Embedding of hyperbolic spaces in the product of trees

TL;DR

This paper demonstrates how hyperbolic spaces can be embedded into products of trees, revealing geometric relationships and limitations in such embeddings.

Contribution

It establishes the existence of quasi-isometric embeddings of hyperbolic spaces into products of trees for dimensions n≥2, and shows impossibility for certain embeddings involving T×R^m.

Findings

Hyperbolic space H^n embeds into T^n for n≥2.

No quasi-isometric embedding of H^2 into T×R^m exists.

Embeddings depend critically on the dimension and structure of the target space.

Abstract

We show that for each n\ge 2 there is a quasi-isometric embedding of the hyperbolic space H^n in the product T^n=Tx...xT of n copies of a (simplicial) metric tree T. On the other hand, we prove that there is no quasi-isometric embedding H^2 --> TxR^m for any metric tree T and any m\ge 0.