Quasi-isometric rigidity of three manifold groups

TL;DR

This paper proves that finitely generated Kleinian groups and three-manifold groups are quasi-isometrically rigid, meaning their large-scale geometric structures uniquely determine their algebraic properties.

Contribution

It establishes the quasi-isometric rigidity for classes of Kleinian and three-manifold groups, a significant advancement in geometric group theory.

Findings

Finitely generated Kleinian groups are quasi-isometrically rigid.

Three-manifold groups are quasi-isometrically rigid.

Large-scale geometry determines algebraic structure in these groups.

Abstract

We provide a proof that the classes of finitely generated Kleinian groups and of three-manifold groups are quasi-isometrically rigid.