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

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