We study a family of metrics on Euclidean space that generalize the left-invariant metric of the SOL group and the metric of the logarithmic model of Hyperbolic space. Suppose G is a connected, simply connected, Heintze group of abelian type with diagonalizable derivation or the horospherical product of two such groups. With this hypothesis, G is isometric to Euclidean space with a metric of the type considered. Using techniques from differential geometry and integration on Riemannian manifolds, we found bounds on the growth rate of geodesic balls. We found an asymptotic approximation formula for volume growth of metrics of negative curvature on Heintze abelian groups of diagonal type and derived a formula for the volume entropy of metrics in this family that applies to horospherical products of these groups. With help of this formula, we solved a conjecture about the volume entropy of a family of 3-manifolds that interpolates between the SOL group and hyperbolic space.
Rene Israel Garcia-Lara (Fri,) studied this question.