Embedding products of trees into higher rank

TL;DR

This paper demonstrates that products of hyperbolic planes and 3-regular trees can be embedded into higher-rank symmetric spaces and Euclidean buildings, extending previous results with purely geometric methods.

Contribution

It establishes new quasi-isometric and bi-Lipschitz embeddings of product spaces into symmetric spaces and Euclidean buildings of the same rank, generalizing prior work.

Findings

Embedding of hyperbolic plane products into symmetric spaces

Bi-Lipschitz embedding of tree products into Euclidean buildings

Extension of Fisher–Whyte's results to non Bruhat–Tits buildings

Abstract

We show that there exists a quasi-isometric embedding of the product of $n$ copies of $\mathbb{H}_{\mathbb{R}}^2$ into any symmetric space of non-compact type of rank $n$, and there exists a bi-Lipschitz embedding of the product of $n$ copies of the $3$-regular tree $T_3$ into any thick Euclidean building of rank $n$ with co-compact affine Weyl group. This extends a previous result of Fisher--Whyte. The proof is purely geometrical, and the result also applies to the non Bruhat--Tits buildings.