Quasi-isometric co-Hopficity of non-uniform lattices in rank-one   semi-simple Lie groups
  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. 
Ilya Kapovich and Anton Lukyanenko
math.GT math.GR