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

2024/10/14 by René García-Lara, Garcia-Lara, Rene · 1 citation
Mathematics · Medicine · #28A75 (Secondary) #58C35 (Primary) 51M25 #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematical and Theoretical Epidemiology and Ecology Models #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2410.10631

openalex publication_date 2024/10/14 · openalex created_date 2024/10/20 · openalex updated_date 2026/07/28

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.

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• Volume growth of horospheres in diagonalizable Heintze groups

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