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

TL;DR

This paper investigates the structure of quasi-isometric embeddings between symmetric spaces and lattices in higher rank Lie groups, extending rigidity results and providing examples where rigidity fails.

Contribution

It extends rigidity results for quasi-isometric embeddings to semisimple Lie groups and decomposes embeddings into products of irreducible cases.

Findings

Quasi-isometric embeddings can be decomposed into products of embeddings into irreducible symmetric spaces.

Rigidity results hold for embeddings between symmetric spaces of the same rank.

Counterexamples show cases where rigidity does not apply, including flats mapping into multiple flats.

Abstract

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.