JournalscmhVol. 81 , No. 4DOI 10.4171/cmh/80

Metrics on diagram groups and uniform embeddings in a Hilbert space

  • Goulnara N. Arzhantseva

    Universität Wien, Austria
  • V. S. Guba

    Vologda State University, Russian Federation
  • Mark V. Sapir

    Vanderbilt University, Nashville, United States
Metrics on diagram groups and uniform embeddings in a Hilbert space cover

Abstract

We give first examples of finitely generated groups having an intermediate, with values in (0,1)(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 FF is equal to 1/21/2, the Hilbert space compression of ZZ\mathbb{Z}\wr\mathbb{Z} is between 1/21/2 and 3/43/4, and the Hilbert space compression of Z(ZZ)\mathbb{Z}\wr(\mathbb{Z}\wr\mathbb{Z}) is between 0 and 1/21/2. In general, we find a relationship between the growth of HH and the Hilbert space compression of ZH\mathbb{Z}\wr H.