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](http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CCAQFjAA&url=http%3A%2F%2Fmath.nyu.edu%2F~bkleiner%2Fsymm.pdf&ei=6Ne2VI-5O4GPgwSi4oDoBQ&usg=AFQjCNFa3zbDknDrMBUgHJsBY9ObDYl7pA&sig2=ux5QribcDgJZMyQA3uWbfA&bvm=bv.83640239,d.eXY).)


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](http://arxiv.org/abs/math/0410048) 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.



Embedding Euclidean buildings into products of trees
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](http://arxiv.org/pdf/1407.0445v2.pdf) 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 $X$. 


