# gr.group concept – Hilbert area compression of lamplighter over lamplighter teams Answer

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## gr.group concept – Hilbert area compression of lamplighter over lamplighter teams

$$C_2 wr mathbb{Z}$$ is the lamplighter group however I’m at present wanting on the lamplighter group with this group as a abject area.

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

[EDIT: I just noticed this is mentioned as an open question by Naor & Peres in $$L^p$$ Compression, traveling salesmen and stable walks. I also changed the upcoming paragraphs which gave a false estimate]

Note that the Cayley graph of the group $$F wr (F wr mathbb{Z})$$ are usually not Liouville (the very fact it has hurry exponent 1 is too given by Naor & Peres in Embeddings of Discrete Groups and the Speed of Random
Walks
at (8)). So its compression exponent is at most $$tfrac{1}{2}$$ (and I’m inclined to cerebrate it has precisely this exponent).

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