Metrics on diagram groups and uniform embeddings in a Hilbert space

Details Metrics on diagram groups and uniform embeddings in a Hilbert space Metrics on diagram groups and uniform embeddings in a Hilbert space

2004-11-26

arxiv.org/abs/math/0411605v4

Mathematics

Group Theory

20 pages, Theorem 1.13 and Lemma 3.7. are new

Scientific paper

We give first examples of finitely generated groups having an intermediate, with values in (0,1), Hilbert space compression (which is a numerical parameter measuring the distortion required to embed a metric space into Hilbert space). These groups include certain diagram groups. In particular, we show that the Hilbert space compression of Richard Thompson's group $F$ is equal to 1/2, the Hilbert space compression of the restricted wreath product $Z\wr Z$ is between 1/2 and 3/4, and the Hilbert space compression of $Z\wr (Z\wr Z)$ is between 0 and 1/2. In general, we find a relationship between the growth of $H$ and the Hilbert space compression of $Z\wr H$.

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