Quasi-isometric embeddings of non-uniform lattices

Abstract

Let $G$ and $G^{\prime }$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\Gamma <G$ and $\Lambda <G^{\prime }$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $\Gamma$ into $\Lambda$ 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.