  We study quasi-isometric embeddings of symmetric spaces and non-uniform
irreducible lattices in semisimple higher rank Lie groups. We show that any
quasi-isometric embedding between symmetric spaces of the same rank can be
decomposed into a product of quasi-isometric embeddings into irreducible
symmetric spaces. We thus extend earlier rigidity results about quasi-isometric
embeddings to the setting of semisimple Lie groups. We also present some
examples when the rigidity does not hold, including first examples in which
every flat is mapped into multiple flats.
