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}]
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