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$C_2 \wr \mathbb{Z}$ 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?

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