Discreteness properties of translation numbers in solvable groups

We dene a group to be translation proper if it carries a left-invariant metric in which the translation numbers of the non-torsion elements are nonzero and translation discrete if they are bounded away from zero. The main results of this paper are that a translation proper solvable group of nite virtual cohomological dimension is metabelian-by-nite, and that a translation discrete solvable group of nite virtual cohomological dimension, m, is a nite extension of Zm. 1.