It is well-known that expanders are hard to embed into Hilbert (or $\ell^p$) spaces - any embedding of an expander with $n$ vertices has distortion $\Omega(\log n)$.


Can anyone provide a reference (or a quick argument) that the same holds for any "nice" CAT(0) space (like the hyperbolic space) instead of Hilbert space? I'm not giving a precise definition of "nice" here, but this is discussed e.g. in Gromov's "Random Walk in Random Groups".



Embedding expanders in CAT(0) spaces
Here is a class of Hadamard spaces in which, I think, no expander embeds:
Consider a Hadamard space $X$, 
which is also "FDSCBB" in the sense 
Burago, Gromov, and Perel'man (see [http://seven.ihes.fr/~gromov/PDF/3[86].pdf](http://seven.ihes.fr/~gromov/PDF/3%5B86%5D.pdf) section 7).


By Corollary 7.10 in [BGP] The tangent cone has non-negative curvature in Alexandrov sense.
By Theorem 3.19 from [Bridson & Haeflinger] the tangent cone is also CAT(0), which together means it is Euclidean.


So given a finite subset $S\subset X$, let $x\_0\in X$ be its center of mass. 


Since the tangent cone around $x\_0$ is Euclidean we can have a map into Hilbert space $f:S\cup\{x\_0\} \to H$ such that $f(x\_0)=0$ for every $s\in S$, 
$\|f(s)\|\_2= d(x\_0,s)$, and the angles in the image are preserved, .i.e. $\angle\_0 (f(s),f(s')) = \angle\_{x\_0}(s,s')$ for every $s,s'\in S$.


The CAT(0) property implies that the mapping $f$ is 1-Lipschitz on $S$.
$f$ also have the "average distortion property", i.e.,
$$ \sum\_{s,s'\in S} \|f(s)-f(s')\|\_2^2 \ge \frac14 \sum\_{s,s'\in S} d(s,s')^2 .$$


Together this implies the ``Poincar\'e inequality" for any graph with respect to Hilbert space transfers to $X$ with a loss of factor of 4, 
see for example <http://www.cims.nyu.edu/~naor/homepage%20files/spectral-compare.pdf> page 5.


We are left prove the average distortion property of $f$.
This is probably standard but since I'm not aware of it, I'll sketch the argument. 


For $\lambda\in[0,1]$,
denote by $\lambda S \subset X$ the subset where each $s\in S$ is replaced with the (unique) point $\lambda s$ which is of distance
$\lambda d(x\_0,s)$ from $x\_0$ along the geodesic connecting $x\_0$ and $s$.
Observe that $x\_0$ is also the center of mass of $\lambda S$.
Define $f\_\lambda:\lambda S\to H$ by $f\_\lambda(\lambda s)=\lambda f(s)$.
When $\lambda $ is small $f\_\lambda$ preserves distances upto arbitrary precision times $\lambda$, and so, denoting $y\_0$ the barycenter of $f\_\lambda(\lambda S)$


$$ \sum\_{s,s'\in S} \|f(s)-f(s')\|\_2^2 =\lambda^{-2} \sum\_{s,s'\in S} \|f\_\lambda(\lambda s)-f(\lambda s')\|\_2^2 =
\lambda^{-2} 2|S| \sum\_{s\in S} \|f\_\lambda(\lambda s)-y\_0\|\_2^2
\ge \lambda^{-2} |S| \sum\_{s\in S} \|f\_\lambda(\lambda s)\|\_2^2=|S| \sum\_{s\in S} d(x\_0,s)^2
\ge \frac14 \sum\_{s,s'\in S} d(s,s')^2 .$$


Update: 
Upon reading the literature more carefully, similar arguments were done before. For example, Naor & Silberman In [arXiv:1005.4084](http://arxiv.org/abs/1005.4084) have basically the same argument. See
Proposition 4.9 and Corollary 4.10 there.


