coordinates of $\phi(v)$ that are not constant functions of
$v$. We denote the set of subsets of size $k$ of a set $S$ by $\binom{S}{k}$.

######  Definition 1.4 (Faces and face maps).

Let $m\leq n$ be non-negative integers. An $m$_-face_ of $\\{0,1\\}^{n}$
(short for $m$-dimensional face) is a set $F\subset\\{0,1\\}^{n}$ specified by
fixing $n-m$ coordinates of $v\in\\{0,1\\}^{n}$, that is
$F=\\{v\in\\{0,1\\}^{n}:v\scalebox{0.8}{$(i)$}=x\scalebox{0.8}{$(i)$}\;\;\forall\,i\in
I\\}$, for some fixed set $I\in\binom{[n]}{n-m}$ and $x\in\\{0,1\\}^{I}$. We
call a morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ an $m$-_face map_ if
$\phi$ is injective and the image $\operatorname{Im}(\phi)$ is an $m$-face.
Equivalently, $\phi$ is an $m$-face map if $|J_{\phi}(i)|=1$ for every
$i\in[m]$.

For instance, the map $\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},0,1-v\scalebox{0.8}{$(2)$}\big{)}$ from
Example 1.3 is a 2-face map and $\operatorname{Im}(\phi)$ is the 2-face
$\\{w\in\\{0,1\\}^{3}:w\scalebox{0.8}{$(2)$}=0\\}$. The map
$\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},1-v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(2)$}\big{)}$
is an injective morphism, but is not a face map.

We shall often specify an $m$-face $F\subset\\{0,1\\}^{n}$ by the data
$(I,x)$. We shall then refer to the cardinality of $I$ as the _codimension_ of
$F$, denoted by $\operatorname{codim}(F)$. For later use, let us also reserve
the special notation $\phi_{F}$ for the $m$-face map that has image equal to
$F$ in the “simplest way”. More precisely, $\phi_{F}$ will always denote the
$m$-face map such that for all $j<j^{\prime}$ in $J_{\phi_{F}}$ we have
$i(j)<i(j^{\prime})$ in $[m]$, and
$\phi_{F}(v)\scalebox{0.8}{$(j)$}=v\scalebox{0.8}{$(i(j))$}$ for all $j\in
J_{\phi_{F}}$. For instance if
$F=\\{w\in\\{0,1\\}^{3}:w\scalebox{0.8}{$(2)$}=0\\}$ then
$\phi_{F}:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},0,v\scalebox{0.8}{$(2)$}\big{)}$. Note that
if $m=n$ then $\phi_{F}$ is just the identity in
$\operatorname{Aut}(\\{0,1\\}^{n})$.

######  Definition 1.5 (Face restriction).

Let $F\subset\\{0,1\\}^{n}$ be an $m$-face and let $g$ be a map defined on
$\\{0,1\\}^{n}$. The _face restriction_ of $g$ to $F$ is the map
$g\operatorname{\circ}\phi_{F}$ on $\\{0,1\\}^{m}$.

###  2 Definition of a nilspace

######  Definition 2.1.

A _nilspace_ is a set $\operatorname{X}$ together with a collection of sets
$\operatorname{C}^{n}(\operatorname{X})\subset\operatorname{X}^{\\{0,1\\}^{n}}$,
for each non-negative integer $n$, satisfying the following axioms:

  1. (i)

(Composition) For every morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ and
every $\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X})$, we have
$\operatorname{c}\operatorname{\circ}\phi\in\operatorname{C}^{m}(\operatorname{X})$.

  2. (ii)

(Ergodicity)
$\operatorname{C}^{1}(\operatorname{X})=\operatorname{X}^{\\{0,1\\}}$.

  3. (iii)

(Corner completion) Let
$\operatorname{c}^{\prime}:\\{0,1\\}^{n}\setminus\\{1^{n}\\}\to\operatorname{X}$
be such that every restriction of $\operatorname{c}^{\prime}$ to an
$(n-1)$-face containing $0^{n}$ is in
$\operatorname{C}^{n-1}(\operatorname{X})$. Then there exists
$\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X})$ such that
$\operatorname{c}(v)=\operatorname{c}^{\prime}(v)$ for all $v\neq 1^{n}$.

The elements of $\operatorname{C}^{n}(\operatorname{X})$ are referred to as
the $n$_-cubes_ (or $n$-dimensional cubes) on $\operatorname{X}$. A map
$\operatorname{c}^{\prime}$ satisfying the premise of the third axiom is
called an $n$_-corner_. We shall call an $n$-cube $\operatorname{c}$
satisfying the third axiom for $\operatorname{c}^{\prime}$ a _completion_ of
$\operatorname{c}^{\prime}$.

######  Definition 2.2.

A $k$_-step nilspace_ (or $k$-_nilspace_) is a nilspace such that every
$(k+1)$-corner has a unique completion.

As a first manipulation of the axioms, one may check that the set underlying a
0-step nilspace, if non-empty, must be a singleton.

Occasionally we will have to consider spaces $\operatorname{X}$ for which one
of the last two axioms above may fail. We shall have to deal with such spaces
explicitly only from Chapter 2 onwards, so we postpone their discussion until
then, recording only the following definition for now.

######  Definition 2.3.

A _cubespace_ is a set $\operatorname{X}$ together with a collection of sets
$\operatorname{C}^{n}(\operatorname{X})\subset\operatorname{X}^{\\{0,1\\}^{n}}$,
for each $n\geq 0$, satisfying the composition axiom from Definition 2.1 and
such that $\operatorname{C}^{0}(\operatorname{X})=\operatorname{X}$. A
cubespace $\operatorname{X}$ is said to be _$k$ -fold ergodic_ if
$\operatorname{C}^{k}(\operatorname{X})=\operatorname{X}^{\\{0,1\\}^{k}}$.

The nilspace axioms can be seen as variants of the axioms defining the notion
of _parallelepiped structures_ , notion introduced by Host and Kra in [12].
For instance, the corner completion axiom appears in an equivalent form in
[12, Definition 4]. There is in fact a sense in which the concept of
parallelepiped structures is equivalent to that of a nilspace. We shall detail
this in Subsection 2.
The term ‘nilspace’ may seem deceptive at first, in that it is not clear from
the axioms that a nilspace should be related to anything involving nilpotency.
It is a nontrivial fact that there is indeed such a relation. A first explicit
description of this phenomenon appeared in the setting of ergodic theory, in
the work of Host and Kra [11]. The notion of cubes on a filtered group,
treated in the next chapter, originates in that work. In the later paper [12],
Host and Kra initiated the program of understanding the above-mentioned
relation in a more conceptual and general way, starting from the axioms that
define parallelepiped structures. This greater generality consisted especially
in that there was no measure-preserving system underlying the definition of
these structures, unlike in [11]. The results included the determination of a
nilpotent group naturally associated with such a structure (see [12, Section
3.8]). The work of Camarena and Szegedy in [1] can be seen as a continuation
of this program (see the end of Subsection 2).

In the next chapter we begin to study some natural examples of nilspaces,
starting with the cube structures on filtered groups introduced by Host and
Kra. In fact, the results in later chapters will indicate that these filtered
groups, together with their quotients (including nilmanifolds), constitute the
main examples of nilspaces.

## Chapter 1 Main examples of nilspaces

In this chapter we begin our study of nilspaces by examining some natural
examples.
We start in Section 1 with a basic case, namely that of cubes on abelian
groups. We shall provide several characterizations of these cubes and show
that an abelian group together with these cubes satisfies the nilspace axioms.
In Section 2 we shall then generalize this cube structure to any filtered
group (not necessarily commutative) and show that this also yields a nilspace
structure on such groups.
The nilspaces covered in Section 2 have the particular feature that the cubes
of a fixed dimension form a group under pointwise multiplication. Taking
quotients of these nilspaces in certain ways produces new nilspaces that do
not have this group property; this is illustrated in Section 3.

###  1 Standard cubes on abelian groups

Let $(G,+)$ be an abelian group. An $n$-cube on $G$ is a map
$\operatorname{c}:\\{0,1\\}^{n}\to G$ of the form111Strictly speaking, the
term ‘$n$-dimensional parallelepiped’ may be more accurate here, but we shall
use the shorter ‘$n$-cube’ for convenience.

$\operatorname{c}(v)=x+v\scalebox{0.8}{$(1)$}\,h_{1}+\cdots+v\scalebox{0.8}{$(n)$}\,h_{n},$

for some fixed elements $x,h_{1},\ldots,h_{n}\in G$. This definition of cubes
on $G$ is quite familiar in arithmetic combinatorics (it is involved in the
definition of the Gowers uniformity norms, for instance). In Proposition 1.2
below we shall record other equivalent ways to view these cubes. To this end
we use the following function.

######  Definition 1.1.

Let $G$ be an abelian group. We define the function
$\sigma_{2}:G^{\\{0,1\\}^{2}}\to G$ as follows: for $g:\\{0,1\\}^{2}\to G$,
$\sigma_{2}(g):=g\scalebox{0.8}{$(00)$}-g\scalebox{0.8}{$(10)$}+g\scalebox{0.8}{$(11)$}-g\scalebox{0.8}{$(01)$}$.

Note that $\operatorname{c}$ is a $2$-cube on $G$ if and only if
$\sigma_{2}(\operatorname{c})=0$. Note also that if $\operatorname{c}$ is a
$2$-cube on $G$ then from the expression
$\operatorname{c}(v)=x+v\scalebox{0.8}{$(1)$}h_{1}+v\scalebox{0.8}{$(2)$}h_{2}$
it clearly follows that $\operatorname{c}$ can be extended to an affine map
$\mathbb{Z}^{2}\to G$, namely the map $(n_{1},n_{2})\mapsto
x+n_{1}h_{1}+n_{2}h_{2}$. The converse holds also.

These remarks generalize in a straightforward way to $n$-cubes for $n>2$, as
follows.

######  Proposition 1.2.

Let $\operatorname{c}:\\{0,1\\}^{n}\to G$ be a map. The following properties
are equivalent.

  1. (i)

There exist $x,h_{1},h_{2},\ldots,h_{n}\in G$ such that for every
$v\in\\{0,1\\}^{n}$ we have

$\operatorname{c}(v)=x+v\cdot
h:=x+v\scalebox{0.8}{$(1)$}h_{1}+\cdots+v\scalebox{0.8}{$(n)$}h_{n}.$ (1)

  2. (ii)

The map $\operatorname{c}$ extends to an affine homomorphism
$\mathbb{Z}^{n}\to G$.

  3. (iii)

Every $2$-face map $\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{n}$ satisfies

$\sigma_{2}(\operatorname{c}\operatorname{\circ}\phi)=0.$ (2)

We call the map $\operatorname{c}$ a (standard) $n$_-cube_ on $G$ if it
satisfies any of these properties.

###### Proof.

The equivalence of ($i$) and ($ii$) is clear. It is also clear that ($ii$)
implies ($iii$), since every 2-face map $\phi$ is by definition the
restriction to $\\{0,1\\}^{2}$ of an affine homomorphism
$\mathbb{Z}^{2}\to\mathbb{Z}^{n}$, and so
$\operatorname{c}\operatorname{\circ}\phi$ is the restriction of an affine
homomorphism $\mathbb{Z}^{2}\to G$, i.e. a 2-cube. If $\operatorname{c}$
satisfies ($iii$), then let $x:=\operatorname{c}(0^{n})$ and for each
$i\in[n]$ let $h_{i}:=\operatorname{c}(e_{i})-x$, where $e_{i}$ denotes the
$i$-th vector in the standard basis of $\mathbb{R}^{n}$. We claim that
$\operatorname{c}$ satisfies (1) with this choice of elements
$x,h_{1},\ldots,h_{n}$. This can be shown by induction on $|v|$. For $|v|=0,1$
the property holds by definition of $x$ and $h_{i}$, so let $|v|>1$ and
suppose that $\operatorname{c}$ satisfies ($ii$) for all $v^{\prime}$ with
$|v^{\prime}|<|v|$. Now there clearly is a 2-face-map
$\phi:\\{0,1\\}^{2}\to\\{0,1\\}^{n}$ such that $v=\phi(1^{2})$ and
$|\phi(w)|<|v|$ for all $w\neq 1^{2}$, so we have by induction
$\operatorname{c}(\phi(w))=x+\phi(w)\cdot h$ for each
$w\in\\{0,1\\}^{2}\setminus\\{1^{2}\\}$. By (2), we then have
$\operatorname{c}(\phi(11))=\operatorname{c}(\phi(10))+\operatorname{c}(\phi(01))-\operatorname{c}(\phi(00))=x+(\phi(10)+\phi(01)-\phi(00))\cdot
h=x+v\cdot h$ as required. ∎

The cubes characterized by this proposition yield a basic example of a
nilspace.

######  Proposition 1.3.

An abelian group together with the collection of all standard $n$-cubes on $G$
(for each non-negative integer $n$) is a 1-step nilspace.

###### Proof.

The ergodicity axiom holds (from (1) say). Composition is also clear (by
property $(ii)$ from Proposition 1.2, for instance). The unique corner
completion for $n\geq 2$ follows from the fact that, by (1), a standard
$n$-cube is determined by its values at elements $v\in\\{0,1\\}^{n}$ with
$|v|\leq 1$. Thus, given an $n$-corner $\operatorname{c}^{\prime}$ on $G$,
letting $x=\operatorname{c}^{\prime}(0^{n})$ and
$h_{i}=\operatorname{c}^{\prime}(e_{i})-x$ for each $i\in[n]$, the unique
$n$-cube $\operatorname{c}$ completing $\operatorname{c}^{\prime}$ is given by
the formula
$\operatorname{c}(v)=x+v\scalebox{0.8}{$(1)$}h_{1}+\cdots+v\scalebox{0.8}{$(n)$}h_{n}$.
∎

######  Remark 1.4.

For an element $g\in G$ and a set $F\subset\\{0,1\\}^{n}$, let $g^{F}$ denote
the element of $G^{\\{0,1\\}^{n}}$ defined by $g^{F}(v)=g$ if $v\in F$ and
$0_{G}$ otherwise. Then it follows from (1) that the standard $n$-cubes form
the subgroup of $G^{\\{0,1\\}^{n}}$ generated by the ‘constant elements’
$x^{\\{0,1\\}^{n}}$ and the elements $h_{i}^{F_{i}}$, $i\in[n]$, where $F_{i}$
is the face $\\{v\in\\{0,1\\}^{n}:v\scalebox{0.8}{$(i)$}=1\\}$ of codimension
1. (In fact each $n$-cube is generated like this in a unique way.) In the next
subsection we shall generalize the standard cubes using this group-theoretic
viewpoint.

######  Remark 1.5.

In Proposition 1.2 one can replace ‘Every 2-face map’ with ‘Every morphism’
without affecting the equivalence. We omit the details here, as a more general
remark of this kind will be made later (see Remark 2.29).

###  2 Filtered groups of degree $k$ as $k$-step nilspaces

In this section the main objectives are the following. First we shall extend
the notion of standard cubes to any group and show that these cubes form a
nilspace structure; see Proposition 2.8. We shall then seek alternative
characterizations of these cubes to obtain a generalization of Proposition
1.2. This will be obtained eventually as Proposition 2.28, and the task will
involve in particular a treatment of polynomial maps between filtered groups,
in Subsection 2.

####  1 Cubes on a general filtered group

Let us begin by recalling the definition of a filtered group. Given a group
$G$, the commutator $[g,h]$ of two elements $g,h\in G$ is defined by
$[g,h]=g^{-1}h^{-1}gh$. Thus $gh=hg[g,h]$. Given two subgroups $H_{1},H_{2}$
of $G$, we denote by $[H_{1},H_{2}]$ the subgroup of $G$ generated by all the
commutators $[h_{1},h_{2}]$, $h_{i}\in H_{i}$.

######  Definition 2.1.

A _filtration_ on a group $G$ is a sequence
$G_{\bullet}=(G_{i})_{i=0}^{\infty}$ of subgroups of $G$ satisfying $G\geq
G_{0}\geq G_{1}\geq\cdots$ and such that $[G_{i},G_{j}]\subset G_{i+j}$ for
all $i,j\geq 0$. We refer to $(G,G_{\bullet})$ as a _filtered group_. If some
term in the sequence $G_{\bullet}$ is the trivial subgroup
$\\{\mathrm{id}_{G}\\}$, then the _degree_ of the filtration, denoted by
$\deg(G_{\bullet})$, is the smallest integer $k$ such that
$G_{k+1}=\\{\mathrm{id}_{G}\\}$. The filtered group $(G,G_{\bullet})$ is then
said to be _of degree_ $k$.

We shall usually assume that $G_{0}=G_{1}=G$, except in some clearly indicated
places. Note that $G_{i}$ is a normal subgroup of $G$ for each $i\geq 0$. We
shall denote by $\pi_{i}$ the quotient homomorphism $G\to G/G_{i}$.
One can always take as a filtration the lower central series on $G$, and for
many purposes this filtration is the natural one to use. However, other
filtrations do arise naturally (see for instance Subsection 4), and in some
settings, typically quantitative ones, it is convenient to work with a general
filtration [6]. We shall do so here, thus for the remainder of this section we
suppose that an arbitrary filtration $G_{\bullet}$ has been fixed for the
given group $G$.

The general cube structure on a filtered group that we shall study originated
in the work of Host and Kra [11]. The definition relies on the following
notion, which builds up on Remark 1.4.

######  Definition 2.2 (Face groups).

Given a face $F$ in $\\{0,1\\}^{n}$ and an element $g\in G$, we write $g^{F}$
for the element of $G^{\\{0,1\\}^{n}}$ defined by $g^{F}(v)=g$ if $v\in F$ and
$\mathrm{id}_{G}$ otherwise. We define

$G_{(F)}=\\{g^{F}:g\in G_{\operatorname{codim}(F)}\\}\leq G^{\\{0,1\\}^{n}}.$

We refer to these subgroups $G_{(F)}$ as the _face groups_ in
$G^{\\{0,1\\}^{n}}$.

######  Definition 2.3 (Cubes on a filtered group).

Let $(G,G_{\bullet})$ be a filtered group. The group of $n$-cubes on
$(G,G_{\bullet})$, denoted by $\operatorname{C}^{n}(G_{\bullet})$, is the
subgroup of $G^{\\{0,1\\}^{n}}$ generated by the face groups.

Let us make a remark on the notation $\operatorname{C}^{n}(G_{\bullet})$. On
one hand this involves the same ‘$\operatorname{C}^{n}$’ as in Definition 2.1;
this will be justified by the main result of this subsection, which shows that
$G$ together with $(\operatorname{C}^{n}(G_{\bullet}))_{n\geq 0}$ is a
$k$-step nilspace with $k=\deg(G_{\bullet})$. On the other hand, as we shall
see, these cubes do depend significantly on the filtration, which is why the
notation specifies the filtration rather than just the group. For instance, if
$G$ is abelian and $G_{\bullet}$ is the lower central series $G_{0}=G_{1}=G$,
$G_{2}=\\{\mathrm{id}_{G}\\}$, then $\operatorname{C}^{n}(G_{\bullet})$
consists of the standard $n$-cubes seen in the previous section, while if
$G_{\bullet}$ is the degree-$k$ filtration $G_{0}=G_{1}=\cdots=G_{k}$,
$G_{k+1}=\\{\mathrm{id}_{G}\\}$ then we obtain a rather different but still
very natural nilspace structure (see Subsection 4).

Note that it is immediate from the definition that
$\operatorname{C}^{n}(G_{\bullet})$ is globally invariant under composition
with an automorphism of $\\{0,1\\}^{n}$.

We now give a more explicit description of these cubes, in terms of certain
‘coefficients’, that generalizes the expression
$\operatorname{c}(v)=x+v\scalebox{0.8}{$(1)$}h_{1}+\cdots+v\scalebox{0.8}{$(n)$}h_{n}$
for standard abelian cubes. To that end we use the following notion.

######  Definition 2.4 (Upper faces).

An _upper face_ of $\\{0,1\\}^{n}$ is a set of the form

$F(v)=\\{u\in\\{0,1\\}^{n}:\operatorname{Supp}(u)\supseteq\operatorname{Supp}(v)\\}$

for some $v\in\\{0,1\\}^{n}$. Note that
$\operatorname{codim}(F)=|\operatorname{Supp}(v)|$. We order the upper faces
of $\\{0,1\\}^{n}$ by writing $F(u)<F(v)$ if $u<v$ in the colex order, thus

$F_{0}=\\{0,1\\}^{n}=F(0^{n})<F((1,0,\ldots,0))<\cdots<F(1^{n})=\\{1^{n}\\}=F_{2^{n}-1}.$

Note that given two upper faces $F_{i}<F_{j}$, their intersection is an upper
face $F_{k}\geq F_{j}$, and
$\operatorname{codim}(F_{k})\leq\operatorname{codim}(F_{i})+\operatorname{codim}(F_{j})$.

As observed in Remark 1.4, any standard abelian cube is a unique sum of upper-
face group elements for faces of codimension at most 1. This can be
generalized as follows.

######  Lemma 2.5 (Unique factorization of cubes).

An element $\operatorname{c}\in G^{\\{0,1\\}^{n}}$ lies in
$\operatorname{C}^{n}(G_{\bullet})$ if and only if it has a unique
factorization of the form

$\operatorname{c}=g_{0}^{F_{0}}g_{1}^{F_{1}}\cdots g_{2^{n}-1}^{F_{2^{n}-1}},$
(3)

with $g_{i}\in G_{\operatorname{codim}(F_{i})}$ $\forall\,i$.

Recall that for each face $F\subset\\{0,1\\}^{n}$ there is unique data
$I\subset[n],\,x\in\\{0,1\\}^{I}$ such that

$F=\\{v\in\\{0,1\\}^{n}:v\scalebox{0.8}{$(i)$}=x\scalebox{0.8}{$(i)$}\;\;\forall\,i\in
I\\}.$

Upper faces are precisely those such that the corresponding element $x$ has
all entries equal to 1.

###### Proof.

First we claim that every element
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ can be written as a
product of face-group elements $g^{F}$ where $F$ is an _upper_ face, in other
words that $\operatorname{C}^{n}(G_{\bullet})$ is generated just by the upper-
face groups $G_{(F_{i})}$. This follows from the fact that every face-group
element $g^{F}$ is a product of upper-face group elements. Indeed, the
indicator function $1_{F}:\\{0,1\\}^{n}\to\\{0,1\\}$ can always be written as
an integer combination of indicator functions of upper faces of codimension at
most $\operatorname{codim}(F)$ (this can be proved by induction on
$\operatorname{codim}(F)$).

Our task is thus reduced to showing that any product of finitely many upper-
face group elements has the claimed unique factorization. This has been done
in several places, for instance in [5, Appendix E], but we recall the argument
here for completeness.

Observe that if $F<F^{\prime}$ are two upper faces, then
$G_{(F^{\prime})}\cdot G_{(F)}\subset G_{(F)}\cdot G_{(F^{\prime})}\cdot
G_{(F\cap F^{\prime})}$. Indeed, for every $g\in
G_{\operatorname{codim}(F)},g^{\prime}\in
G_{\operatorname{codim}(F^{\prime})}$, we have
${g^{\prime}}^{F^{\prime}}\;g^{F}=g^{F}\;{g^{\prime}}^{F^{\prime}}\;[{g^{\prime}}^{F^{\prime}},g^{F}]$,
where $[{g^{\prime}}^{F^{\prime}},g^{F}]=[g^{\prime},g]^{F^{\prime}\cap F}\in
G_{(F^{\prime}\cap F)}$. Using this fact we can obtain the desired
rearrangement by first moving all elements of $G_{(F_{0})}$ to the left of the
product, then using the closure of this group to combine them as a single
element $g_{0}^{F_{0}}$; next we move to the left all $G_{(F_{1})}$ elements
in the remaining product, to put them next to $g_{0}^{F_{0}}$, and combine
these to obtain $g_{1}^{F_{1}}$. Continuing this way yields the claimed
factorization.

Uniqueness follows from (3). Indeed, note first that
$\operatorname{Supp}(v_{i})\supset\operatorname{Supp}(v)$ implies $v_{i}\geq
v$. This implies that for any upper face $F(v_{i})$, the only other upper
faces $F(v)$ containing $v_{i}$ are those such that
$\operatorname{Supp}(v)\subset\operatorname{Supp}(v_{i})$ and $v<v_{i}$. From
(3) we can therefore determine the coefficients $g_{i}$ by induction as
follows:

$g_{0}=\operatorname{c}(v_{0}),\text{ and for each
}i>0,\;g_{i}=\Big{(}\prod_{\begin{subarray}{c}j<i:\;\\\
\operatorname{Supp}(v_{j})\subset\operatorname{Supp}(v_{i})\end{subarray}}g_{j}^{F_{j}}\Big{)}^{-1}\cdot\operatorname{c}\,(v_{i}),$
(4)

where the order from left to right in the product is increasing in the colex
order of $v_{j}$. ∎

For $j\in[n]$, let $\\{0,1\\}^{n}_{\leq\,j}:=\\{v\in\\{0,1\\}^{n}:|v|\leq
j\\}$.

######  Corollary 2.6.

If $G_{\bullet}$ has degree $d$, then for every
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$, every value
$\operatorname{c}(v)$ with $|v|>d$ is a word in the values
$\operatorname{c}(v)$ with $|v|\leq d$. In particular, the map
$\operatorname{c}$ is entirely determined by its restriction to
$\\{0,1\\}^{n}_{\leq\,d}$.

###### Proof.

Let $F_{i}$ be an upper face with $\operatorname{codim}(F_{i})=|v_{i}|\leq d$.
By (4), $g_{i}$ is a word in the values $\operatorname{c}(v_{j})$ with
$v_{j}\leq v_{i}$ with
$\operatorname{Supp}(v_{j})\subset\operatorname{Supp}(v_{i})$. On the other
hand any $g_{i}$ with $\operatorname{codim}(F_{i})>d$ is trivial by
assumption. Hence we are done, by uniqueness of (3). ∎

This corollary will be used to show that $\operatorname{C}^{n}(G_{\bullet})$
satisfies the corner-completion axiom. The following result shows that these
cubes also satisfy the composition axiom.

######  Lemma 2.7.

Let $\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ and let
$\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ be a morphism. Then
$\operatorname{c}\operatorname{\circ}\phi\in\operatorname{C}^{m}(G_{\bullet})$.

Given a map $g$ defined on $\\{0,1\\}^{n}$, for each $i\in\\{0,1\\}$ we define
$g(\cdot,i)$ on $\\{0,1\\}^{n-1}$ by $g(v,i)=g((v,i))$, where
$(v,i):=\big{(}v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n-1)$},i\big{)}\in\\{0,1\\}^{n}$.

###### Proof.

Let us first assume that $\phi$ is injective, thus $\phi$ is a bijection
$\\{0,1\\}^{m}\to\operatorname{Im}(\phi)$. For every $v\in\\{0,1\\}^{m}$, we
have

$\operatorname{c}(\phi(v))=\prod_{i=0}^{2^{n}-1}g_{i}^{F_{i}}(\phi(v))=\prod_{i:F_{i}\cap\operatorname{Im}(\phi)\neq\emptyset}g_{i}^{F_{i}}(\phi(v)).$

For each face $F_{i}$ such that
$F_{i}\cap\operatorname{Im}(\phi)\neq\emptyset$, with data $(I_{i},x_{i})$,
fix some $y_{i}=\phi(v_{i})\in\operatorname{Im}(\phi)$ such that
${y_{i}}|_{I_{i}}=x_{i}$. Let $\beta$ denote the function $J_{\phi}\to[m]$,
$j\mapsto i$ for all $j\in J_{\phi}(i)$ (recall (1)). We can then see that
$F^{\prime}_{i}:=\phi^{-1}(F_{i}\cap\operatorname{Im}(\phi))$ is a face of
$\\{0,1\\}^{m}$ of codimension at most $\operatorname{codim}(F_{i})$. Indeed,
$F_{i}^{\prime}$ is the face with data $I^{\prime}=\beta(J_{\phi}\cap I_{i})$
and $x^{\prime}=\phi^{-1}(y_{i})|_{I^{\prime}}={v_{i}}|_{I^{\prime}}$. (To see
this, note that $v\in F_{i}^{\prime}$ if and only if
$\phi(v)\in\operatorname{Im}(\phi)\cap F_{i}$, if and only if
$\phi(v)\scalebox{0.8}{$(j)$}=x_{i}\scalebox{0.8}{$(j)$}$ for each $j\in
I_{i}$, and this in turn holds if and only if
$\phi(v)\scalebox{0.8}{$(j)$}=y_{i}\scalebox{0.8}{$(j)$}$ for each $j\in
J_{\phi}\cap I_{i}$. Thus we have $v\in F_{i}^{\prime}$ if and only if
$\phi(v)|_{J_{\phi}\cap I_{i}}=\phi(v_{i})|_{J_{\phi}\cap I_{i}}$, and this
holds if and only if $v|_{I^{\prime}}=v_{i}|_{I^{\prime}}$.)
Therefore we have

$\operatorname{c}(\phi(v))=\prod_{i:F_{i}\cap\operatorname{Im}(\phi)\neq\emptyset}g_{i}^{F_{i}\cap\operatorname{Im}(\phi)}(\phi(v))=\prod_{i:F_{i}\cap\operatorname{Im}(\phi)\neq\emptyset}g_{i}^{\phi^{-1}(F_{i}\cap\operatorname{Im}(\phi))}(v)=\prod_{i:F_{i}\cap\operatorname{Im}(\phi)\neq\emptyset}g_{i}^{F_{i}^{\prime}}(v),$

hence $\operatorname{c}\operatorname{\circ}\phi$ is a product of face-group
elements and is therefore in $\operatorname{C}^{m}(G_{\bullet})$.

Now suppose that there is a set $S\subset[m]$ of indices of coordinates of $v$
on which $\phi(v)$ does not depend. We show by induction on $|S|$ that
$\operatorname{c}\operatorname{\circ}\phi\in\operatorname{C}^{m}(G_{\bullet})$.
The case $|S|=0$ corresponds to $\phi$ being injective. If $|S|>0$, fix any
$s\in S$, and note that it suffices to show that
$\operatorname{c}\operatorname{\circ}\phi\operatorname{\circ}\theta\in\operatorname{C}^{m}(G_{\bullet})$
for the automorphism $\theta$ permuting coordinate-indices $s$ and $m$, so we
may assume that $s=m$. Thus
$g^{\prime}:=\operatorname{c}\operatorname{\circ}\phi$ is a map
$\\{0,1\\}^{m}\to G$ such that $g^{\prime}(\cdot,0)=g^{\prime}(\cdot,1)$ is an
element of $\operatorname{C}^{m-1}(G_{\bullet})$ (by induction), with
coefficients $g_{i}$. We claim that
$g^{\prime}\in\operatorname{C}^{m}(G_{\bullet})$. In fact, $g^{\prime}$ can be
checked to be the $m$-cube with the following coefficients: if
$v_{i}^{\prime}\in\\{0,1\\}^{m}$ has $v_{i}^{\prime}\scalebox{0.8}{$(m)$}=0$,
then $g_{i}^{\prime}$ equals the coefficient $g_{j}$ corresponding to
$v_{j}=(v_{i}^{\prime}\scalebox{0.8}{$(1)$},v_{i}^{\prime}\scalebox{0.8}{$(2)$},\ldots,v_{i}^{\prime}\scalebox{0.8}{$(m-1)$})$,
and otherwise $g_{i}^{\prime}=\mathrm{id}_{G}$. ∎

We can now give the main result of this subsection.

######  Proposition 2.8.

Let $(G,G_{\bullet})$ be a filtered group. Then $G$ together with
$(\operatorname{C}^{n}(G_{\bullet}))_{n\geq 0}$ is a nilspace. It is a
$k$-step nilspace if and only if $\deg(G_{\bullet})\leq k$.

###### Proof.

Ergodicity holds if and only if the first two terms of $G_{\bullet}$ are
$G_{0}=G_{1}=G$. The composition axiom follows from Lemma 2.7. To prove the
completion axiom, we argue by induction on $d=\deg(G_{\bullet})$. For $d=1$
the group $G$ is abelian and completion was proved for Proposition 1.3. Let
$d>1$ and let $G^{\prime}=G/G_{d}$, with filtration
$G^{\prime}_{\bullet}=(G_{i}/G_{d})$ of degree at most $d-1$. By induction
$\operatorname{C}^{n}(G^{\prime}_{\bullet})$ satisfies the completion axiom.
Suppose that $\operatorname{c}^{\prime}$ is an $n$-corner on $G$. Then
$\pi_{d}\operatorname{\circ}\operatorname{c}^{\prime}$ is clearly an
$n$-corner on $G/G_{d}$, so it has a completion, which by surjectivity of
$\operatorname{c}\mapsto\pi_{d}\operatorname{\circ}\operatorname{c}$ equals
$\pi_{d}\operatorname{\circ}\tilde{\operatorname{c}}$ for some
$\tilde{\operatorname{c}}\in\operatorname{C}^{n}(G_{\bullet})$. We shall now
produce a sequence
$\operatorname{c}_{0},\operatorname{c}_{1},\ldots,\operatorname{c}_{d}$ of
elements of $\operatorname{C}^{n}(G_{\bullet})$ such that
$\operatorname{c}_{j}$ agrees with $\operatorname{c}^{\prime}$ on all of
$\\{0,1\\}^{n}_{\leq\,j}$.

First, we obtain $\operatorname{c}_{0}$ such that
$\operatorname{c}_{0}(0^{n})=\operatorname{c}^{\prime}(0^{n})$ and
$\pi_{d}\operatorname{\circ}\operatorname{c}_{0}=\pi_{d}\operatorname{\circ}\operatorname{c}^{\prime}$
(just left-multiplying every entry of $\tilde{\operatorname{c}}$ by
$\operatorname{c}(0^{n})\,\tilde{\operatorname{c}}(0^{n})^{-1}\in G_{d}$). Now
for $1\leq j\leq d$ suppose that a cube $\operatorname{c}_{j-1}$ has been
constructed such that
$\pi_{d}\operatorname{\circ}\operatorname{c}_{j-1}=\pi_{d}\operatorname{\circ}\operatorname{c}^{\prime}$
and $\operatorname{c}_{j-1}(v)=\operatorname{c}^{\prime}(v)$ for all
$v\in\\{0,1\\}^{n}_{<\,j}$. We shall obtain $\operatorname{c}_{j}$ by
correcting the differences between $\operatorname{c}_{j-1}$ and
$\operatorname{c}^{\prime}$ at elements $v$ with $|v|=j$. Note that
$\pi_{d}\operatorname{\circ}\operatorname{c}_{j-1}=\pi_{d}\operatorname{\circ}\operatorname{c}^{\prime}$
implies that these discrepancies involve only elements of $G_{d}$, that is for
any such $v$ there is $g_{v}\in G_{d}\subset G_{j}$ such that
$\operatorname{c}_{j-1}(v)g_{v}=\operatorname{c}(v)$. Letting $F(v)$ be the
corresponding upper face (of codimension $j$), we therefore have
$g_{v}^{F(v)}\in G_{(F(v))}$. We define
$\operatorname{c}_{j}=\operatorname{c}_{j-1}\cdot\prod_{v:|v|=j}g_{v}^{F(v)}$.
This lies in $\operatorname{C}^{n}(G_{\bullet})$, and agrees with
$\operatorname{c}$ on $\\{0,1\\}^{n}_{\leq\,j}$ as required (note that each
$F(v)$ intersects $\\{0,1\\}^{n}_{\leq\,j}$ only at $v$).

We have thus obtained a cube
$\operatorname{c}_{d}\in\operatorname{C}^{n}(G_{\bullet})$ that agrees with
$\operatorname{c}^{\prime}$ on all of $\\{0,1\\}^{n}_{\leq\,d}$. This
agreement is now easily extended to every $(n-1)$-face containing $0^{n}$ (and
thus to all of $\\{0,1\\}^{n}\setminus\\{1^{n}\\}$) applying composition and
Corollary 2.6. ∎

The general nilspace structure described in Proposition 2.8 is the one that we
shall consider by default on a filtered group. Thus, from now on, given such a
group $(G,G_{\bullet})$, by “a cube on $G$” we shall always mean an element of
$\operatorname{C}^{n}(G_{\bullet})$ for some $n\geq 0$. There is, however, a
slightly more general version of this cube structure, that can also be useful.

######  Definition 2.9.

Let $(G,G_{\bullet})$ be a filtered group, let $i_{1},\dots,i_{n}$ be non-
negative integers, and let $F=F(v)$ be an upper face in $\\{0,1\\}^{n}$. We
denote by $G_{(F)}^{(i_{1},\ldots,i_{n})}$ the subgroup of $G^{\\{0,1\\}^{n}}$
consisting of elements $g^{F}$ where $g\in
G_{\sum_{j\in\operatorname{Supp}v}i_{j}}$. We denote by
$\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$ the subgroup of
$G^{\\{0,1\\}^{n}}$ generated by the groups $G_{(F)}^{(i_{1},\ldots,i_{n})}$.

This notion appeared already in [7, Definition B.2]. (Note that
$\operatorname{C}^{n}(G_{\bullet})=\operatorname{C}^{n}_{(1,\dots,1)}(G_{\bullet})$.)
Arguing as in the proof of Lemma 2.5, we obtain the following result.

######  Lemma 2.10.

An element $\operatorname{c}\in G^{\\{0,1\\}^{n}}$ lies in
$\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$ if and only if it
has a unique factorization of the form

$\operatorname{c}=g_{0}^{F_{0}}g_{1}^{F_{1}}\cdots g_{2^{n}-1}^{F_{2^{n}-1}},$
(5)

with $g_{i}\in G_{\sum_{j\in\operatorname{Supp}v_{i}}i_{j}}$ $\forall\,i$.

We now wish to complete the picture concerning the cube structures
$\operatorname{C}^{n}(G_{\bullet})$ by showing that they satisfy an analogue
of Proposition 1.2. We have already done this for part $(i)$ of that
proposition (in Lemma 2.5). For part $(ii)$, we need an appropriate analogue
of an affine homomorphism $\mathbb{Z}^{n}\to G$. A suitable notion turns out
to be that of a _polynomial map_ from $\mathbb{Z}^{n}$ to $G$. This motivates
the discussion of general polynomial maps between filtered groups, given in
the next subsection. We shall see in particular that these maps are precisely
the morphisms that preserve the nilspace structures formed by these cubes.

####  2 Nilspace morphisms between filtered groups: polynomial maps

######  Definition 2.11 (Nilspace morphism).

Let $\operatorname{X},\operatorname{Y}$ be nilspaces. A map
$g:\operatorname{X}\to\operatorname{Y}$ is a _nilspace morphism_ if for every
$n\geq 0$, for every cube
$\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X})$ we have
$g\operatorname{\circ}\operatorname{c}\in\operatorname{C}^{n}(\operatorname{Y})$.
We denote the set of these morphisms by
$\hom(\operatorname{X},\operatorname{Y})$.

In the case of two filtered groups $(G,G_{\bullet})$, $(H,H_{\bullet})$ we
shall denote the set of morphisms between the corresponding nilspaces by
$\hom(H_{\bullet},G_{\bullet})$.

######  Definition 2.12 (Polynomial maps).

Let $(G,G_{\bullet})$ and $(H,H_{\bullet})$ be filtered groups. Given a map
$g:H\to G$, and $h\in H$, we define the map $\partial_{h}g:H\to G$ by
$\partial_{h}g(x)=g(x)^{-1}g(xh)$. We say that $g$ is a _polynomial map_
(adapted to $H_{\bullet},G_{\bullet}$) if for every sequence of non-negative
integers $i_{1},i_{2},\ldots,i_{n}$ and elements $h_{j}\in H_{i_{j}}$,
$j\in[n]$, we have
$\partial_{h_{1}}\partial_{h_{2}}\cdots\partial_{h_{n}}g(x)\in
G_{i_{1}+\cdots+i_{n}}$ for all $x\in H$. The set of such maps is denoted by
$\operatorname{poly}(H_{\bullet},G_{\bullet})$.

######  Example 2.13.

For $H=\mathbb{Z}$ with the lower central series, the corresponding set of
polynomial maps is denoted by $\operatorname{poly}(\mathbb{Z},G_{\bullet})$
and its elements are called _polynomial sequences_ on $G$ (one can add
‘adapted to $G_{\bullet}$’ when the filtration needs to be specified) [6, 16].
If $H$ is an abelian group with lower central series $H_{\bullet}$, and $G$ is
abelian with $G_{\bullet}$ being the degree $d$ filtration
$G_{0}=\cdots=G_{d}=G$, $G_{d+1}=\\{\mathrm{id}_{G}\\}$, then
$\operatorname{poly}(H_{\bullet},G_{\bullet})$ is the set of maps $g$ such
that for every $x\in H,h=(h_{1},\ldots,h_{d+1})\in H^{d+1}$ we have
$\sum_{v\in\\{0,1\\}^{d+1}}(-1)^{|v|}g(x+v\cdot h)=0$. For
$H=\mathbb{R}/\mathbb{Z}$ these are the ‘globally polynomial phase functions’,
familiar in arithmetic combinatorics; see for instance [4, §3].

The main result of this section is the following important fact.

######  Theorem 2.14.

Let $(G,G_{\bullet})$, $(H,H_{\bullet})$ be filtered groups. Then

$\hom(H_{\bullet},G_{\bullet})=\operatorname{poly}(H_{\bullet},G_{\bullet}).$
(6)

This result yields a functor from the category of filtered groups with
polynomial maps to the category of nilspaces with nilspace morphisms, namely
the functor that sends a filtered group $(G,G_{\bullet})$ to the nilspace
$\big{(}G,(\operatorname{C}^{n}(G_{\bullet}))_{n\geq 0}\big{)}$ and sends a
map in $\operatorname{poly}(H_{\bullet},G_{\bullet})$ to the same map viewed
as a morphism in $\hom(H_{\bullet},G_{\bullet})$. (This functor is forgetful;
see Remark 4.2.)

Theorem 2.14 has the following remarkable consequence.

######  Corollary 2.15.

For any filtered groups $(G,G_{\bullet})$, $(H,H_{\bullet})$, the set
$\operatorname{poly}(H_{\bullet},G_{\bullet})$ with pointwise multiplication
is a group.

This corollary follows from (6) together with the fact that
$\hom(H_{\bullet},G_{\bullet})$ is a group under pointwise multiplication,
which is a special case of the following lemma.

######  Lemma 2.16.

Let $\operatorname{X}$ be a nilspace and let $(G,G_{\bullet})$ be a filtered
group. Then $\hom(\operatorname{X},G_{\bullet})$ is a group under pointwise
multiplication.

This lemma follows clearly from the fact that the cubes
$\operatorname{C}^{n}(G_{\bullet})$ form a group.
Corollary 2.15 was first established for polynomial sequences. This special
case is known as the Lazard-Leibman theorem, it was proved by Leibman in [16],
and earlier, for Lie groups, by Lazard [15]. The more general case of
polynomial maps between groups was then proved by Leibman in [17]. These
original proofs did not go via the charaterization of polynomial maps as
nilspace morphisms, as we do here. A special case of this characterization
appeared in [6, Proposition 6.5], and the general version appeared (with
slightly different language) in [7, Theorem B.3].
Another nice consequence of Theorem 2.14 is the following result, which gives
the desired generalization of part (ii) of Proposition 1.2.

######  Corollary 2.17.

A map $\operatorname{c}:\\{0,1\\}^{n}\to G$ is in
$\operatorname{C}^{n}(G_{\bullet})$ if and only if it extends to a polynomial
map $g\in\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$.

Note that throughout this section the implicit filtration on $\mathbb{Z}^{n}$
is just the lower central series $G_{0}=G_{1}=\mathbb{Z}^{n}$,
$G_{2}=\\{0^{n}\\}$. Let us establish this corollary before turning to the
proof of the theorem.

###### Proof of Corollary 2.17.

The ‘if’ direction is the easiest. Note first that the map $\theta$ that
embeds $\\{0,1\\}^{n}$ in $\mathbb{Z}^{n}$ in the natural way is trivially in
$\operatorname{C}^{n}(\mathbb{Z}^{n})$. Hence if
$\operatorname{c}=g\operatorname{\circ}\theta$ for
$g\in\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$, then, since by the
theorem $g$ is a morphism of cubes, we have
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$.

For the ‘only if’ direction, given an $n$-cube $\operatorname{c}$, we write
the unique factorization

$\operatorname{c}(v)=g_{0}^{F_{0}}(v)\;g_{1}^{F_{1}}(v)\;\cdots\;g_{2^{n}-1}^{F_{2^{n}-1}}(v),$

and we then define the map $g:\mathbb{Z}^{n}\to G$ by

$g(\mathbf{t})=g(t_{1},\ldots,t_{n})=\prod_{j=0}^{2^{n}-1}g_{j}^{\binom{\mathbf{t}}{v_{j}}},$

where
$\binom{\mathbf{t}}{v_{j}}:=\binom{t_{1}}{v_{j}\scalebox{0.8}{$(1)$}}\binom{t_{2}}{v_{j}\scalebox{0.8}{$(2)$}}\cdots\binom{t_{n}}{v_{j}\scalebox{0.8}{$(n)$}}$,
and where $v_{j}$ is the element determining the upper face $F_{j}=F(v_{j})$
(recall Definition 2.4). Note that the restriction of $g$ to $\\{0,1\\}^{n}$
equals $\operatorname{c}$. Observe also that each map $\mathbf{t}\mapsto
g_{j}^{\binom{\mathbf{t}}{v_{j}}}$ is in
$\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$ (this is easily seen from
the definition) and so, by the group property for
$\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$ (which follows from Theorem
2.14) we have $g\in\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$ as well. ∎

To prove Theorem 2.14, we shall use the following construction, which will
also play a crucial role in the next chapter.

######  Definition 2.18 (Arrows).

Let $\operatorname{X}$ be a set, let $n,k\in\mathbb{N}$, and let us denote
elements of $\\{0,1\\}^{n+k}$ by pairs $(v,w)$,
$v\in\\{0,1\\}^{n},w\in\\{0,1\\}^{k}$. Given two maps
$\operatorname{c}_{0},\operatorname{c}_{1}:\\{0,1\\}^{n}\to\operatorname{X}$,
we define the _$n$ -dimensional $k$-arrow_, denoted by
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$, to be the
following map:

$\displaystyle\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}\quad:\quad\\{0,1\\}^{n+k}$
$\displaystyle\to$ $\displaystyle\quad\operatorname{X}$
$\displaystyle(v,w)\hskip 14.22636pt$ $\displaystyle\mapsto$
$\displaystyle\left\\{\begin{array}[]{lr}\operatorname{c}_{0}(v),&w\neq
1^{k}\\\ \operatorname{c}_{1}(v),&w=1^{k}\end{array}\right..$

Usually, the dimension of the cubes
$\operatorname{c}_{0},\operatorname{c}_{1}$ (and hence of the arrow
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$) will be clear
from the context, and we shall then call
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$ just ‘the
$k$-arrow of $\operatorname{c}_{0},\operatorname{c}_{1}$’.

We shall see in Chapter 2 that arrows yield a useful general construction of
new nilspaces from old ones. For now we shall use arrows just for cubes on
filtered groups.
The following result provides a convenient way to check whether an arrow on a
filtered group is itself a cube. Given a filtration $G_{\bullet}$ and
$\ell\in\mathbb{N}$, we denote by $G_{\bullet}^{+\ell}$ the shifted
filtration, with $i$-th term $G_{i+\ell}$.

######  Lemma 2.19.

Let $(G,G_{\bullet})$ be a filtered group, let
$\operatorname{c}_{0},\operatorname{c}_{1}:\\{0,1\\}^{n}\to G$, and let
$k\in\mathbb{N}$. Then
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$ lies in
$\operatorname{C}^{n+k}_{(i_{1},\ldots,i_{n+k})}(G_{\bullet})$ if and only if
$\operatorname{c}_{0}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$
and
$\operatorname{c}_{0}^{-1}\operatorname{c}_{1}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet}^{+\ell})$,
where $\ell=\sum_{j=n+1}^{n+k}i_{j}$.

###### Proof.

If $f=\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$ is in
$\operatorname{C}^{n+k}_{(i_{1},\ldots,i_{n+k})}(G_{\bullet})$ then by (5) we
have that
$\operatorname{c}_{0}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$.
Moreover, from (5) and the definition of $f$ we see that every coefficient
$g_{(v,w)}$ of $f$ corresponding to an upper face $F((v,w))$ of
$\\{0,1\\}^{n+k}$ with $w\neq 0^{k},1^{k}$ is trivial. It follows that for
$v\in\\{0,1\\}^{n}$ we have

$\operatorname{c}_{0}^{-1}(v)\operatorname{c}_{1}(v)=f(v,0^{k})^{-1}\,f(v,1^{k})=\prod_{u\in\\{0,1\\}^{n}}g_{(u,1^{k})}^{F((u,1^{k}))}(v,1^{k}),$

where $g_{(u,1^{k})}\in
G_{\sum_{j\in\operatorname{Supp}u}i_{j}+\sum_{j=n+1}^{n+k}i_{j}}$, so
$\operatorname{c}_{0}^{-1}\,\operatorname{c}_{1}$ is indeed in
$\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet}^{+\ell})$ as claimed.

For the converse, note that the factorizations for $\operatorname{c}_{0}$ and
$\operatorname{c}_{0}^{-1}\,\operatorname{c}_{1}$ given by (5) combine to give
a factorization of the form (5) for
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$. ∎

We now prove Theorem 2.14 in two steps.

######  Lemma 2.20.

We have
$\hom(H_{\bullet},G_{\bullet})\subset\operatorname{poly}(H_{\bullet},G_{\bullet})$.

###### Proof.

We have to show that for every $g\in\hom(H_{\bullet},G_{\bullet})$, for every
integer $n\geq 0$, integers $i_{1},\dots,i_{n}\geq 0$ and elements $h_{j}\in
H_{i_{j}}$, $j\in[n]$, we have $\partial_{h_{1}}\cdots\partial_{h_{n}}g(x)\in
G_{i_{1}+\cdots+i_{n}}$ for all $x\in H$. This is clear for $n=0$. For $n>0$,
by induction it suffices to show that for every $i\geq 0$ and $h\in H_{i}$ we
have $\partial_{h}g\in\hom(H_{\bullet},G_{\bullet}^{+i})$. Given any
$\operatorname{c}\in\operatorname{C}^{m}(H_{\bullet})$, we have by Lemma 2.19
that $\langle\operatorname{c},\operatorname{c}\cdot
h\rangle_{i}\in\operatorname{C}^{m+i}(H_{\bullet})$. We then have
$g\operatorname{\circ}\langle\operatorname{c},\operatorname{c}\cdot
h\rangle_{i}\in\operatorname{C}^{m+i}(G_{\bullet})$. But
$g\operatorname{\circ}\langle\operatorname{c},\operatorname{c}\cdot
h\rangle_{i}=\langle
g\operatorname{\circ}\operatorname{c},g\operatorname{\circ}(\operatorname{c}\cdot
h)\rangle_{i}$, and then this being in $\operatorname{C}^{m+i}(G_{\bullet})$
and $g\operatorname{\circ}\operatorname{c}$ being in
$\operatorname{C}^{m}(G_{\bullet})$ implies by Lemma 2.19 that
$(g\operatorname{\circ}\operatorname{c})^{-1}\,\cdot\,g\operatorname{\circ}(\operatorname{c}\cdot
h)\in\operatorname{C}^{m}(G_{\bullet}^{+i})$. Since
$(g\operatorname{\circ}\operatorname{c})^{-1}\,\cdot\,g\operatorname{\circ}(\operatorname{c}\cdot
h)=(\partial_{h}g)\operatorname{\circ}\operatorname{c}$, the proof is
complete. ∎

We now prove a result which implies the opposite inclusion
$\hom(H_{\bullet},G_{\bullet})\supset\operatorname{poly}(H_{\bullet},G_{\bullet})$.

######  Lemma 2.21.

For every integer $n\geq 0$ the following holds. Given any filtered groups
$(G,G_{\bullet})$, $(H,H_{\bullet})$ and any map
$g\in\operatorname{poly}(H_{\bullet},G_{\bullet})$, for any
$i_{1},\ldots,i_{n}\geq 0$ and any
$\operatorname{c}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(H_{\bullet})$
we have
$g\operatorname{\circ}\operatorname{c}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$.

Thus polynomial maps conserve in fact the more general cube structures from
Definition 2.9. Note that the last two lemmas combined tell us that nilspace
morphisms also conserve these structures.

###### Proof.

(We follow an argument from [7, Appendix B].) We argue again by induction on
$n$. Given $n>0$ let
$\operatorname{c}\in\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(H_{\bullet})$
and note that, letting
$\operatorname{c}_{0}:=\operatorname{c}(\cdot,0),\operatorname{c}_{1}:=\operatorname{c}(\cdot,1)$,
by Lemma 2.19 we have
$\operatorname{c}_{0}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet})$
and
$\operatorname{c}_{0}^{-1}\operatorname{c}_{1}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet}^{+i_{n}})$.
We want to show that $g\operatorname{\circ}\operatorname{c}$ lies in
$\operatorname{C}^{n}_{(i_{1},\ldots,i_{n})}(G_{\bullet})$. Since
$(g\operatorname{\circ}\operatorname{c})(\cdot,t)=g\operatorname{\circ}\operatorname{c}_{t}$
for $t=0,1$, and by induction
$g\operatorname{\circ}\operatorname{c}_{0}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(G_{\bullet})$,
by Lemma 2.19 it suffices to show that
$(g\operatorname{\circ}\operatorname{c}_{0})^{-1}g\operatorname{\circ}\operatorname{c}_{1}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(G_{\bullet}^{+i_{n}})$.
Now
$\operatorname{c}_{0}^{-1}\operatorname{c}_{1}=h_{1}^{F_{1}}h_{2}^{F_{2}}\cdots
h_{t}^{F_{t}}$ for upper-face group elements
$h_{j}^{F_{j}}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet}^{+i_{n}})$.
Moreover, note that
$(g\operatorname{\circ}\operatorname{c}_{0})^{-1}g\operatorname{\circ}\operatorname{c}_{1}$
has the telescoping expansion

$\displaystyle(g\operatorname{\circ}\operatorname{c}_{0})^{-1}\;\;g\operatorname{\circ}\big{(}\operatorname{c}_{0}h_{1}^{F_{1}}\big{)}\;g\operatorname{\circ}\big{(}\operatorname{c}_{0}h_{1}^{F_{1}}\big{)}^{-1}\;\;g\operatorname{\circ}\Big{(}\big{(}\operatorname{c}_{0}h_{1}^{F_{1}}\big{)}h_{2}^{F_{1}}\Big{)}\;g\operatorname{\circ}\Big{(}\big{(}\operatorname{c}_{0}h_{1}^{F_{1}}\big{)}h_{2}^{F_{1}}\Big{)}^{-1}\cdots$
$\displaystyle\hskip 113.81102pt\cdots
g\operatorname{\circ}\Big{(}\operatorname{c}_{0}h_{1}^{F_{1}}h_{2}^{F_{2}}\cdots
h_{t-1}^{F_{t-1}}\Big{)}\;g\operatorname{\circ}\Big{(}\operatorname{c}_{0}h_{1}^{F_{1}}h_{2}^{F_{2}}\cdots
h_{t-1}^{F_{t-1}}\Big{)}^{-1}g\operatorname{\circ}\operatorname{c}_{1}.$

Therefore, it suffices to show that for every cube
$\operatorname{c}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet})$
and every upper-face-group element
$h^{F}\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet}^{+i_{n}})$,
we have
$(g\operatorname{\circ}\operatorname{c})^{-1}g\operatorname{\circ}(\operatorname{c}h^{F})\in\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(G_{\bullet}^{+i_{n}})$
.
Let
$f(v):=(g\operatorname{\circ}\operatorname{c})^{-1}g\operatorname{\circ}(\operatorname{c}h^{F})(v)$,
and note that this equals $\partial_{h}g(\operatorname{c}(v))$ for $v\in F$
and $\mathrm{id}_{G}$ otherwise. Let $w\in\\{0,1\\}^{n-1}$ be the element with
support $I$ of size $k$ such that $F=F(w)$. By composing with an automorphism
(permuting the indices $i_{j}$ accordingly) we can assume that $I=[n-k,n-1]$
and then $f$ is the $(n-1)$-dimensional arrow
$\langle\mathrm{id}_{G},(\partial_{h}g)\operatorname{\circ}\operatorname{c}\rangle_{k}$.
By Lemma 2.19, $f$ is therefore in
$\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(G_{\bullet}^{+i_{n}})$ if and
only if $(\partial_{h}g)\operatorname{\circ}\operatorname{c}$ restricted to
$\\{0,1\\}^{n-1-k}$ lies in
$\operatorname{C}^{n-1-k}_{(i_{1},\ldots,i_{n-1-k})}(G_{\bullet}^{+i_{n}+\sum_{j\in
I}i_{j}})$. Since $h\in G_{\sum_{j\in I}i_{j}}$, we have that $\partial_{h}g$
is a $G_{\sum_{j\in I}i_{j}}$-valued polynomial map on $H$, so by the
induction hypothesis $\partial_{h}g$ maps
$\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(H_{\bullet}^{+i_{n}})$ into
$\operatorname{C}^{n-1}_{(i_{1},\ldots,i_{n-1})}(G_{\bullet}^{+i_{n}+\sum_{j\in
I}i_{j}})$ under composition. In particular, since $\operatorname{c}$
restricted to $\\{0,1\\}^{n-1-k}$ lies in
$\operatorname{C}^{n-k-1}_{(i_{1},\ldots,i_{n-1})|_{[n-1]\setminus
I}}(H_{\bullet}^{+i_{n}})$, we have
$(\partial_{h}g)\operatorname{\circ}\operatorname{c}|_{\\{0,1\\}^{n-1-k}}\in\operatorname{C}^{n-1-k}_{(i_{1},\ldots,i_{n-1-k})}(G_{\bullet}^{+i_{n}+\sum_{j\in
I}i_{j}})$ as required. ∎

In the next subsection we complete the generalization of Proposition 1.2 by
giving the analogue of part $(iii)$. This analogue will consist in a
characterization of cubes on filtered groups in terms of certain equations
that generalize (2). Recall that Corollary 2.6 tells us that if $G$ has degree
$d$ then the value of every $(d+1)$-cube at every point $v\in\\{0,1\\}^{d+1}$
is a word in the values at the other points. The equations just mentioned will
tell us explicitly what this word is. This third characterization of cubes on
filtered groups will be very convenient to describe an important type of
nilspace structure on an abelian group, namely the so-called _degree- $k$
structure_ (see Subsection 4).

####  3 Defining cubes on a filtered group in terms of equations

We shall use the following generalization of the function $\sigma_{2}$ from
Definition 1.1.

######  Definition 2.22.

Let $G$ be a group. We define the function $\sigma_{n}:G^{\\{0,1\\}^{n}}\to G$
recursively, for each integer $n\geq 0$, as follows. Let $\sigma_{0}(g):=g$
for any $g\in G=G^{\\{0,1\\}^{0}}$, and for $n>0$, given $g:\\{0,1\\}^{n}\to
G$, let

$\sigma_{n}(g):=\sigma_{n-1}\big{(}g(\cdot,1)\big{)}^{-1}\;\sigma_{n-1}\big{(}g(\cdot,0)\big{)}.$
(8)

We have $\sigma_{1}(g)=g_{1}^{-1}\,g_{0}$,
$\sigma_{2}(g)=g_{01}^{-1}\;g_{11}\;g_{10}^{-1}\;g_{00}$, and so on. Expanding
the product $\sigma_{n}(g)$ and reading right to left (say), we see that it is
an alternating product taken along a Gray code (or reflective binary code) on
the discrete $n$-cube.

######  Definition 2.23.

The _Gray order_ $\gamma_{n}:\\{0,1\\}^{n}\to[0,2^{n}-1]$ is defined by
induction on $n$ as follows. For $n=1$ we set
$\gamma_{1}(0)=0,\gamma_{1}(1)=1$. For $n>1$ we set
$\gamma_{n}(v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n)$})=\gamma_{n-1}(v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n-1)$})$
if $v\scalebox{0.8}{$(n)$}=0$, and if $v\scalebox{0.8}{$(n)$}=1$ we set
$\gamma_{n}(v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n)$})=2^{n}-\gamma_{n-1}(v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n-1)$})$
.

We note the expression
$\sigma_{n}(g)=\prod_{j=0}^{2^{n}-1}g(\gamma_{n}^{-1}(j))^{(-1)^{j}}$. We will
have more use below for expression (8), which is helpful in some inductive
arguments. In fact, that expression leads to a third viewpoint on $\sigma_{n}$
described in the following lemma, which will be crucial below.

######  Lemma 2.24.

Let $G$ be a group and let $g:\mathbb{Z}^{n}\to G$. Then

$\sigma_{n}(g)=\partial_{e_{n}}\partial_{e_{n-1}}\cdots\partial_{e_{1}}g(0^{n}).$
(9)

Here $\\{e_{i}:i\in[n]\\}$ is the standard basis of $\mathbb{R}^{n}$.

###### Proof.

For $n=1$ we clearly have $\sigma_{1}(g)=\partial_{e_{1}}g(0)$. Suppose that
$n>1$, suppose the claim holds for $m<n$ and let $g:\mathbb{Z}^{n}\to G$. Then
by induction $g(\cdot,0)$ and $g(\cdot,1)$ satisfy

$\sigma_{n-1}(g(\cdot,i))=\partial_{e_{n-1}}\cdots\partial_{e_{1}}g(0^{n-1},i),\quad
i=0,1.$

By (8) we then have

$\displaystyle\sigma_{n}(g)$ $\displaystyle=$
$\displaystyle\sigma_{n-1}(g(\cdot,1))^{-1}\;\sigma_{n-1}(g(\cdot,0))$
$\displaystyle=$
$\displaystyle\big{(}\partial_{e_{n-1}}\cdots\partial_{e_{1}}g(0^{n-1},1)\;\big{)}^{-1}\;\partial_{e_{n-1}}\cdots\partial_{e_{1}}g(0^{n-1},0)$
$\displaystyle=$
$\displaystyle\partial_{e_{n}}\partial_{e_{n-1}}\cdots\partial_{e_{1}}g(0^{n}).$

∎

Note that the increasing sequence of elements of $\\{0,1\\}^{n}$ taken in the
Gray order gives a Hamiltonian path in the graph of 1-faces on this cube,
starting from $0^{n}$.

The main result of this section is the following.

######  Proposition 2.25.

Let $(G,G_{\bullet})$ be a filtered group. We have
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ if and only if for
every $m$-face map $\phi$ into $\\{0,1\\}^{n}$ we have
$\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)\in G_{m}$.

Thus, the $n$-cubes on $(G,G_{\bullet})$ are the maps
$\operatorname{c}:\\{0,1\\}^{n}\to G$ that satisfy the equation

$\pi_{m}\operatorname{\circ}\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)=\mathrm{id}$

for every face map $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$.

We split the proof of Proposition 2.25 into two parts. Firstly, we establish
the following stronger statement in one direction.

######  Lemma 2.26.

If $\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ then for every
morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ we have
$\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)\in G_{m}$.

###### Proof.

Lemma 2.7 implies
$\operatorname{c}\operatorname{\circ}\phi\in\operatorname{C}^{m}(G_{\bullet})$.
Corollary 2.17 gives $g\in\operatorname{poly}(\mathbb{Z}^{m},G)$ such that
$\operatorname{c}\operatorname{\circ}\phi(v)=g(v)$ for all
$v\in\\{0,1\\}^{m}$. The result then follows from (9) and the definition of
$\operatorname{poly}(\mathbb{Z}^{m},G_{\bullet})$. ∎

Secondly, we have the following result in the converse direction.

######  Lemma 2.27.

Suppose that $\operatorname{c}:\\{0,1\\}^{n}\to G$ satisfies
$\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)\in G_{m}$ for every
$m$-face map $\phi$ into $\\{0,1\\}^{n}$. Then
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$.

###### Proof.

We argue by induction on the degree $d$ of $G_{\bullet}$. For $d=1$, $G$ is
abelian and the claim follows from (the proof of) Proposition 1.2, so let
$d>1$ and suppose that the lemma holds for every filtered group of degree less
than $d$. Let $G^{\prime}=G/G_{d}$, and let
$G^{\prime}_{\bullet}=(G_{i}/G_{d})_{i\geq 0}$, a filtration of degree at most
$d-1$. Suppose that $\operatorname{c}:\\{0,1\\}^{n}\to G$ satisfies the
premise of the lemma. Then $\pi_{d}\operatorname{\circ}\operatorname{c}$
satisfies the premise in $(G^{\prime},G^{\prime}_{\bullet})$, so
$\pi_{d}\operatorname{\circ}\operatorname{c}\in\operatorname{C}^{n}(G^{\prime}_{\bullet})$.

Note that $\operatorname{c}\mapsto\pi_{d}\operatorname{\circ}\operatorname{c}$
is a surjection
$\operatorname{C}^{n}(G_{\bullet})\to\operatorname{C}^{n}(G^{\prime}_{\bullet})$
(this follows from the surjectivity of $\pi_{d}:G\to G^{\prime}$ combined with
(3), or just with Definition 2.3). Hence there exists
$\operatorname{c}^{\prime}\in\operatorname{C}^{n}(G_{\bullet})$ such that
$\pi_{d}\operatorname{\circ}\operatorname{c}=\pi_{d}\operatorname{\circ}\operatorname{c}^{\prime}$.
We shall now produce a sequence
$\operatorname{c}_{0},\operatorname{c}_{1},\ldots,\operatorname{c}_{d}$ of
elements of $\operatorname{C}^{n}(G_{\bullet})$ such that
$\operatorname{c}_{j}$ agrees with $\operatorname{c}$ on all of
$\\{0,1\\}^{n}_{\leq\,j}$.

First, we obtain $\operatorname{c}_{0}$ such that
$\operatorname{c}_{0}(0^{n})=\operatorname{c}(0^{n})$ and
$\pi_{d}\operatorname{\circ}\operatorname{c}_{0}=\pi_{d}\operatorname{\circ}\operatorname{c}$,
by left-multiplying every entry of $\operatorname{c}^{\prime}$ by
$\operatorname{c}(0^{n})\,\operatorname{c}^{\prime}(0^{n})^{-1}\in G_{d}$. Now
for $1\leq j\leq d$ suppose that $\operatorname{c}_{j-1}$ has been constructed
such that
$\pi_{d}\operatorname{\circ}\operatorname{c}_{j-1}=\pi_{d}\operatorname{\circ}\operatorname{c}$
and $\operatorname{c}_{j-1}(v)=\operatorname{c}(v)$ for all
$v\in\\{0,1\\}^{n}_{\leq j-1}$. We shall obtain $\operatorname{c}_{j}$ by
correcting the discrepancies between $\operatorname{c}_{j-1}$ and
$\operatorname{c}$ at elements $v$ with $|v|=j$. Note that
$\pi_{d}\operatorname{\circ}\operatorname{c}_{j-1}=\pi_{d}\operatorname{\circ}\operatorname{c}$
implies that these discrepancies involve only elements of $G_{d}$, that is for
any such $v$ there is $g_{v}\in G_{d}\subset G_{j}$ such that
$\operatorname{c}_{j-1}(v)g_{v}=\operatorname{c}(v)$. Letting $F(v)$ be the
corresponding upper face (of codimension $j$), we therefore have
$g_{v}^{F(v)}\in G_{(F(v))}$. We define
$\operatorname{c}_{j}=\operatorname{c}_{j-1}\cdot\prod_{v:|v|=j}g_{v}^{F(v)}$.
This lies in $\operatorname{C}^{n}(G_{\bullet})$, and agrees with
$\operatorname{c}$ on $\\{0,1\\}^{n}_{\leq\,j}$ as required (note that each
$F(v)$ intersects $\\{0,1\\}^{n}_{\leq\,j}$ only at $v$).

We have thus obtained $\operatorname{c}_{d}$. We now claim that the agreement
between $\operatorname{c}_{d}$ and $\operatorname{c}$ extends to all of
$\\{0,1\\}^{n}$, and so
$\operatorname{c}=\operatorname{c}_{d}\in\operatorname{C}^{n}(G_{\bullet})$.
Note that the above process cannot be continued in order to prove this claim,
because now the new elements $g_{v}^{F(v)}$ that we would multiply by are not
guaranteed to lie in $\operatorname{C}^{n}(G_{\bullet})$ anymore, since now
$\operatorname{codim}(F(v))>d$. However, we have that $\operatorname{c}$ and
$\operatorname{c}_{d}$ both satisfy the premise of the lemma (for
$\operatorname{c}_{d}$ this follows from Lemma 2.26) and agree on
$\\{0,1\\}^{n}_{\leq d}$. This can be used to complete the proof, as follows.
First let us show that for any $v$ with $|v|=d+1$ we have
$\operatorname{c}(v)=\operatorname{c}_{d}(v)$. Since $|v|=d+1$, we can find a
$(d+1)$-face map $\phi$ into $\\{0,1\\}^{n}$ with $\phi(0^{d+1})=v$ and
$|\phi(w)|\leq d$ otherwise. Using the premise we then have
$\operatorname{c}(v)=\operatorname{c}\operatorname{\circ}\phi(0^{d+1})=\operatorname{c}\operatorname{\circ}\phi(0^{d+1})\;\sigma_{d+1}(\operatorname{c}\operatorname{\circ}\phi)^{-1}=\operatorname{c}_{d}\operatorname{\circ}\phi(0^{d+1})\;\sigma_{d+1}(\operatorname{c}_{d}\operatorname{\circ}\phi)^{-1}=\operatorname{c}_{d}(v)$.
We can now use a similar argument to see that
$\operatorname{c}(v)=\operatorname{c}_{d}(v)$ for $|v|=d+2$, then for
$|v|=d+3$, and so on, concluding finally that
$\operatorname{c}=\operatorname{c}_{d}\in\operatorname{C}^{n}(G_{\bullet})$. ∎

Let us record the generalization of Proposition 1.2 that we have finally
obtained.

######  Proposition 2.28.

Let $(G,G_{\bullet})$ be a filtered group, and let
$\operatorname{c}:\\{0,1\\}^{n}\to G$. We have
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ if and only if one of
the following equivalent statements holds:

  1. (i)

There is a unique expression
$\operatorname{c}=g_{0}^{F_{0}}g_{1}^{F_{1}}\cdots g_{2^{n}-1}^{F_{2^{n}-1}}$
with $g_{i}\in G_{\operatorname{codim}(F_{i})}$ for each $i$.

  2. (ii)

The map $\operatorname{c}$ extends to a polynomial map in
$\operatorname{poly}(\mathbb{Z}^{n},G_{\bullet})$.

  3. (iii)

For every $m$-face map $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ we have
$\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)\in G_{m}$.

######  Remark 2.29.

One may replace “every $m$-face map” in property $(iii)$ above by “every
morphism”. Indeed, if the property holds with every $m$-face map, then the
proposition tells us that $\operatorname{c}$ is a cube. But then, given any
morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$, by composition
$\operatorname{c}\operatorname{\circ}\phi$ is also a cube, so applying
property $(iii)$ using as an $m$-face map the identity map on $\\{0,1\\}^{m}$,
we deduce that $\sigma_{m}(\operatorname{c}\operatorname{\circ}\phi)\in G_{m}$
as required.

####  4 Cubes of degree $k$ on abelian groups

In this section we record a special case of the previous nilspace structures
that will play an important role in the sequel. This is a $k$-step nilspace
structure defined on an abelian group.

######  Definition 2.30.

Let $\operatorname{Z}$ be an abelian group. The _degree- $k$ structure_ on
$\operatorname{Z}$, denoted by
$\operatorname{\mathcal{D}}_{k}(\operatorname{Z})$, is the nilspace formed by
$\operatorname{Z}$ together with the cubes
$\operatorname{C}^{n}(\operatorname{Z}_{\bullet(k)})$, $n\in\mathbb{N}$, where
$\operatorname{Z}_{\bullet(k)}$ denotes the maximal filtration of degree $k$
on $\operatorname{Z}$, namely the filtration with
$\operatorname{Z}_{0}=\cdots=\operatorname{Z}_{k}=\operatorname{Z}$,
$\operatorname{Z}_{k+1}=\\{0_{\operatorname{Z}}\\}$.

Note that, by Proposition 2.28, we have

$\operatorname{C}^{n}(\operatorname{\mathcal{D}}_{k}(\operatorname{Z}))=\\{\operatorname{c}:\\{0,1\\}^{n}\to\operatorname{Z}~{}|~{}\textrm{for
every face map
}\phi:\\{0,1\\}^{k+1}\to\\{0,1\\}^{n},\;\sigma_{k+1}(\operatorname{c}\operatorname{\circ}\phi)=0\\},$
(10)

where in this abelian setting we have
$\sigma_{k+1}(\operatorname{c}\operatorname{\circ}\phi)=\sum_{v\in\\{0,1\\}^{k+1}}(-1)^{|v|}\operatorname{c}\operatorname{\circ}\phi(v)$.
Note also that the nilspace $\operatorname{\mathcal{D}}_{k}(\operatorname{Z})$
is $k$-fold ergodic.

###  3 Quotients of filtered groups of degree $k$ as $k$-step nilspaces

The purpose of this section is to enlarge further our family of basic
examples, to include some natural nilspaces that do not have a group
structure. Given a filtered group $(G,G_{\bullet})$ and a subgroup $\Gamma$ of
$G$, the set of cosets $G/\Gamma$ may not have a group operation ($\Gamma$ may
not be a normal subgroup), but one can still define a cube structure on it by
projecting the cubes in $\operatorname{C}^{n}(G_{\bullet})$ down to
$G/\Gamma$.

######  Proposition 3.1.

Let $(G,G_{\bullet})$ be a filtered group of degree $k$, let $\Gamma$ be a
subgroup of $G$, and let $\pi_{\Gamma}:G\to G/\Gamma$ be the canonical
projection. Then $\operatorname{X}=G/\Gamma$ together with the sets

$\operatorname{C}^{n}(\operatorname{X})=\\{\pi_{\Gamma}\operatorname{\circ}\operatorname{c}:\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})\\}=(\operatorname{C}^{n}(G_{\bullet})\cdot\Gamma^{\\{0,1\\}^{n}})\,/\,\Gamma^{\\{0,1\\}^{n}}$
(11)

is a $k$-step nilspace. We shall refer to such nilspaces as _coset nilspaces_.

###### Proof.

The composition axiom follows clearly from that for $(G,G_{\bullet})$, and
ergodicity follows from the transitivity of the action of $G$ on $G/\Gamma$.
To check the completion axiom, one can argue again by induction on the degree
$d$. For $d=1$, $G/\Gamma$ is just an abelian group. Let $d>1$, let
$\operatorname{c}^{\prime}$ be an $n$-corner on $\operatorname{X}=G/\Gamma$
and let $\overline{\pi_{d}}$ be the projection
$\operatorname{X}\to\operatorname{X}^{\prime}:=G^{\prime}/\Gamma^{\prime}$,
where $\pi_{d}$ denotes the quotient homomorphism $G\to G^{\prime}=G/G_{d}$,
and $\Gamma^{\prime}=(\Gamma\cdot G_{d})/G_{d}$. We thus have the following
commutative diagram.

${G}$${G^{\prime}}$${\operatorname{X}=G/\Gamma\;\;}$${\;\;\operatorname{X}^{\prime}=G^{\prime}/\Gamma^{\prime}}$$\scriptstyle{\pi_{d}}$$\scriptstyle{\pi_{\Gamma}}$$\scriptstyle{\pi_{\Gamma^{\prime}}}$$\scriptstyle{\overline{\pi_{d}}}$

The surjectivity of the map
$\operatorname{C}^{n}(G_{\bullet})\to\operatorname{C}^{n}(G_{\bullet}^{\prime})$,
$\operatorname{c}\mapsto\pi_{d}\operatorname{\circ}\operatorname{c}$ implies
the surjectivity of
$\operatorname{C}^{n}(\operatorname{X})\to\operatorname{C}^{n}(\operatorname{X}^{\prime})$,
$\operatorname{c}\mapsto\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}$.
Note that $G^{\prime}$ with $G^{\prime}_{\bullet}=(G_{i}/G_{d})_{i}$ is a
filtered group of degree $d-1$. By induction there exists a cube in
$\operatorname{C}^{n}(\operatorname{X}^{\prime})$ completing
$\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}^{\prime}$, and by the
surjectivity just mentioned there exists
$\operatorname{c}_{0}\in\operatorname{C}^{n}(\operatorname{X})$ such that this
completion is $\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}_{0}$,
that is we have
$\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}_{0}(v)=\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}^{\prime}(v)$
for all $v\neq 1^{n}$. By definition of
$\operatorname{C}^{n}(\operatorname{X})$ there exists
$\tilde{\operatorname{c}}_{0}\in\operatorname{C}^{n}(G_{\bullet})$ such that
$\operatorname{c}_{0}=\pi_{\Gamma}\operatorname{\circ}\tilde{\operatorname{c}}_{0}$.
Choose also any map $\tilde{\operatorname{c}}^{\prime}\in
G^{\\{0,1\\}^{n}\setminus\\{1^{n}\\}}$ such that
$\pi_{\Gamma}\operatorname{\circ}\tilde{\operatorname{c}}^{\prime}=\operatorname{c}^{\prime}$.
Then
$\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}_{0}(v)=\overline{\pi_{d}}\operatorname{\circ}\operatorname{c}^{\prime}(v)$
means that
$\tilde{\operatorname{c}}_{0}(v)G_{d}\Gamma=\tilde{\operatorname{c}}^{\prime}(v)G_{d}\Gamma$.
This implies that for each $v\neq 1^{n}$ there exists $g_{v}\in G_{d}$ such
that
$\pi_{\Gamma}(\tilde{\operatorname{c}}_{0}(v)g_{v})=\operatorname{c}^{\prime}(v)$.
We can now carry out the same process as in the proof of Proposition 2.8 to
correct $\tilde{\operatorname{c}}_{0}$ to a new cube
$\tilde{\operatorname{c}}\in\operatorname{C}^{n}(G_{\bullet})$ satisfying
$\pi_{\Gamma}(\tilde{\operatorname{c}}(v))=\operatorname{c}^{\prime}(v)$ for
all $v\neq 1^{n}$. Then
$\operatorname{c}=\pi_{\Gamma}\operatorname{\circ}\tilde{\operatorname{c}}$ is
a completion of $\operatorname{c}^{\prime}$. ∎

The coset nilspace construction is important in that it captures the algebraic
structure of a large class of nilspaces, namely _filtered nilmanifolds_. From
a purely algebraic point of view, filtered nilmanifolds are indeed examples of
coset nilspaces, however they also have crucial topological properties
afforded by strong topological assumptions on $G$ and $\Gamma$, namely that
$G$ is a connected nilpotent Lie group and $\Gamma$ is a discrete, cocompact
subgroup of $G$. Nilmanifolds play a crucial role in the topological part of
the theory of nilspaces, which studies the _compact nilspaces_ mentioned in
the introduction (see Chapter 3 of [1]; see also [5, 6, 7] for more
information on nilmanifolds). In these notes we shall only look at the
algebraic properties of such spaces. It turns out that many of these general
properties are often fully illustrated already by 2-step nilspaces. Among
these, the following well-known example can be used as a source of intuition
for several concepts and tools treated in Chapter 2.

######  Example 3.2 (The Heisenberg nilmanifold).

Consider the Heisenberg group

$H=\begin{pmatrix}1&\mathbb{R}&\mathbb{R}\\\\[1.00006pt]
&1&\mathbb{R}\\\\[1.00006pt]
&&1\end{pmatrix}:=\Big{\\{}\begin{pmatrix}1&x_{1}&x_{3}\\\\[1.00006pt]
&1&x_{2}\\\\[1.00006pt] &&1\end{pmatrix}:x_{i}\in\mathbb{R}\Big{\\}},$

let $\Gamma=\begin{pmatrix}1&\mathbb{Z}&\mathbb{Z}\\\\[1.00006pt]
&1&\mathbb{Z}\\\\[1.00006pt] &&1\end{pmatrix}$, and let $H_{\bullet}$ denote
the lower central series on $H$, that is the filtration with
$H_{(2)}=\begin{pmatrix}1&0&\mathbb{R}\\\\[1.00006pt] &1&0\\\\[1.00006pt]
&&1\end{pmatrix}$ and $H_{(3)}=\\{\mathrm{id}_{H}\\}$. Let $\operatorname{X}$
be the nilspace consisting of the set $H/\Gamma$ together with the cube sets
$\operatorname{C}^{n}(H_{\bullet})$ projected on $H/\Gamma$. By Proposition
3.1 we have that $\operatorname{X}$ is a 2-step nilspace. We can identify the
set $H/\Gamma$ with the fundamental domain $[0,1)^{3}$ for the right action of
$\Gamma$ on $H$.

###  4 Motivation for a general algebraic characterization

Looking back at the examples of nilspaces that we have treated so far, we see
that while not all of them have a group structure, on all of them there is an
_action_ by some filtered group $G$, and the cubes on the nilspace are the
cubes in $\operatorname{C}^{n}(G_{\bullet})$ composed with this action. For
instance, in the case of a quotient $G/\Gamma$ as in the last section, the
action is defined by $(g,x\Gamma)\mapsto(gx)\Gamma$, and a map
$\\{0,1\\}^{n}\to G/\Gamma$ is a cube if and only if it is of the form
$v\mapsto\operatorname{c}(v)x\Gamma$ for some cube
$\operatorname{c}\in\operatorname{C}^{n}(G_{\bullet})$ and some coset
$x\Gamma$. We can also observe that, whether $\operatorname{X}$ consists of a
group or a coset space, we can always view $\operatorname{X}$ algebraically as
a type of principal bundle with fibre isomorphic to an abelian group. For
instance, if $\operatorname{X}$ consists of $G/\Gamma$ for a filtered group
$(G,G_{\bullet})$ of degree $d$, then the abelian group
$\operatorname{Z}=G_{d}/(G_{d}\cap\Gamma)$ has a well-defined free action on
$G/\Gamma$. Then, identifying points of $G/\Gamma$ lying in the same orbit of
$\operatorname{Z}$ gives a projection (or bundle map) from $G/\Gamma$ to
$(G/G_{d})/((\Gamma\cdot G_{d})/G_{d})$. This observation can be iterated,
expressing $G/G_{d}$ in turn as the same type of bundle using the abelian
group $G_{d-1}/G_{d}$, and so on.
In the next chapter we present a description of the above kind for a general
nilspace. More precisely, on one hand we formalize the observation above using
the notion of an _abelian bundle_ (Definition 2.17) and we present a central
theorem from [1] that describes a general nilspace $\operatorname{X}$ as an
iterated abelian bundle (Subsection 3). On the other hand, we shall define a
certain non-trivial group $G$ acting on $\operatorname{X}$, called the
_translation group_ , which comes naturally equipped with a filtration
$G_{\bullet}$, such that for each $n$ every cube in
$\operatorname{C}^{n}(G_{\bullet})$ composed with this action is a cube in
$\operatorname{C}^{n}(\operatorname{X})$ (Subsection 4). (Note however that,
unlike for the nilspaces in this chapter, in general not every cube on
$\operatorname{X}$ will be of this form; see Remark 2.39.) To motivate this
general description further, we close this chapter by establishing the
simplest case of the description, namely, the following characterization of a
1-step nilspace.

######  Proposition 4.1.

Let $\operatorname{X}$ be a $1$-step nilspace. Then there is an abelian group
$\operatorname{Z}$ such that $\operatorname{X}$ is a principal homogeneous
space of $\operatorname{Z}$ and the cubes on $\operatorname{X}$ are the
projections of the degree-1 cubes on $\operatorname{Z}$.

###### Proof.

Fix a point $e\in\operatorname{X}$. The abelian group $\operatorname{Z}$ is
the set $\operatorname{X}$ together with the binary operation $+$ defined as
follows: given $x,y\in\operatorname{X}$, we let $x+y$ be the unique element
$z\in\operatorname{X}$ such that the map $\operatorname{c}:\\{0,1\\}^{2}\to X$
with values
$\operatorname{c}(00)=e,\operatorname{c}(10)=x,\operatorname{c}(01)=y,\operatorname{c}(11)=z$
is in $\operatorname{C}^{2}(\operatorname{X})$. We claim that this is a
commutative group operation. To check commutativity, let
$\theta\in\operatorname{Aut}(\\{0,1\\}^{2})$ be the symmetry permuting $01$
and $10$. Then by composition we have
$\operatorname{c}\operatorname{\circ}\theta\in\operatorname{C}^{2}(\operatorname{X})$,
so by uniqueness of completion we must have $x+y=y+x$. For associativity,
suppose that $x,y,z\in\operatorname{X}$ and consider the following $3$-corner
$\operatorname{c}^{\prime}:\\{0,1\\}^{3}\setminus\\{1^{3}\\}\to\operatorname{X}$.

$z$$x+z$$e$$x$$y+z$$y$$x+y$

This has a unique completion $\operatorname{c}$. Considering the morphisms
$\phi_{1}:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(2)$})$
and $\phi_{2}:\\{0,1\\}^{2}\to\\{0,1\\}^{3}$,
$v\mapsto(v\scalebox{0.8}{$(1)$},v\scalebox{0.8}{$(2)$},v\scalebox{0.8}{$(2)$})$,
we then have

$(x+y)+z=\operatorname{c}\operatorname{\circ}\phi_{1}(11)=\operatorname{c}(111)=\operatorname{c}\operatorname{\circ}\phi_{2}(11)=x+(y+z).$

Finally, given $x$, the corner $\operatorname{c}^{\prime}(00)=x$,
$\operatorname{c}^{\prime}(01)=\operatorname{c}^{\prime}(10)=e$ has a
completion $y$, which then satisfies $x+y=e$. This abelian group
$\operatorname{Z}$, obtained by fixing a point $e$ in $\operatorname{X}$,
clearly has a free and transitive action on $\operatorname{X}$, and the cubes
on $\operatorname{X}$ are precisely the compositions of degree-1 cubes on
$\operatorname{Z}$ with this action. ∎

Henceforth, the expression ‘affine abelian group with the degree-1 structure’
(or ‘degree-1 abelian torsor’) will refer to the structure in the above
result, that is, a principal homogeneous space of some abelian group
$\operatorname{Z}$, with a nilspace structure given by the standard cubes on
$\operatorname{Z}$ composed with the $\operatorname{Z}$-action on $e$.

######  Remark 4.2.

Note that if we start with an abelian group $\operatorname{Z}$ and then view
it as a 1-step nilspace, then in doing so we forget which element is the
identity. In other words, to think of $\operatorname{Z}$ as a 1-step nilspace
amounts to viewing it as an _affine_ abelian group. Fixing any element $e$ of
the set underlying $\operatorname{Z}$, and applying the argument in the above
proof to the 1-step nilspace structure, we recover the original group
structure on $\operatorname{Z}$, but only up to a shift. This is an example of
the forgetfulness of the functor from the category of filtered groups with
polynomial maps to the category of nilspaces with morphisms, mentioned after
Theorem 2.14. Alternatively, one could work in the category of _rooted_
nilspaces (i.e. nilspaces with a distinguished point). Such nilspaces are
considered for instance in [18, 19].

## Chapter 2 Algebraic characterization of nilspaces

###  1 Some basic notions

In this section we collect several tools providing different ways to construct
new cubes or new nilspaces out of old ones.

####  1 Cubespaces, simplicial completion, and concatenation

Let us recall the definition of a cubespace from Chapter Notes on nilspaces:
algebraic aspects.

######  Definition 1.1.

A _cubespace_ is a set $\operatorname{X}$ together with a collection of sets
$\operatorname{C}^{n}(\operatorname{X})\subset\operatorname{X}^{\\{0,1\\}^{n}}$,
$n\geq 0$, satisfying the composition axiom from Definition 2.1 and such that
$\operatorname{C}^{0}(\operatorname{X})=\operatorname{X}$.

######  Definition 1.2.

Given cubespaces $\operatorname{X},\operatorname{Y}$, their _product_ is the
cubespace consisting of the cartesian product
$\operatorname{X}\times\operatorname{Y}$ with the cube sets
$\operatorname{C}^{n}(\operatorname{X}\times\operatorname{Y}):=\operatorname{C}^{n}(\operatorname{X})\times\operatorname{C}^{n}(\operatorname{Y})$
for each $n\geq 0$. We say that $\operatorname{Y}$ is a _subcubespace_ of
$\operatorname{X}$ if
$\operatorname{C}^{n}(\operatorname{Y})\subset\operatorname{C}^{n}(\operatorname{X})$
for every $n\geq 0$.

Clearly, the product of two $k$-nilspaces is a $k$-nilspace.

######  Definition 1.3.

Let $P$ be a cubespace and $Q$ be a subcubespace of $P$. We say that $Q$ has
the _extension property_ in $P$ if for every non-empty nilspace
$\operatorname{X}$ and every morphism $g^{\prime}:Q\to\operatorname{X}$ there
is a morphism $g:P\to\operatorname{X}$ with $g|_{Q}=g^{\prime}$.

If $S$ is a finite set and $h$ is a subset of $S$ then we denote by
$\\{0,1\\}^{S}_{h}$ the set of elements of $\\{0,1\\}^{S}$ that are supported
on $h$. Note that $\\{0,1\\}^{S}_{h}$ can be viewed as the discrete cube of
dimension $|h|$.

######  Definition 1.4.

Let $S$ be a finite set and let $H$ be a set system in $S$ (a collection of
subsets of $S$). The collection of all cube morphisms
$g:\\{0,1\\}^{n}\to\\{0,1\\}^{S}_{h}$, $n\geq 0,\,h\in H$, defines a cubespace
structure on $P=\bigcup_{h\in H}\\{0,1\\}^{S}_{h}$. Cubespaces arising this
way will be called _simplicial cubespaces_.

In this definition $P$ is a subcubespace of $\\{0,1\\}^{S}$. Note that by
composing such morphisms $g$ with discrete-cube morphisms, and invoking the
composition axiom, we can deduce that for every $h$ in $H$, for every
$h^{\prime}\subset h$, every $n$-cube morphism
$\phi:\\{0,1\\}^{n}\to\\{0,1\\}^{S}_{h^{\prime}}$ is a cube in the above
cubespace. It follows that we can assume without loss of generality that $H$
is a downset, or simplicial complex, i.e. that it satisfies the following
property: if $h\in H$ and $h^{\prime}\subset h$ then $h^{\prime}\in H$.

######  Lemma 1.5 (Simplicial completion).

Let $S$ be a finite set and $P$ be a simplicial cubespace corresponding to a
set system $H$ in $S$. Then $P$ has the extension property in $\\{0,1\\}^{S}$.

###### Proof.

As noted above, we may assume that $H$ is a simplicial complex, so in
particular the empty set is in $H$. If $H$ is the whole power set $2^{S}$ then
there is nothing to prove. Otherwise, starting from some set $S\not\in H$ we
can go downward in the lattice $2^{S}$ (passing to subsets) until we find a
set $h^{\prime}$ that is not in $H$ and such that every proper subset of
$h^{\prime}$ is contained in $H$. The new system
$H^{\prime}=H\cup\\{h^{\prime}\\}$ is also a simplicial complex. Let
$g:\bigcup_{h\in H}\\{0,1\\}_{h}^{S}\to\operatorname{X}$ be a morphism into
some nilspace $\operatorname{X}$. The restriction of $g$ to
$\\{0,1\\}^{h^{\prime}}\setminus\\{1^{h^{\prime}}\\}$ is an
$|h^{\prime}|$-corner. By the corner-completion axiom, we can therefore extend
$g$ to a morphism $\bigcup_{h\in
H^{\prime}}\\{0,1\\}^{S}_{h}\to\operatorname{X}$. By repeating this procedure
we can extend $g$ to all of $\\{0,1\\}^{S}$. ∎

We shall now describe a basic way to form a new cube from two given cubes on a
nilspace.

######  Definition 1.6 (Adjacent maps and concatenations).

Given a map $f$ defined on $\\{0,1\\}^{n}$, recall that for each
$i\in\\{0,1\\}$ we define $f(\cdot,i)$ on $\\{0,1\\}^{n-1}$ by
$f(v,i)=f((v,i))$, where
$(v,i):=\big{(}v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n-1)$},i\big{)}\in\\{0,1\\}^{n}$.
A map $f_{1}$ on $\\{0,1\\}^{n}$ is said to be _adjacent_ to another such map
$f_{2}$ if $f_{1}(\cdot,1)=f_{2}(\cdot,0)$, and we then write $f_{1}\prec
f_{2}$. We say that $f$ is a _concatenation_ of $f_{1},f_{2},\ldots,f_{\ell}$
if $f(\cdot,0)=f_{1}(\cdot,0)$, $f(\cdot,1)=f_{\ell}(\cdot,1)$, and
$f_{i}\prec f_{i+1}$ for $i\in[\ell-1]$.

######  Lemma 1.7.

Let $\operatorname{X}$ be a nilspace and suppose that
$\operatorname{c}_{1},\operatorname{c}_{2}$ are cubes in
$\operatorname{C}^{n}(\operatorname{X})$ with
$\operatorname{c}_{1}\prec\operatorname{c}_{2}$. Then the concatenation of
$\operatorname{c}_{1},\operatorname{c}_{2}$ is also in
$\operatorname{C}^{n}(\operatorname{X})$.

###### Proof.

Let $S=[n+1]$, $h_{1}=[n]$, $h_{2}=[n+1]\setminus\\{n\\}$ and
$H=\\{h_{1},h_{2}\\}$. Let $P$ be the simplicial cubespace corresponding to
$H$. For $i=1,2$, let $\phi_{i}$ be an invertible morphism
$\\{0,1\\}^{n+1}_{h_{i}}\to\\{0,1\\}^{n}$ such that the maps
$\operatorname{c}_{i}\operatorname{\circ}\phi_{i}$ agree on
$\\{0,1\\}^{n+1}_{h_{1}}\cap\\{0,1\\}^{n+1}_{h_{2}}=\\{0,1\\}^{n+1}_{[n-1]}$.
(We can take
$\phi_{1}:(v\scalebox{0.8}{$(1)$},\dots,v\scalebox{0.8}{$(n)$},0)\mapsto(v\scalebox{0.8}{$(1)$},\dots,v\scalebox{0.8}{$(n-1)$},1-v\scalebox{0.8}{$(n)$})$
and
$\phi_{2}:(v\scalebox{0.8}{$(1)$},\dots,v\scalebox{0.8}{$(n-1)$},0,v\scalebox{0.8}{$(n+1)$})\mapsto(v\scalebox{0.8}{$(1)$},\dots,v\scalebox{0.8}{$(n-1)$},v\scalebox{0.8}{$(n+1)$})$.)
We can then define $f:P\to\operatorname{X}$ by
$f|_{\\{0,1\\}^{n+1}_{h_{i}}}=\operatorname{c}_{i}\operatorname{\circ}\phi_{i}$.
Then $f$ is a morphism, so by Lemma 1.5 it extends to a morphism
$f^{\prime}:\\{0,1\\}^{n+1}\to\operatorname{X}$. Consider the morphism
$\phi:\\{0,1\\}^{n}\to\\{0,1\\}^{n+1},\,v\mapsto\big{(}v\scalebox{0.8}{$(1)$},\ldots,v\scalebox{0.8}{$(n-1)$},1-v\scalebox{0.8}{$(n)$},v\scalebox{0.8}{$(n)$}\big{)}$,
and note that the concatenation of $\operatorname{c}_{1},\operatorname{c}_{2}$
is $f^{\prime}\operatorname{\circ}\phi$. By the composition axiom, the result
follows. ∎

Sometimes we shall have to consider spaces $\operatorname{X}$ for which the
ergodicity axiom is not satisfied. As a first application of concatenations,
let us show that such a space can always be partitioned into components each
of which satisfies the ergodicity axiom.

######  Lemma 1.8.

Suppose that $\operatorname{X}$ satisfies the axioms in Definition 2.1 with
ergodicity replaced by the weaker condition
$\operatorname{C}^{0}(\operatorname{X})=\operatorname{X}$. Then
$\operatorname{X}$ can be decomposed into a disjoint union of nilspaces.

###### Proof.

We define a relation on $\operatorname{X}$ by writing $x\sim y$ if the map
$\operatorname{c}:\\{0,1\\}\to\operatorname{X}$ with $\operatorname{c}(0)=x$
and $\operatorname{c}(1)=y$ is in $\operatorname{C}^{1}(\operatorname{X})$.
This is an equivalence relation, indeed reflexivity follows from
$\operatorname{C}^{0}(\operatorname{X})=\operatorname{X}$ and composition with
$\mathbf{0}$, symmetry follows also from composition, and transitivity follows
from Lemma 1.7. On any equivalence class $\operatorname{X}^{\prime}$ the cubes
in $\bigcup_{n}\operatorname{C}^{n}(\operatorname{X})$ with values in
$\operatorname{X}^{\prime}$ form an ergodic nilspace. (Note that composition
implies that the completion of an $\operatorname{X}^{\prime}$-valued corner is
also $\operatorname{X}^{\prime}$-valued.) ∎

Note that if $G$ is a group and a map $g:\\{0,1\\}^{n}\to G$ is the
concatenation of two other such maps $g_{1}\prec g_{2}$, then by (8) we have
the following equality (useful in particular when $G$ is abelian):

$\sigma_{n}(g)=\sigma_{n}(g_{2})\;\sigma_{n}(g_{1}).$ (1)

We end the subsection with a basic lemma that can be helpful to reduce
questions about injective morphisms to ones about face maps. For a morphism
$\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$, recall from (2) that
$J=J_{\phi}\subset[n]$ is the set of indices of coordinates of $\phi(v)$ that
genuinely depend on one of the coordinates of $v\in\\{0,1\\}^{m}$. We know
that $\phi$ is injective if and only if $|J|\geq m$.

######  Lemma 1.9.

Let $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ be an injective morphism with
$|J_{\phi}|>m$. Then there is an automorphism $\theta$ of $\\{0,1\\}^{m}$ such
that $\phi\operatorname{\circ}\theta$ is the concatenation of at most four
injective morphisms $\phi_{i}$ with $|J_{\phi_{i}}|<|J_{\phi}|$ for all $i$.

###### Proof.

We have $|J_{\phi}(i)|\geq 2$ for some $i$. Let $\theta$ be the automorphism
that transposes coordinates $i$ and $m$. Relabelling
$\phi\operatorname{\circ}\theta$ as $\phi$ we can now assume that $i=m$. We
distinguish two cases. In the first case, both maps $\tau_{0},\tau_{1}$
(recall the paragraph before Lemma 1.2) occur among the varying coordinates
$\phi(v)\scalebox{0.8}{$(j)$}$, $j\in J_{\phi}(m)$. In the second case only
one such map occurs.

For the first case, define $\phi_{1}$ by changing in $\phi(v)|_{J_{\phi}(m)}$
all maps $\tau_{0}$ to $\mathbf{0}$, and define $\phi_{2}$ by changing in
$\phi(v)|_{J_{\phi}(m)}$ all maps $\tau_{1}$ to $\mathbf{0}$. Then we have
$|J_{\phi_{1}}|,|J_{\phi_{2}}|<|J_{\phi}|$, and $\phi$ is the concatenation of
$\phi_{1}\prec\phi_{2}$.

For the second case, suppose that only $\tau_{0}$ occurs. Fix any $j\in
J_{\phi}(m)$, then define $\phi_{1}$ from $\phi$ by switching $\tau_{0}$ to
$\mathbf{0}$ at index $j$, define $\phi_{2}$ from $\phi$ by switching
$\tau_{0}$ to $\tau_{1}$ at every index in $J_{\phi}(m)\setminus\\{j\\}$, and
define $\phi_{3}$ from $\phi$ by switching $\tau_{0}$ to $\mathbf{1}$ at $j$.
Observe that $\phi$ is the concatenation of
$\phi_{1}\prec\phi_{2}\prec\phi_{3}$. We also have
$|J_{\phi_{1}}|,|J_{\phi_{3}}|$ both less than $|J_{\phi}|$ and, although
$|J_{\phi_{2}}|=|J_{\phi}|$, the first case above applies to $\phi_{2}$, so we
are done. (If only $\tau_{1}$ occurs, a similar argument works.) ∎

####  2 Nilspaces and parallelepiped structures

Before we continue the development of the theory, in this subsection we pause
to illustrate the use of some of the tools from the previous subsection. We
shall apply these tools to treat a natural question concerning nilspaces,
namely the question of the relation between the nilspace axioms from [1] and
the axioms from [12] defining parallelepiped structures. In the introduction,
we mentioned that the former axioms are a variant of the latter, and here we
shall detail this remark.

In [12] the notion of an abstract parallelepiped structure was introduced. It
was defined explicitly only for parallelepipeds of dimension 2 and 3 but, as
noted at the end of [12], the generalization to higher dimensions is clear. We
formulate this generalization below using notation from previous sections.
Recall in particular the notation
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}$ for $k$-arrows
from Definition 2.18, the notation $\phi_{F}$ for the face-restriction maps
from Definition 1.5, and the notation $\operatorname{Aut}(\\{0,1\\}^{n})$ for
the group of automorphisms of the discrete cube, from Definition 1.1.

######  Definition 1.10 (Parallelepiped structures).

Let $\operatorname{X}$ be a set and let $n$ be a positive integer. We say that
a finite sequence $(P_{m})_{m=1}^{n}$ of sets
$P_{m}\subset\operatorname{X}^{\\{0,1\\}^{m}}$ is an _$n$ -parallelepiped
structure_ on $\operatorname{X}$ if $P_{1}=\operatorname{X}^{\\{0,1\\}}$ and
the following axioms are satisfied for every $m\in[2,n]$:

  1. (i)

(Face restrictions) For every $p\in P_{m}$ and every $(m-1)$-face
$F\subset\\{0,1\\}^{m}$, we have $p\operatorname{\circ}\phi_{F}\in P_{m-1}$.

  2. (ii)

(Symmetries) For every $p\in P_{m}$ and every
$\theta\in\operatorname{Aut}(\\{0,1\\}^{m})$, we have
$p\operatorname{\circ}\theta\in P_{m}$.

  3. (iii)

(Equivalence relation) The relation $\approx_{P_{m}}$ on $P_{m-1}$, defined by
$p\approx_{P_{m}}p^{\prime}$ if and only if $\langle
p,p^{\prime}\rangle_{1}\in P_{m}$, is an equivalence relation.

  4. (iv)

(Closing property) Let
$p^{\prime}:\\{0,1\\}^{m}\setminus\\{1^{m}\\}\to\operatorname{X}$ be such that
for every $(m-1)$-face $F$ containing $0^{m}$ we have
$p^{\prime}\operatorname{\circ}\phi_{F}\in P_{m-1}(\operatorname{X})$. Then
there exists $p\in P_{m}(\operatorname{X})$ such that $p(v)=p^{\prime}(v)$ for
all $v\neq 1^{m}$.

These axioms are the natural generalizations to dimension $n$ of the four
axioms in [12, Definition 4]. Note that the fourth axiom above is just the
corner-completion axiom from Definition 2.1 stated for the elements of
$P_{m}$.

To relate parallelepiped structures to nilspaces, we want to show that the
sets $P_{m}$ above can be viewed as cube sets forming a nilspace structure on
$\operatorname{X}$. However, note that on a nilspace $\operatorname{X}$ there
is a cube set $\operatorname{C}^{n}(\operatorname{X})$ defined for _every_
$n\in\mathbb{N}$, whereas an $n$-parallelepiped structure defines
parallelepipeds only of dimension up to $n$. Nevertheless, we can consider
what we shall call a _sequence of parallelepiped structures_ on a set
$\operatorname{X}$, namely an infinite sequence $(P_{m})_{m\in\mathbb{N}}$
such that for every $n\in\mathbb{N}$ the initial segment $(P_{m})_{m\in[n]}$
is an $n$-parallelepiped structure on $\operatorname{X}$. We then have the
following result.

######  Proposition 1.11.

Let $\operatorname{X}$ be a set and for each $n\in\mathbb{N}$ let
$P_{n}\subset\operatorname{X}^{\\{0,1\\}^{n}}$. Then
$\big{(}\operatorname{X},(P_{n})_{n\in\mathbb{N}}\big{)}$ is a nilspace if and
only if $(P_{n})_{n\in\mathbb{N}}$ is a sequence of parallelepiped structures
on $\operatorname{X}$.

###### Proof.

To see the forward implication, note first that the composition axiom for the
nilspace immediately implies axioms $(i)$ and $(ii)$ above for each $m$, and
trivially implies also axiom $(iv)$ since we are assuming the completion
axiom. Moreover, axiom $(iii)$ above is also satisfied, indeed the reflexivity
and symmetry of the relation $\approx_{P_{m}}$ are again direct consequences
of the composition axiom, and the transitivity is a consequence of the
concatenation property (Lemma 1.7).

To see the backward implication, the only difficulty is in checking that the
composition axiom holds, that is, if $p\in P_{n}$ and
$\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ is a morphism, then
$p\operatorname{\circ}\phi\in P_{m}$. We check this in several steps as
follows.

First let us suppose that $\phi$ is an $m$-face map, with image an $m$-face
$F\subset\\{0,1\\}^{n}$. Note that we may then assume that $\phi$ is just the
restriction map $\phi_{F}$, by axiom $(ii)$ above. We now prove that
$p\operatorname{\circ}\phi_{F}\in P_{m}$ by induction on $n-m$, as follows. If
$n=m+1$ then we have $p\operatorname{\circ}\phi_{F}\in P_{m}$ immediately by
axiom $(i)$. If $n>m+1$ then let $F^{\prime}$ be an $(n-1)$-face in
$\\{0,1\\}^{n}$ that contains $F$, and note that by axiom $(i)$ we have
$p\operatorname{\circ}\phi_{F^{\prime}}\in P_{n-1}$. Since $\phi_{F^{\prime}}$
is injective we can take its inverse map
$\phi_{F^{\prime}}^{-1}:F^{\prime}\to\\{0,1\\}^{n-1}$. Note that
$\phi_{F^{\prime}}^{-1}\operatorname{\circ}\phi_{F}$ is a face map
$\\{0,1\\}^{m}\to\\{0,1\\}^{n-1}$. Hence, by induction, we have that
$P_{m}\ni(p\operatorname{\circ}\phi_{F^{\prime}})\operatorname{\circ}(\phi_{F^{\prime}}^{-1}\operatorname{\circ}\phi_{F})=p\operatorname{\circ}\phi_{F}$,
as required.

Our next step is to check that $p\operatorname{\circ}\phi\in P_{m}$ if $\phi$
is an injective morphism. Thus we still have $m\leq n$, we assume that
$|J_{\phi}(i)|\geq 1$ for every $i\in[m]$, and we recall that we have equality
here for every $i$ if and only if $\phi$ is an $m$-face map. We shall prove
that $p\operatorname{\circ}\phi\in P_{m}$ by induction on
$|J_{\phi}|=\sum_{i\in[m]}|J_{\phi}(i)|$. The basic case $|J_{\phi}|=m$ is
clear by the previous paragraph. Suppose then that $|J_{\phi}|>m$. By Lemma
1.9 there is $\theta\in\operatorname{Aut}(\\{0,1\\}^{m})$ and morphisms
$\phi_{1},\ldots,\phi_{k}:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$, $k\leq 4$, such that
$\phi\operatorname{\circ}\theta$ is the concatenation of
$\phi_{1}\prec\cdots\prec\phi_{k}$ and $|J_{\phi_{i}}|<|J_{\phi}|$ for each
$i$. We have by axiom $(ii)$ above that $p\operatorname{\circ}\phi\in P_{m}$
if and only if $p\operatorname{\circ}\phi\operatorname{\circ}\theta\in P_{m}$,
so for our purposes we may assume that $\theta$ is the identity. Thus $\phi$
is the concatenation of $\phi_{1}\prec\cdots\prec\phi_{k}$, and therefore
$p\operatorname{\circ}\phi$ is the concatenation of
$(p\operatorname{\circ}\phi_{1})\prec\cdots\prec(p\operatorname{\circ}\phi_{k})$.
By induction, each map $p\operatorname{\circ}\phi_{i}$ is in $P_{m}$. On the
other hand, if two maps in $P_{m}$ are adjacent, then by the transitivity of
the relation $\approx_{P_{m}}$ their concatenation is also in $P_{m}$.
Applying this to the maps $p\operatorname{\circ}\phi_{i}$ we deduce that
$p\operatorname{\circ}\phi\in P_{m}$.

Finally, we check that $p\operatorname{\circ}\phi\in P_{m}$ for _every_
morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$. Let $\ell=\ell(\phi)$ denote
the number of indices $i\in[m]$ such that $J_{\phi}(i)=\emptyset$. We shall
prove that $p\operatorname{\circ}\phi\in P_{m}$ by induction on $\ell$. The
basic case $\ell=0$ is clear by the previous paragraph, since $\phi$ is then
injective. If $\ell>0$, then $J_{\phi}(i)=\emptyset$ for some $i$, and note
that we may assume that $i=m$ (using an automorphism of $\\{0,1\\}^{m}$ if
necessary, as in the previous paragraph). But then letting $F$ denote the
$(m-1)$-face $\\{v_{m}=0\\}\subset\\{0,1\\}^{m}$, we have
$\ell(\phi\operatorname{\circ}\phi_{F})<\ell(\phi)$, so by induction we have
$p\operatorname{\circ}\phi\operatorname{\circ}\phi_{F}\in P_{m-1}$. Since
$\phi(v)$ does not depend on $v_{m}$, we also have that
$p\operatorname{\circ}\phi=\langle
p\operatorname{\circ}\phi\operatorname{\circ}\phi_{F},p\operatorname{\circ}\phi\operatorname{\circ}\phi_{F}\rangle_{1}$.
Hence, by the reflexivity of $\approx_{P_{m}}$, it follows that
$p\operatorname{\circ}\phi\in P_{m}$. ∎

######  Remark 1.12.

In [12] Host and Kra use the term _strong parallelepiped structure_ to refer
to an $n$-parallelepiped structure for which we also have that for each
$p^{\prime}$ satisfying the premise of the closing property, there is a
_unique_ $p$ satisfying the conclusion. Parallelepiped structures without this
uniqueness in the closing property are called _weak_ parallelepiped
structures. In light of Proposition 1.11, we have that
$\big{(}\operatorname{X},(P_{n})_{n\in\mathbb{N}}\big{)}$ is a $k$-step
nilspace if and only if $(P_{n})_{n\in\mathbb{N}}$ is a sequence of
parallelepiped structures on $\operatorname{X}$ and $(P_{n})_{n=1}^{k+1}$ is
strong.

As we mentioned in the introduction, in [12] a relation was established
between parallelepiped structures and nilpotent groups. Indeed, Host and Kra
showed that for every strong parallelepiped structure of dimension 3 on a set
$\operatorname{X}$ there is a group acting on $\operatorname{X}$ by bijections
that preserve the parallelepipeds in a certain natural sense, and this group
is 2-step nilpotent (see [12, Proposition 12]). In light of Proposition 1.11,
this result of Host and Kra is applicable to any 2-step nilspace. One of the
central ways in which the results in [1] continue the program started in [12]
is by providing a generalization, for higher-step nilspaces, of this relation
to nilpotent groups. This generalization is detailed in Subsection 4 below,
where we treat the _translation group_ of a nilspace.

####  3 Tricubes and the tricube composition

Here we shall describe another operation that gives a very useful construction
of new cubes on a nilspace. To this end we shall use special examples of
cubespaces, called tricubes. These objects play an important role in the
theory. Their associated operation, called the tricube composition, can be
viewed in a certain sense as an analogue, for cubes on nilspaces, of the
convolution operation for functions.

######  Definition 1.13.

The _tricube_ of dimension $n$ is the set $T_{n}=\\{-1,0,1\\}^{n}$ together
with the following cubespace structure. First we define, for each
$v\in\\{0,1\\}^{n}$, the map

$\operatorname{\Psi}_{v}:\\{0,1\\}^{n}\to
T_{n},\quad\operatorname{\Psi}_{v}\big{(}w\scalebox{0.8}{$(1)$},w\scalebox{0.8}{$(2)$},\dots,w\scalebox{0.8}{$(n)$}\big{)}\scalebox{0.8}{$(j)$}=(2v\scalebox{0.8}{$(j)$}-1)(1-w\scalebox{0.8}{$(j)$}).$
(2)

Thus $\operatorname{\Psi}_{v}$ is an injective map embedding $\\{0,1\\}^{n}$
into $T_{n}$ as the subcube with base point $\operatorname{\Psi}_{v}(0^{n})$
equal to the ‘outer point’
$(2v\scalebox{0.8}{$(1)$}-1,2v\scalebox{0.8}{$(2)$}-1,\ldots,2v\scalebox{0.8}{$(n)$}-1)$
of $T_{n}$, and with $\operatorname{\Psi}_{v}(1^{n})=0^{n}$. A map
$\operatorname{c}:\\{0,1\\}^{m}\to T_{n}$ is declared to be a cube if there
exists a morphism $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ and some element
$v\in\\{0,1\\}^{n}$ such that
$\operatorname{c}=\operatorname{\Psi}_{v}\operatorname{\circ}\phi$.

Thus an $m$-cube on $T_{n}$ is just a map $\\{0,1\\}^{m}\to T_{n}$ that
factors as a cube-morphism through one of the embeddings
$\operatorname{\Psi}_{v}$. Note that this cubespace structure on $T_{n}$ is
the smallest one such that all the maps $\operatorname{\Psi}_{v}$ are cubes.
Note also the following basic fact.

######  Lemma 1.14.

For each $n\in\mathbb{N}$, the cubespace $T_{n}$ is the product cubespace
$(T_{1})^{n}$.

Given two maps $f_{1},f_{2}:S\to\operatorname{X}$ we write $f_{1}\times f_{2}$
for the map $S\to\operatorname{X}\times\operatorname{X}$,
$s\mapsto(f_{1}(s),f_{2}(s))$.

###### Proof.

Let $k_{0},k_{1}\in\mathbb{N}$, let $n=k_{0}+k_{1}$, and note that for every
$v_{0}\in\\{0,1\\}^{k_{0}},v_{1}\in\\{0,1\\}^{k_{1}}$ we have from (2) that
$\operatorname{\Psi}_{v_{0}}\times\operatorname{\Psi}_{v_{1}}(w)=\operatorname{\Psi}_{(v_{0},v_{1})}(w)$,
$w\in\\{0,1\\}^{n}$. Every map $\operatorname{c}:\\{0,1\\}^{m}\to T_{n}$ can
be written $\operatorname{c}_{0}\times\operatorname{c}_{1}$ for
$\operatorname{c}_{i}:\\{0,1\\}^{m}\to T_{k_{i}}$. Then we have
$\operatorname{c}\in\operatorname{C}^{m}(T_{n})$ if and only if for every
$v=(v_{0},v_{1})\in\\{0,1\\}^{n}$ there is a morphism
$\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ such that
$\operatorname{c}=\operatorname{\Psi}_{v}\operatorname{\circ}\phi$. But $\phi$
can also be written $\phi_{0}\times\phi_{1}$ and is a morphism if and only if
each $\phi_{i}$ is, whence $\operatorname{c}$ is an $m$-cube on $T_{n}$ if and
only if each map $\operatorname{c}_{i}\in\operatorname{C}^{m}(T_{k_{i}})$. ∎

To form a new cube on a nilspace $\operatorname{X}$ using $T_{n}$, we shall
typically take $2^{n}$ cubes
$\operatorname{c}_{v}\in\operatorname{C}^{n}(\operatorname{X})$, one for each
$v\in\\{0,1\\}^{n}$, and take the morphism $\phi:T_{n}\to\operatorname{X}$
defined as follows: for each $u\in T_{n}$, we take any embedding
$\operatorname{\Psi}_{v}$ containing $u$ in its image, and then we let
$\phi(u)=\operatorname{c}_{v}(\operatorname{\Psi}_{v}^{-1}(u))$. Note that
this is a well-defined map provided that the cubes $\operatorname{c}_{v}$ can
indeed be “glued together into $T_{n}$”, that is for every $v,v^{\prime}$ the
maps
$\operatorname{c}_{v}\operatorname{\circ}\operatorname{\Psi}_{v}^{-1},\operatorname{c}_{v^{\prime}}\operatorname{\circ}\operatorname{\Psi}_{v^{\prime}}^{-1}$
agree on the intersection of the images of
$\operatorname{\Psi}_{v},\operatorname{\Psi}_{v^{\prime}}$. Note that if
$\phi$ is thus well-defined then it is also a cubespace morphism. Given all
this, the new cube on $\operatorname{X}$ is then obtained by taking the values
of $\phi$ on the ‘outer points’ of $T_{n}$, i.e. the elements of
$\\{-1,1\\}^{n}$, via the following map.

######  Definition 1.15 (Outer-point map).

We denote by $\omega_{n}$ the embedding $\\{0,1\\}^{n}\to T_{n}$ defined by
$\omega_{n}(v)=\operatorname{\Psi}_{v}(0^{n})=(2v\scalebox{0.8}{$(1)$}-1,\ldots,2v\scalebox{0.8}{$(n)$}-1)$.

The new cube mentioned above is thus $\phi\operatorname{\circ}\omega_{n}$.
That this map is indeed a cube is a nontrivial fact, which we record as
follows.

######  Lemma 1.16 (Tricube composition).

Let $\operatorname{X}$ be a nilspace and let $f:T_{n}\to\operatorname{X}$ be a
morphism. Then the composition $f\operatorname{\circ}\omega_{n}$ is in
$\operatorname{C}^{n}(\operatorname{X})$.

The point of this result lies in that $\omega_{n}$ is not a cube on $T_{n}$
(if it were a cube then the result would be trivial, since $f$ is a morphism).
The idea of the proof is that, while the cube structure on $T_{n}$ is not rich
enough for $\omega_{n}$ to be in $\operatorname{C}^{n}(T_{n})$, there is an
injective morphism $q:T_{n}\to\\{0,1\\}^{2n}$ such that, firstly,
$q\operatorname{\circ}\omega_{n}$ is a morphism of discrete cubes, and
secondly $q(T_{n})$ has the extension property, so that
$f\operatorname{\circ}q^{-1}$ extends to a cube in
$\operatorname{C}^{2n}(\operatorname{X})$, and therefore
$f\operatorname{\circ}\omega_{n}=f\operatorname{\circ}q^{-1}\operatorname{\circ}q\operatorname{\circ}\omega_{n}$
is a cube on $\operatorname{X}$. Let us detail this further.

First we describe the embedding of $T_{n}$, which will also be useful later
on.

######  Lemma 1.17.

Let $\lambda:\\{-1,0,1\\}\to\\{0,1\\}^{2}$ be the function $\lambda(1)=(1,0)$,
$\lambda(0)=(0,0)$, $\lambda(-1)=(0,1)$. Then
$q=\lambda^{n}:v\mapsto\big{(}\lambda(v\scalebox{0.8}{$(1)$}),\ldots,\lambda(v\scalebox{0.8}{$(n)$})\big{)}$
is an injective morphism $T_{n}\to\\{0,1\\}^{2n}$. Moreover, $q(T_{n})$ is
simplicial in $\\{0,1\\}^{2n}$.

###### Proof.

The map $\lambda$ is easily checked to be a morphism, and it is also easily
checked that products of morphisms are morphisms relative to the product
cubespace structure, whence $q$ is indeed a morphism. To see that $q(T_{n})$
is simplicial, suppose that $w=q(v)\in q(T_{n})$ and $u\in\\{0,1\\}^{2n}$ has
$\operatorname{Supp}u\subset\operatorname{Supp}w$. Let $v^{\prime}$ be
obtained from $v$ by switching to 0 every entry $v\scalebox{0.8}{$(j)$}$ such
that the indices corresponding to $\lambda(v\scalebox{0.8}{$(j)$})$ are not in
the support of $u$, i.e. $\\{2j-1,2j\\}\not\subset\operatorname{Supp}u$. Then
$u=q(v^{\prime})$. ∎

###### Proof of Lemma 1.16.

Since $q(T_{n})$ is simplicial in $\\{0,1\\}^{2n}$ and
$f\operatorname{\circ}q^{-1}:q(T_{n})\to\operatorname{X}$ is a morphism
(relative to $q(T_{n})$ as a subcubespace of $\\{0,1\\}^{2n}$), by Lemma 1.5
the map $f\operatorname{\circ}q^{-1}$ extends to a cube
$\operatorname{c}:\\{0,1\\}^{2n}\to\operatorname{X}$. Since
$f\operatorname{\circ}\omega_{n}=\operatorname{c}\operatorname{\circ}(q\operatorname{\circ}\omega_{n})$,
it now suffices to show that
$q\operatorname{\circ}\omega_{n}:\\{0,1\\}^{n}\to\\{0,1\\}^{2n}$ is a
morphism. To see this, note that $\lambda$ restricted to $\\{-1,1\\}$ equals
the map $x\mapsto(\frac{1+x}{2},\frac{1-x}{2})$. For each $j\in[n]$, the
$j$-th coordinate of $\omega_{n}(v)$ is $2v\scalebox{0.8}{$(j)$}-1$, whence
$\lambda(\omega_{n}(v)\scalebox{0.8}{$(j)$})=(v\scalebox{0.8}{$(j)$},1-v\scalebox{0.8}{$(j)$})=(\tau_{0}(v\scalebox{0.8}{$(j)$}),\tau_{1}(v\scalebox{0.8}{$(j)$}))$,
where $\tau_{0},\tau_{1}$ are the identity and reflection introduced before
Lemma 1.2. It follows that $q\operatorname{\circ}\omega_{n}$ is a morphism
$\\{0,1\\}^{n}\to\\{0,1\\}^{2n}$, indeed we have

$q\operatorname{\circ}\omega_{n}(v)=\Big{(}\,\big{(}\tau_{0}(v\scalebox{0.8}{$(1)$}),\tau_{1}(v\scalebox{0.8}{$(1)$})\big{)}\,,\,\big{(}\tau_{0}(v\scalebox{0.8}{$(2)$}),\tau_{1}(v\scalebox{0.8}{$(2)$})\big{)}\,,\ldots,\,\big{(}\tau_{0}(v\scalebox{0.8}{$(n)$}),\tau_{1}(v\scalebox{0.8}{$(n)$})\big{)}\,\Big{)}.\qed$

######  Remark 1.18.

Given Lemma 1.17, we may give an alternative definition of tricubes. More
precisely, if we identify $\\{0,1\\}^{2n}$ with $(\\{0,1\\}^{2})^{n}$ the
natural way, then thanks to the embedding $q$ we may redefine $T_{n}$ as the
subcubespace of $\\{0,1\\}^{2n}$ of the form $\\{(1,0),(0,0),(0,1)\\}^{n}$.
This alternative definition would be natural in the sense that the cubespace
structure on the tricube is easily described as the restriction of the one on
$\\{0,1\\}^{2n}$. In particular, the proof of Lemma 1.16 itself relies
essentially on this identification of $T_{n}$ with a simplicial subcubespace
of $\\{0,1\\}^{2n}$. However, the cube structure on $T_{n}$ as described in
Definition 1.13 is also intuitively clear and, in addition, we shall make much
more use in the sequel of the intuition and notation related to viewing
$T_{n}$ as $\\{-1,0,1\\}^{n}$. There may be one way in which this view could
be misleading, namely, that one could think that morphisms
$T_{n}\to\operatorname{X}$ extend automatically to morphisms from more general
subsets of $\mathbb{Z}^{n}$ into $\operatorname{X}$; let us emphasize that
this is not the case.

####  4 Arrow spaces

Recall from Definition 2.18 that for two maps
$f_{0},f_{1}:\\{0,1\\}^{n}\to\operatorname{X}$, the corresponding $k$-arrow is
the map

$\langle
f_{0},f_{1}\rangle_{k}:\\{0,1\\}^{n+k}\to\operatorname{X},\;\;(v,w)\;\mapsto\;\left\\{\begin{array}[]{lr}f_{0}(v),&w\neq
1^{k}\\\ f_{1}(v),&w=1^{k}\end{array}\right..$

Note that taking the $1$-arrow is an invertible operation
$\operatorname{X}^{\\{0,1\\}^{n}}\times\operatorname{X}^{\\{0,1\\}^{n}}\to\operatorname{X}^{\\{0,1\\}^{n+1}}$,
$f_{0}\times f_{1}\mapsto\langle f_{0},f_{1}\rangle_{1}$, with inverse the map
$f\mapsto f(\cdot,0)\times f(\cdot,1)$, which we denote by
$\operatorname{\wp}_{n}$.

######  Definition 1.19 (Arrow spaces).

Let $\operatorname{X}$ be a cubespace. For each positive integer $k$, the _$k$
-th arrow space_ over $\operatorname{X}$, denoted by
$\operatorname{X}\Join_{k}\operatorname{X}$, is the cubespace defined on the
cartesian product $\operatorname{X}\times\operatorname{X}$ by letting
$\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X}\Join_{k}\operatorname{X})$
if and only if
$\operatorname{c}=\operatorname{c}_{0}\times\operatorname{c}_{1}$ with
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{k}\in\operatorname{C}^{n+k}(\operatorname{X})$.

######  Example 1.20.

Let us illustrate this construction in the case of filtered groups. If
$\operatorname{X}$ consists of a filtered group $(G,G_{\bullet})$ with cube
sets $\operatorname{C}^{n}(G_{\bullet})$, $n\geq 0$, then it follows from
Lemma 2.19 that a product map $\operatorname{c}_{0}\times\operatorname{c}_{1}$
is in $\operatorname{C}^{n}(\operatorname{X}\Join_{k}\operatorname{X})$ if and
only if $\operatorname{c}_{0},\operatorname{c}_{1}$ are both in
$\operatorname{C}^{n}(G_{\bullet})$ and the map
$v\mapsto\operatorname{c}_{0}(v)^{-1}\operatorname{c}_{1}(v)$ is in
$\operatorname{C}^{n}(G_{\bullet}^{+k})$ (here recall that given a filtration
$G_{\bullet}$ we denote by $G_{\bullet}^{+k}$ the shifted filtration, which
has $i$-th term $G_{i+k}$). In particular, if $(G,G_{\bullet})$ is abelian
with the lower central series, then
$\operatorname{c}_{0}\times\operatorname{c}_{1}$ is in
$\operatorname{C}^{n}(\operatorname{X}\Join_{1}\operatorname{X})$ if and only
if $\operatorname{c}_{0}$ is a standard $n$-cube on $G$ and
$\operatorname{c}_{1}$ is a translate of $\operatorname{c}_{0}$, i.e. for some
$h\in G$ we have $\operatorname{c}_{1}(v)=\operatorname{c}_{0}(v)+h$ for all
$v$. Still in this abelian case, for $k\geq 2$ we have
$\operatorname{c}_{0}\times\operatorname{c}_{1}\in\operatorname{C}^{n}(\operatorname{X}\Join_{k}\operatorname{X})$
if and only if $\operatorname{c}_{1}=\operatorname{c}_{0}$ is a standard
$n$-cube on $G$.

Arrow spaces with $k>1$ will be used starting in Subsection 4, for now we use
mainly the case $k=1$.

Note that every cube
$\operatorname{c}\in\operatorname{C}^{n+1}(\operatorname{X})$ is a $1$-arrow
of $n$-cubes, indeed
$\operatorname{c}=\langle\operatorname{c}(\cdot,0),\operatorname{c}(\cdot,1)\rangle_{1}$,
where each map $\operatorname{c}(\cdot,i)$ is in
$\operatorname{C}^{n}(\operatorname{X})$ by the composition axiom. Therefore,
we have

$\operatorname{C}^{n}(\operatorname{X}\Join_{1}\operatorname{X})=\operatorname{\wp}_{n}(\operatorname{C}^{n+1}(\operatorname{X}))\subset\operatorname{C}^{n}(\operatorname{X})\times\operatorname{C}^{n}(\operatorname{X}).$
(3)

For $k>1$, not every cube in $\operatorname{C}^{n+k}(\operatorname{X})$ is a
$k$-arrow of $n$-cubes. Denote by $\textrm{Ar}_{k}^{n}(\operatorname{X})$ the
subset of $\operatorname{X}^{\\{0,1\\}^{n+k}}$ that is the image of the set of
maps $f_{0}\times f_{1}$, $f_{i}\in\operatorname{X}^{\\{0,1\\}^{n}}$ under
$f_{0}\times f_{1}\mapsto\langle f_{0},f_{1}\rangle_{k}$. Then the latter map
is clearly invertible on $\textrm{Ar}_{k}^{n}(\operatorname{X})$, with inverse
denoted by $\operatorname{\wp}_{n,k}$ (thus
$\operatorname{\wp}_{n,1}=\operatorname{\wp}_{n}$). Then we have

$\operatorname{C}^{n}(\operatorname{X}\Join_{k}\operatorname{X})=\operatorname{\wp}_{n,k}(\operatorname{C}^{n+k}(\operatorname{X})\cap\textrm{Ar}_{k}^{n}(\operatorname{X}))\subset\operatorname{C}^{n}(\operatorname{X})\times\operatorname{C}^{n}(\operatorname{X}).$
(4)

######  Lemma 1.21.

Let $\operatorname{X}$ be a $k$-step nilspace and let $i\geq 1$. Then
$\operatorname{X}\Join_{i}\operatorname{X}$ is a $k$-step nilspace, not
necessarily ergodic. If $\operatorname{X}$ is $\ell$-fold ergodic then
$\operatorname{X}\Join_{i}\operatorname{X}$ is $(\ell-i)$-fold ergodic.

###### Proof.

Let $\operatorname{Y}=\operatorname{X}\Join_{i}\operatorname{X}$. We first
check the composition axiom. If
$\operatorname{c}_{0}\times\operatorname{c}_{1}$ is an $n$-cube on
$\operatorname{Y}$ and $\phi:\\{0,1\\}^{m}\to\\{0,1\\}^{n}$ is a morphism,
then
$(\operatorname{c}_{0}\times\operatorname{c}_{1})\operatorname{\circ}\phi=(\operatorname{c}_{0}\operatorname{\circ}\phi)\times(\operatorname{c}_{1}\operatorname{\circ}\phi)$
satisfies
$\langle\operatorname{c}_{0}\operatorname{\circ}\phi,\operatorname{c}_{1}\operatorname{\circ}\phi\rangle_{i}=\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{i}\operatorname{\circ}\phi^{\prime}:\\{0,1\\}^{m+i}\to\operatorname{X}$,
where $\phi^{\prime}:\\{0,1\\}^{m+i}\to\\{0,1\\}^{n+i}$,
$(v,w)\mapsto(\phi(v),w)$ is a morphism. Hence composition for
$\operatorname{C}^{n}(\operatorname{Y})$ follows from composition for
$\operatorname{C}^{n+i}(\operatorname{X})$.

We now check the completion axiom. Let
$\operatorname{c}^{\prime}:\\{0,1\\}^{n}\setminus\\{1^{n}\\}\to\operatorname{Y}$
be an $n$-corner, and let
$\operatorname{c}_{0}^{\prime},\operatorname{c}_{1}^{\prime}:\\{0,1\\}^{n}\setminus\\{1^{n}\\}\to\operatorname{X}$
be the maps such that
$\operatorname{c}^{\prime}=\operatorname{c}_{0}^{\prime}\times\operatorname{c}_{1}^{\prime}$.
Then by assumption for each $(n-1)$-dimensional lower-face
$F\subset\\{0,1\\}^{n}$ we have
$\operatorname{c}^{\prime}\operatorname{\circ}\phi_{F}=(\operatorname{c}_{0}^{\prime}\operatorname{\circ}\phi_{F})\times(\operatorname{c}_{1}^{\prime}\operatorname{\circ}\phi_{F})\in\operatorname{C}^{n-1}(\operatorname{Y})$,
that is
$\langle\operatorname{c}_{0}^{\prime}\operatorname{\circ}\phi_{F},\operatorname{c}_{1}^{\prime}\operatorname{\circ}\phi_{F}\rangle_{i}\in\operatorname{C}^{n-1+i}(\operatorname{X})$.
In particular $\operatorname{c}_{0}^{\prime},\operatorname{c}_{1}^{\prime}$
are both $(n-1)$-corners on $\operatorname{X}$. Let $\operatorname{c}_{0}$ be
a completion of $\operatorname{c}_{0}^{\prime}$. Let
$\operatorname{c}^{\prime\prime}:\\{0,1\\}^{n+i}\setminus\\{1^{n+i}\\}\to\operatorname{X}$
be the map defined by
$\operatorname{c}^{\prime\prime}(v,w)=\operatorname{c}_{0}(v)$ if $w\neq
1^{i}$, and
$\operatorname{c}^{\prime\prime}(v,1^{i})=\operatorname{c}_{1}^{\prime}(v)$
for $v\neq 1^{n}$. We claim that $\operatorname{c}^{\prime\prime}$ is an
$(n+i)$-corner on $\operatorname{X}$. To see this, fix any $j\in[n+i]$ and
consider the $(n+i-1)$-dimensional lower face
$F_{j}^{\prime}=\\{u\in\\{0,1\\}^{n+i}:u\scalebox{0.8}{$(j)$}=0\\}$. If
$j\in[n]$, then
$\operatorname{c}^{\prime\prime}\operatorname{\circ}\phi_{F^{\prime}_{j}}=\langle\operatorname{c}_{0}^{\prime}\operatorname{\circ}\phi_{F},\operatorname{c}_{1}^{\prime}\operatorname{\circ}\phi_{F}\rangle_{i}$,
where $F$ is the $(n-1)$-dimensional lower face of $\\{0,1\\}^{n}$ obtained by
deleting the last $i$ coordinates from elements of $F^{\prime}_{j}$. Hence
$\operatorname{c}^{\prime\prime}\operatorname{\circ}\phi_{F^{\prime}_{j}}\in\operatorname{C}^{n-1+i}(\operatorname{X})$
for such values of $j$. If $j\in[n+1,n+i]$, then
$\operatorname{c}^{\prime\prime}\operatorname{\circ}\phi_{F^{\prime}_{j}}(v,w)=\operatorname{c}_{0}(v)$
for all $w\neq 1^{i-1}$. But this equals the cube $\operatorname{c}_{0}$
composed with the morphism $(v,w)\mapsto v$, so we also have
$\operatorname{c}^{\prime\prime}\operatorname{\circ}\phi_{F^{\prime}_{j}}\in\operatorname{C}^{n-1+i}(\operatorname{X})$
in this case. Thus $\operatorname{c}^{\prime\prime}$ is indeed an
$(n+i)$-corner on $\operatorname{X}$ and so it can be completed to a
$(n+i)$-cube, which by construction has the form
$\langle\operatorname{c}_{0},\operatorname{c}_{1}\rangle_{i}$ for some
$n$-cube $\operatorname{c}_{1}$ completing $\operatorname{c}_{1}^{\prime}$.
Hence $\operatorname{c}_{0}\times\operatorname{c}_{1}$ completes
$\operatorname{c}^{\prime}$.
Note that if $\operatorname{X}$ is $k$-step then the above $n$-cubes
$\operatorname{c}_{i}$ are the unique completions of
$\operatorname{c}_{i}^{\prime}$ for $i=0,1$ respectively, and therefore the
completion of $\operatorname{c}^{\prime}$ is unique, so that
$\operatorname{Y}$ is also $k$-step.

The last claim in the lemma is clear, for if every map
$\\{0,1\\}^{\ell}\to\operatorname{X}$ is a cube, then in particular $\langle
f_{0},f_{1}\rangle_{i}\in\operatorname{C}^{\ell}(\operatorname{X})$ for every
$f_{0},f_{1}:\\{0,1\\}^{\ell-i}\to\operatorname{X}$. ∎

The last construction that we define in this section is roughly speaking a
means to ‘differentiate’ the cube structure on $\operatorname{X}$ with respect
to a fixed point $x\in\operatorname{X}$, in such a way that from a $k$-step
nilspace we obtain a $(k-1)$-step nilspace.

######  Definition 1.22.

Let $\operatorname{X}$ be a nilspace and let $x\in\operatorname{X}$. We define
the nilspace $\partial_{x}\operatorname{X}$ to be the set $\operatorname{X}$
with the cubespace structure inherited by embedding $\operatorname{X}$ in
$\operatorname{X}\Join_{1}\operatorname{X}$ via the map $y\mapsto(x,y)$. Thus
we declare $\operatorname{c}:\\{0,1\\}^{n}\to\operatorname{X}$ to be in
$\operatorname{C}^{n}(\partial_{x}\operatorname{X})$ if the map111Here we
denote by $x$ the map on $\\{0,1\\}^{n}$ with constant value $x$.
$x\times\operatorname{c}:\\{0,1\\}^{n}\to\operatorname{X}\times\operatorname{X}$,
$v\mapsto(x,\operatorname{c}(v))$ is in
$\operatorname{C}^{n}(\operatorname{X}\Join_{1}\operatorname{X})$, that is if
the 1-arrow $\langle x,\operatorname{c}\rangle_{1}$ is in
$\operatorname{C}^{n+1}(\operatorname{X})$.

The following lemma relates higher iterations of $\partial_{x}$ to higher-
level arrow spaces.

######  Lemma 1.23.

We have
$\operatorname{c}\in\operatorname{C}^{n}(\partial_{x}^{i}\operatorname{X})$ if
and only if
$x\times\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X}\Join_{i}\operatorname{X})$.

###### Proof.

This follows from the fact that if we iterate $i$ times the operation of
taking a 1-arrow $\operatorname{c}\mapsto\langle
x,\operatorname{c}\rangle_{1}$ then we obtain the $i$-arrow $\langle
x,\operatorname{c}\rangle_{i}$. ∎

We now show that $\partial_{x}(\operatorname{X})$ is a $(k-1)$-step nilspace,
rather than $k$-step like $\operatorname{X}\Join_{1}\operatorname{X}$. The
reason for the decrease in the step is that, while completing a $k$-corner on
$\operatorname{X}\Join_{1}\operatorname{X}$ amounted to completing a
$(k+1)$-cube minus a 1-face on $\operatorname{X}$, completing a $k$-corner on
$\partial_{x}(\operatorname{X})$ amounts to completing a full $(k+1)$-corner
on $\operatorname{X}$.

######  Lemma 1.24.

Let $\operatorname{X}$ be a $k$-step nilspace and let $x\in\operatorname{X}$.
Then $\partial_{x}\operatorname{X}$ is a $(k-1)$-step nilspace, not
necessarily ergodic. If $\operatorname{X}$ is $k$-fold ergodic, then
$\partial_{x}\operatorname{X}$ is $(k-1)$-fold ergodic.

###### Proof.

Composition and completion are clearly inherited from
$\operatorname{X}\Join_{1}\operatorname{X}$. Concerning completion, note that
here an $n$-corner in $\partial_{x}\operatorname{X}$ is to be completed to an
$n$-cube $\operatorname{c}$ in
$\operatorname{C}^{n}(\partial_{x}\operatorname{X})$ rather than just in
$\operatorname{C}^{n}(\operatorname{X}\Join_{1}\operatorname{X})$, so
$\operatorname{c}$ must be of the form $x\times\operatorname{c}_{1}$, and is
therefore the completion of the $(n+1)$-corner on $\operatorname{X}$ that
equals the constant function $x$ on the lower-face
$v\scalebox{0.8}{$(n+1)$}=0$ and equals some $n$-corner on
$\\{v:v\scalebox{0.8}{$(n+1)$}=1,\,v\neq 1^{n+1}\\}$. Hence this completion is
unique for $n=k$ as claimed. ∎

######  Example 1.25.

Let $\operatorname{X}$ be a group $G$ with a filtration $G_{\bullet}$ and the
associated cube structure $(\operatorname{C}^{n}(G_{\bullet}))_{n\geq 0}$.
Then for $x=\mathrm{id}_{G}$, using (3) we see that
$\partial_{x}\operatorname{X}$ equals $G$ with the filtration
$G_{\bullet}^{+1}$ and its associated cubes. Note that if $G_{0}=G_{1}=G$ but
$G_{2}$ is a proper subgroup of $G$, then while $\operatorname{X}$ is ergodic,
the nilspace $\partial_{x}\operatorname{X}$ is not.

###  2 $k$-step nilspaces as abelian bundles of degree $k$ acted upon by
filtered groups

In this section we work towards a general decomposition theorem for nilspaces,
which will be eventually obtained as Theorem 2.19.

To begin with, we describe another way to get a new cubespace from a given
one.

######  Definition 2.1 (Quotient cubespace).

Let $\operatorname{X}$ be a cubespace and let $\sim$ be an equivalence
relation on $\operatorname{X}$. The _quotient cubespace_ is the quotient set
$\operatorname{Y}:=\operatorname{X}/\sim$ together with the cubes
$\pi\operatorname{\circ}\operatorname{c}$, for every
$\operatorname{c}\in\operatorname{C}^{n}(\operatorname{X})$, $n\geq 0$, where
$\pi:\operatorname{X}\to\operatorname{Y}$ is the canonical projection.

######  Definition 2.2 (Factor of a nilspace).

Let $\operatorname{X}$ be a nilspace and let $\sim$ be an equivalence relation
on $\operatorname{X}$. If the quotient cubespace $\operatorname{X}/\sim$ is
itself a nilspace, then it is called a _factor_ of $\operatorname{X}$.

In the general decomposition of $\operatorname{X}$ that we shall prove, the
building blocks will be special factors of $\operatorname{X}$ called
_characteristic factors_. These are obtained as quotients of
$\operatorname{X}$ under certain equivalence relations, which are defined in
the following subsection.

####  1 Characteristic factors

The following relations on a nilspace are crucial for the decomposition
theorem. They can be viewed as analogues for nilspaces of the _regionally
proximal relations of order_ $k$ for a dynamical system, which were defined in
[13, §3.3] (see also [12, 14]).

######  Definition 2.3.

Let $\operatorname{X}$ be a nilspace. For each positive integer $k$ we denote
by $\sim_{k}$ the relation on $\operatorname{X}$ defined as follows:

$x\sim_{k}y\;\;\;\Leftrightarrow\;\;\;\exists\,\operatorname{c}_{0},\operatorname{c}_{1}\in\operatorname{C}^{k+1}(\operatorname{X})\;\textrm{
such that
}\;\operatorname{c}_{0}(0^{k+1})=x,\;\;\operatorname{c}_{1}(0^{k+1})=y,\;\;\operatorname{c}_{0}(v)=\operatorname{c}_{1}(v)\;\;\forall\,v\neq
0^{k+1}.$

This relation is reflexive and symmetric. To prove that it is also transitive,
we use the following result.

######  Lemma 2.4.

Two elements $x,y\in\operatorname{X}$ satisfy $x\sim_{k}y$ if and only if the
map $\operatorname{c}:\\{0,1\\}^{k+1}\to\operatorname{X}$ sending $0^{k+1}$ to
$y$ and all other elements $v$ to $x$ is a cube in
$\operatorname{C}^{k+1}(\operatorname{X})$.

###### Proof.

The ‘if’ direction is clear since every constant map
$\\{0,1\\}^{k+1}\to\operatorname{X}$ is a cube. For the converse, suppose that
$\operatorname{c}_{0},\operatorname{c}_{1}\in\operatorname{C}^{k+1}(\operatorname{X})$
satisfy the condition in Definition 2.3, and let $\phi$ be the map from the
tricube $T_{k+1}$ to $\\{0,1\\}^{k+1}$ defined by $\phi=f^{k+1}$, where
$f(-1)=f(1)=1,f(0)=0$ (thus $\phi$ ‘reflects’ each of the $2^{k+1}$ subcubes
of $T_{k+1}$ onto $\\{0,1\\}^{k+1}$). Note that the function
$\operatorname{c}_{0}\operatorname{\circ}\phi$ is a morphism
$T_{k+1}\to\operatorname{X}$, as explained in the previous section (we can
represent this map as $T_{k+1}$ with a reflected copy of
$\operatorname{c}_{0}$ in each of the $2^{k+1}$ sub-cubes). Now observe that,
by our assumption on $\operatorname{c}_{0},\operatorname{c}_{1}$, the map
obtained from $\operatorname{c}_{0}\operatorname{\circ}\phi$ by changing the
value at $1^{k+1}$ from $x$ to $y$ is still a morphism
$T_{k+1}\to\operatorname{X}$, which we denote by $g$. By Lemma 1.16, we then
have
$g\operatorname{\circ}\omega_{k+1}\in\operatorname{C}^{k+1}(\operatorname{X})$,
and the result follows. ∎

######  Corollary 2.5.