1.6. Inverse conjecture over the integers 101

the identity

[x, y, z] = [{x}, {y − x z}, {z}]

where x is the greatest integer part of x, and {x} := x − x ∈ [0, 1) is

the fractional part function. Conclude that G/Γ is topologically equivalent

to the unit cube [0,

1]3

quotiented by the identifications

(0,y,z) ∼ (1,y,z),

(x, 0,z) ∼ (x, 1,z),

(x, y, 0) ∼ (x, {y − x}, 1)

between opposite faces.

Note that by using the projection (x, y, z) → (x, z), we can view the

Heisenberg nilmanifold G/Γ as a twisted circle bundle over

(R/Z)2,

with

the fibers being isomorphic to the unit circle R/Z. Show that G/Γ is not

homeomorphic to

(R/Z)3.

(Hint: Show that there are some non-trivial

homotopies between loops that force the fundamental group of G/Γ to be

smaller than

Z3.)

The logarithm log(Γ) of the discrete cocompact subgroup Γ can be shown

to be a lattice of the Lie algebra g. After a change of basis, one can thus

view the latter algebra as a standard vector space

Rd

and the lattice as

Zd.

Denoting the standard generators of the lattice (and the standard basis

of Rd) as e1,...,ed, we then see that the Lie bracket [ei,ej] of two such

generators must be an integer combination of more generators:

[ei,ej] =

d

k=1

cijkek.

The structure constants cijk describe completely the Lie group structure of

G and Γ. The rational subgroups G≥l can also be described by picking some

generators for log(Γ≥i), which are integer combinations of the e1,...,ed. We

say that the filtered nilmanifold has complexity at most M if the dimension

and degree is at most M, and the structure constants and coeﬃcients of the

generators also have magnitude at most M. This is an admittedly artificial

definition, but for quantitative applications it is necessary to have some

means to quantify the complexity of a nilmanifold.

A polynomial orbit in a filtered nilmanifold G/Γ is a map O : Z → G/Γ

of the form O(n) := g(n)Γ, where g : Z → G is a polynomial sequence. For

instance, any linear orbit O(n) =

gnx,

where x ∈ G/Γ and g ∈ G, is a

polynomial orbit.

Exercise 1.6.16. For any α, β ∈ R, show that the sequence

n → [{−αn}, {αn βn}, {βn}]