Quasi-isometric embeddings of symmetric spaces and lattices: reducible case

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple 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 semi-simple 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.

Automated summary

The study focuses on quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semi-simple higher rank Lie groups, showing 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. It extends earlier rigidity results about quasi-isometric embeddings to the setting of semi-simple Lie groups and provides examples where rigidity does not hold, including cases where every flat is mapped into multiple flats.