Hyperbolic spaces with geometric and geometrically finite quasi-actions are symmetric

TL;DR

The paper proves that certain proper metric spaces quasi-isometric to specific group-related spaces are actually quasi-isometric to rank-one symmetric spaces or the real line.

Contribution

It establishes a rigidity result linking quasi-isometries of metric spaces to symmetric spaces or the real line, based on their relation to finitely generated groups.

Findings

Spaces quasi-isometric to a group and a horoball over a group are symmetric spaces or lines.

The result applies to spaces with geometric and geometrically finite quasi-actions.

Provides a classification of such metric spaces based on their quasi-isometric properties.

Abstract

We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line.