Geometric rigidity of quasi-isometries in horospherical products

TL;DR

This paper demonstrates that quasi-isometries in horospherical products of hyperbolic spaces are close to product maps, extending previous results and introducing new invariants for classification.

Contribution

It generalizes the geometric rigidity of quasi-isometries to a broader class of solvable Lie groups and introduces new quasi-isometric invariants.

Findings

Quasi-isometries are close to product maps in horospherical products.

New invariants for classifying these spaces are established.

The results extend previous rigidity theorems to solvable Lie groups.

Abstract

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in [7]. Our work covers the case of solvable Lie groups of the form R ___ (N 1 x N 2), where N 1 and N 2 are nilpotent Lie groups, and where the action on R contracts the metric on N 1 while extending it on N 2. We obtain new quasi-isometric invariants and classi cations for these spaces.