What is the Hilbert space compression exponent of the standard lamplighter group $\mathbb{Z\_{2}} \wr \mathbb{Z}$? For $\mathbb{Z} \wr \mathbb{Z}$ it is known to be $2/3$ by work of Austin, Naor and Peres, but I couldn't find a reference for the $\mathbb{Z\_{2}}$ case.



Hilbert space compression of the lamplighter group
This is [here](http://arxiv.org/abs/math/0603138): (Tessera 2006, published in CMH 2011): note that lamplighters are among those groups in the class $(\mathcal{L})$ introduced page 3.


