Abstract
We generalize the construction of Arzhantseva, Guentner and Špakula of a box space of the free group which admits a coarse embedding into Hilbert space. We show that for a finitely generated free group, the box space corresponding to the derived \(m\)-series (for any integer \(m\ge 2\)) coarsely embeds into Hilbert space. This gives new examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have Yu’s property A.
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Acknowledgments
The author thanks Alain Valette for enjoyable discussions, and for his very helpful comments on a draft of this paper. The author wishes to thank Antoine Gournay for simplifying part of the proof of Proposition 11, which was previously combinatorial in nature. The author also thanks the anonymous referee for kind comments and suggestions. During the completion of this paper, the author was supported by the Swiss National Science Foundation grant FN 200020-137696/1.