In a recent paper, K. B. Lee introduced the notion of an
infra-solvmanifold of type
(R). These manifolds are completely determined by their fundamental group
Π. Such
a Π is a finite extension of a lattice Γ of a solvable
Lie group of type (R) and this
lattice Γ is called the translational part of Π.
Having fixed an abstract group Π occurring as the fundamental group
of an
infra-solvmanifold of type (R), it seems to be hard
to describe, in a formal algebraic
language, which subgroup of Π is the translational part. In
his paper Lee formulated
a conjecture which would solve this problem, however, we show that this
conjecture
fails. Nevertheless, by defining a concept of eigenvalues for
automorphisms of certain
solvable groups (both Lie groups and discrete groups), we are able to prove
a new
theorem, characterizing completely the translational part of the fundamental
group
of an infra-solvmanifold of type (R).