Quasi-isometric embeddings of non-uniform lattices

TL;DR

The paper proves that quasi-isometric embeddings between certain non-uniform lattices in Lie groups are close to homomorphisms, highlighting conditions under which this rigidity holds or fails due to algebraic reasons.

Contribution

It establishes conditions under which quasi-isometric embeddings of non-uniform lattices are near homomorphisms, extending rigidity results in geometric group theory.

Findings

Quasi-isometric embeddings are often close to homomorphisms in specified settings.

Counterexamples exist where this near-homomorphism property fails due to algebraic reasons.

The results apply to lattices like SL(n,Z) and SL(n,Z[i]) under certain conditions.

Abstract

Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $\Gamma <G$ and $\Lambda < G'$ 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.