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Metrics on diagram groups and uniform embeddings in a Hilbert space

Goulnara Arzhantseva·Victor Guba·Mark Sapir
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Abstract

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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Goulnara Arzhantseva, Victor Guba, Mark Sapir. 2005-04-19. Metrics on diagram groups and uniform embeddings in a Hilbert space. https://arxiv.org/abs/math/0411605

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