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.