Quasi-isometric co-Hopficity of non-uniform lattices in rank-one semi-simple Lie groups

TL;DR

The paper proves that non-uniform lattices in rank-one semi-simple Lie groups, except for the real hyperbolic plane, are quasi-isometrically co-Hopf, meaning every quasi-isometric embedding is essentially surjective.

Contribution

It establishes the quasi-isometric co-Hopf property for a broad class of non-uniform lattices in rank-one semi-simple Lie groups, excluding the real hyperbolic plane.

Findings

Non-uniform lattices in rank-one semi-simple Lie groups are quasi-isometrically co-Hopf.

Every quasi-isometric embedding of such a lattice is coarsely onto.

The result excludes the case of the real hyperbolic plane.

Abstract

We prove that if $G$ is a non-uniform lattice in a rank-one semi-simple Lie group $\ne Isom(\H^2_\R)$ then $G$ is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding $G\to G$ is coarsely onto and thus is a quasi-isometry.