Volume entropy of a family of rank one, split-solvable Lie groups of Abelian type

TL;DR

This paper derives a formula for the volume entropy of a family of metrics on Euclidean space, generalizing known geometries like SOL and hyperbolic space, and applies it to solve a conjecture about 3-manifolds.

Contribution

It introduces a unified approach to compute volume entropy for metrics on Euclidean space related to solvable Lie groups of Abelian type, extending previous models.

Findings

Derived a formula for volume entropy of the metric family.

Solved a conjecture involving 3-manifolds interpolating between SOL and hyperbolic space.

Unified the understanding of geometric structures on Euclidean space with applications to manifold theory.

Abstract

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. In this scenario, G is isometric to Euclidean space with a metric of the type considered. We have derived a formula for the volume entropy of metrics in this family and used it to solve a conjecture related to a family of 3-manifolds that interpolates between the SOL group and hyperbolic space.