Large scale geometry of negatively curved $\R^n \rtimes \R$

TL;DR

This paper classifies negatively curved semi-direct product manifolds of the form ^n times \u00a0, showing that their quasiisometry groups are mostly almost similarities, with boundary maps playing a key role.

Contribution

It provides a comprehensive quasiisometry classification of negatively curved ^n times  manifolds, highlighting the structure of their boundary maps.

Findings

All such manifolds are quasiisometric only if they are biLipschitz to hyperbolic spaces.

Quasiisometries are almost similarities except in the hyperbolic case.

Boundary quasisymmetric maps determine the quasiisometry types.

Abstract

We classify all negatively curved $\R^n \rtimes \R$ up to quasiisometry. We show that all quasiisometries between such manifolds (except when they are biLipschitz to the real hyperbolic spaces) are almost similarities. We prove these results by studying the quasisymmetric maps on the ideal boundary of these manifolds.