Quasiisometries of negatively curved homogeneous manifolds associated with Heisenberg groups

TL;DR

This paper classifies negatively curved homogeneous manifolds related to Heisenberg groups up to quasiisometry, showing most quasiisometries are almost similarities, and analyzes boundary maps to achieve these results.

Contribution

It provides a classification of these manifolds up to quasiisometry and characterizes their quasiisometries as almost similarities, extending understanding of their geometric structure.

Findings

All quasiisometries are almost similarities except for complex hyperbolic spaces.

Classification of manifolds up to quasiisometry based on derivations on Heisenberg algebras.

Analysis of boundary quasisymmetric maps to establish quasiisometry properties.

Abstract

We study quasiisometries between negatively curved homogeneous manifolds associated with diagonalizable derivations on Heisenberg algebras. We classify these manifolds up to quasiisometry, and show that all quasiisometries between such manifolds (except when they are complex hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.