TITLE: Embedding Euclidean buildings into products of trees
QUESTION [8 upvotes]: A Euclidean building has a natural metric space structure. (A definition of Euclidean building can be found on Wikipedia, or, more expansively, in Section 4 of Kleiner-Leeb.)
Question: Is it true that every (finite rank) Euclidean building has a biLipschitz embedding into a finite product of metric $\mathbb{R}$-trees?
It follows from the results of Lang-Schlichenmaier that a Euclidean building $X$ of rank $n$ admits an embedding $f$ into the product $T$ of $n+1$ trees, with the property that for some $C>1$ and $p>0$,
$$ C^{-1}d_X(x,y)^p \leq d_T(f(x),f(y)) \leq Cd_X(x,y)^p.$$
So what I am asking is if there is any obstruction to taking $p=1$ (maybe increasing the number of trees in the product if necessary).
Apologies if the question is foolish, I am not an expert on buildings.

REPLY [6 votes]: The problem of existence/nonexistence of quasi-isometric embeddings $X\to Y$ between symmetric spaces and locally compact Euclidean buildings of rank $\ge 2$ is wide-open in the case when $rank(Y)> rank(X)$ (assuming, of course, that $dim(X)<dim(Y)$ in the setting of symmetric spaces). The most up-to date results and references can be found in this paper paper by Fisher and Whyte from last year.  For instance, it is not unreasonable to expect that each locally finite Euclidean building $X$ admits a QI embedding into the product of $n$ simplicial trees of valence $3$, where $n$ depends on $X$.