  We study a one-parameter family of nonisomorphic solvable Lie groups, which,
when equipped with canonical left-invariant metrics,
$$ds^2=e^{-2z}dx^2+e^{2\alpha z}dy^2+dz^2$$ becomes an interpolation from a
model of the Sol geometry to a model of Hyperbolic Space, with a stop at
$\mathbb{H}^2\times \mathbb{R}$. These Lie groups are also Bianchi groups of
Type VI with orthogonal coordinates. As a continuation of joint work with
Richard Schwartz on Sol, we primarily analyze those Lie groups in our
interpolation with some positive sectional curvature. Our main result is a
characterization of the cut locus at the identity of the group that maximizes
scalar curvature.
