Quasi-isometric embeddings of non-uniform lattices

Let $G$ and $G^{\mathrm{\prime }}$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\mathrm{\Gamma }<G$ and $\mathrm{\Lambda }<G^{\mathrm{\prime }}$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $\mathrm{\Gamma }$ into $\mathrm{\Lambda }$ is at bounded distance from a homomorphism. For example, any quasi-isometric embedding of $SL(n,\mathbb{Z})$ into $SL(n,\mathbb{Z}[i])$ is at bounded distance from a homomorphism. We also include a discussion of some cases when this result is not true for what turn out to be purely algebraic reasons.