Volume growth of horospheres in diagonalizable Heintze groups

TL;DR

This paper investigates the volume growth of horospheres in diagonalizable Heintze groups, showing that their intrinsic geometry classes coincide with isometry and quasi-isometry classes, and explicitly characterizing volume growth when A is not scalar.

Contribution

It provides a detailed analysis of horosphere volume growth in diagonalizable Heintze groups and explicitly characterizes the classes based on volume growth.

Findings

Isometry and quasi-isometry classes of horospheres coincide.

Exactly two classes of horospheres when A is not scalar, distinguished by volume growth.

Explicit calculation of volume growth in these classes.

Abstract

We study the volume growth of horospheres in a Heintze group of the form R ___ A R d with A a diagonal derivation. We conclude that the isometry and quasi-isometry classes of horospheres (with their intrinsic geometry) coincide. Furthermore, if A is not a scalar multiple of the identity, then there are exactly two such classes, characterized by their volume growth, which we calculate explicitly.