# Hilbert space compression of lamplighter over lamplighter groups

$$C_2 \wr \mathbb{Z}$$ is the lamplighter group but I'm currently looking at the lamplighter group with this group as a base space.

Question: Consider the group $$C_2 \wr (C_2 \wr \mathbb{Z})$$, what is its compression exponent?

Note that there is a Cayley graph of the group $$F \wr (F \wr \mathbb{Z})$$ (where $$F$$ is some finite group) which embeds isometrically into (some Cayley graph of) $$\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$$. The latter has compression $$\tfrac{4}{7}$$ (a result of Naor and Peres). So the former has at least this exponent (and I'm inclined to think it has exactly this exponent).

On the other hand $$\mathbb{Z} \wr \mathbb{Z}$$ has compression exponent $$\tfrac{2}{3}$$ while $$F \wr \mathbb{Z}$$ has compression exponent 1. So there might be a discrepancy.

Note that the results of Naor & Peres would give an upper bound. But I could not find an estimate on the speed (or drift) of the random walk on $$C_2 \wr (C_2 \wr \mathbb{Z})$$

Sub-Question: Has the random walk on the group $$C_2 \wr (C_2 \wr \mathbb{Z})$$ the same speed/drift (up to logarithmic factors) as the random walk on $$\mathbb{Z} \wr (\mathbb{Z} \wr \mathbb{Z})$$? If not, what is its speed/drift?