100 1. Higher order Fourier analysis
group G/G≥s (with the degree s 1 filtration (G≥i/G≥s)) by the central
vector space G≥s. Thus one can view degree s filtered nilpotent groups as an
s-fold iterated tower of central extensions by finite-dimensional vector spaces
starting from a point; for instance, the Heisenberg group is an extension of
by R.
We thus see that nilpotent filtered Lie groups are generalisations of vec-
tor spaces (which correspond to the degree 1 case). We now turn to filtered
nilmanifolds, which are generalisations of tori. A degree s filtered nilman-
ifold G/Γ = (G/Γ,G•, Γ) is a filtered degree s nilpotent Lie group G•,
together with a discrete subgroup Γ of G, such that all the subgroups G≥i
in the filtration are rational relative to Γ, which means that the subgroup
Γ≥i := Γ G≥i is a cocompact subgroup of G≥i (i.e., the quotient space
G≥i/Γ≥i is cocompact, or equivalently one can write G≥i = Γ≥i · K≥i for
some compact subset K≥i of G≥i. Note that the subgroups Γ≥i give Γ the
structure of a degree s filtered nilpotent group Γ•.
Exercise 1.6.14. Let G :=
and Γ :=
and let α R. Show that the
subgroup {(x, αx) : x R} of G is rational relative to Γ if and only if α is
a rational number; this may help explain the terminology “rational”.
By hypothesis, the quotient space G/Γ = G≥0/Γ≥0 is a smooth compact
manifold. The space G≥s/Γ≥s is a compact connected abelian Lie group, and
is thus a torus; the degree s filtered nilmanifold G/Γ can then be viewed as
a principal torus bundle over the degree s− 1 filtered nilmanifold G/(G≥sΓ)
with G≥s/Γ≥s as the structure group; thus one can view degree s filtered
nilmanifolds as an s-fold iterated tower of torus extensions starting from a
point. For instance, the Heisenberg nilmanifold
G/Γ :=

1 R R
0 0 1


1 Z Z
0 0 1

is an extension of the two-dimensional torus
by the circle R/Z.
Every torus of some dimension d can be viewed as a unit cube [0,
with opposite faces glued together; up to measure zero sets, the cube then
serves as a fundamental domain for the nilmanifold. Nilmanifolds can be
viewed the same way, but the gluing can be somewhat “twisted”:
Exercise 1.6.15. Let G/Γ be the Heisenberg nilmanifold. If we abbreviate
[x, y, z] :=

1 x y
0 0 1

for all x, y, z R, show that for almost all x, y, z, that [x, y, z] has exactly
one representation of the form [a, b, c] with a, b, c [0, 1], which is given by
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