An Interpolation from Sol to Hyperbolic Space

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, $$d{s}^2=e^{-2z}d{x}^2+e^{2\alpha z}d{y}^2+d{z}^2$$ 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.