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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.

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    $\begingroup$ I think taking $p=1$ is prevented by the Kleiner-Leeb result on quasi-isometric rigidity of Euclidean buildings: see Theorem 1.1.2 in math.nyu.edu/~bkleiner/symm.pdf $\endgroup$
    – Alain Valette
    Commented Jan 14, 2015 at 22:45
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    $\begingroup$ Prof. Valette: Thank you for the response. I don't see how Theorem 1.1.2 in Kleiner-Leeb answers my question, as the quasi-isometries considered in that theorem are assumed to be (coarsely) surjective, whereas I am only interested in an embedding result. (Or did I misunderstand? I reiterate my lack of expertise re: buildings.) $\endgroup$
    – user65993
    Commented Jan 14, 2015 at 23:10
  • $\begingroup$ You're right, I overlooked the surjectivity issue. I leave my comment however, as a very partial contribution. $\endgroup$
    – Alain Valette
    Commented Jan 15, 2015 at 9:49
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    $\begingroup$ It's an open question whether $SL_{3}(\mathbf{Q}_p)$ (or equivalently its Euclidean building) has a quasi-isometric embedding into a product of trees (and more generally $SL_d(\mathbf{Q}_p)$ or $SL_d(\mathbf{R})$ for some/any $d\ge 3$). $\endgroup$
    – YCor
    Commented Jan 15, 2015 at 10:03
  • $\begingroup$ Prof. de Cornulier: Thank you. So it appears that my slightly stronger question may be open as well (or false), but certainly that no proof is known. Do you know of a reference to read more about these specific examples? $\endgroup$
    – user65993
    Commented Jan 15, 2015 at 13:45

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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 $X$.

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  • $\begingroup$ Prof. Kapovich: Thank you very much for the overview and the reference to the Fisher-Whyte paper, which I did not know. I have accepted this answer as a clear summary of the current state of knowledge: my question is open. I thank all the experts who took time to comment. $\endgroup$
    – user65993
    Commented Jan 18, 2015 at 2:29

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