94 1. Higher order Fourier analysis
It turns out (somewhat remarkably) that these notions can be satisfac-
torily generalised to a non-abelian setting, this was first observed by Leib-
man [Le1998, Le2002]. The (now multiplicative) groups H, G need to be
equipped with an additional structure, namely that of a filtration.
Definition 1.6.1 (Filtration). A filtration on a multiplicative group G is a
family (G≥i)i=0

of subgroups of G obeying the nesting property
G G≥0 G≥1 . . .
and the filtration property
[G≥i,G≥j] G≥i+j
for all i, j 0, where [H, K] is the group generated by {[h, k] : h H, k
K}, where [h, k] :=
hkh−1k−1
is the commutator of h and k. We will refer
to the pair G• = (G, (G≥i)i=0)

as a filtered group. We say that an element
g of G has degree i if it belongs to G≥i; thus, for instance, a degree i
and degree j element will commute modulo i + j errors.
In practice we usually have G≥0 = G. As such, we see that [G, G≥j]
G≥j for all j, and so all the G≥j are normal subgroups of G.
Exercise 1.6.2. Define the lower central series
G = G0 = G1 G2 . . .
of a group G by setting G0,G1 := G and Gi+1 := [G, Gi] for i 1. Show
that the lower central series (Gj)j=0

is a filtration of G. Furthermore, show
that the lower central series is the minimal filtration that starts at G, in the
sense that if (G≥j)j=0

is any other filtration with G≥0 = G, then G≥j G≥j
for all j.
Example 1.6.2. If G is an abelian group, and d 0, we define the degree
d filtration (G, d) on G by setting G≥i := G if i d and G≥i = {id} for
i d.
Example 1.6.3. If G• = (G, (G≥i)i=0)

is a filtered group, and k 0, we
define the shifted filtered group G•
+k
:= (G, (G≥i+k)i=0);

this is clearly again
a filtered group.
Definition 1.6.4 (Host-Kra groups). Let G• = (G, (G≥i)i=0)

be a filtered
group, and let k 0 be an integer. The Host-Kra group
HKk(G•)
is the
subgroup of
G{0,1}k
generated by the elements gF with F an arbitrary face
in {0,
1}k
and g an element of G≥k−dim(F ), where gF is the element of
G{0,1}k
whose coordinate at ω is equal to g when ω F and equal to {id} otherwise.
From construction we see that the Host-Kra group is symmetric with
respect to the symmetry group Sk
(Z/2Z)k
of the unit cube {0,
1}k.
We
will use these symmetries implicitly in the sequel without further comment.
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