  This work builds on the foundation laid by Gordon and Wilson in the study of
isometry groups of solvmanifolds, i.e. Riemannian manifolds admitting a
transitive solvable group of isometries. We restrict ourselves to a natural
class of solvable Lie groups called almost completely solvable; this class
includes the completely solvable Lie groups. When the commutator subalgebra
contains the center, we have a complete description of the isometry group of
any left-invariant metric using only metric Lie algebra information.
  Using our work on the isometry group of such spaces, we study quotients of
solvmanifolds. Our first application is to the classification of homogeneous
Ricci soliton metrics. We show that the verification of the Generalized
Alekseevsky Conjecture reduces to the simply-connected case. Our second
application is a generalization of a result of Heintze on the rigidity of
existence of compact quotients for certain homogeneous spaces. Heintze's result
applies to spaces with negative curvature. We remove all the geometric
requirements, replacing them with algebraic requirements on the homogeneous
structure.
