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

Rene Garcia-Lara

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

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arxiv 2410.10631 v1 pith:PWEY7GN5 submitted 2024-10-14 math.DG math.MG

Rene Garcia-Lara This is my paper · ORCID

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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
math.DG 2025-05 conditional novelty 7.0 of 10

In diagonal Heintze groups with non-scalar A, horospheres split into exactly two isometry and quasi-isometry classes, with the non-Euclidean class having volume growth of order r^k, k=(λ1+...+λd)/λ1.

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