Metrics on diagram groups and uniform embeddings in a Hilbert space

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 $Z\wr H$.Comment: 20 pages, Theorem 1.13 and Lemma 3.7. are ne

Limit groups for relatively hyperbolic groups, II: Makanin-Razborov diagrams

Let Gamma be a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. We construct Makanin-Razborov diagrams for Gamma. We also prove that every system of equations over Gamma is equivalent to a finite subsystem, and a number of structural results about Gamma-limit groups.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper54.abs.htm

Presentations of higher dimensional Thompson groups

In a previous paper, we defined a higher dimensional analog of Thompson's group V, and proved that it is simple, infinite, finitely generated, and not isomorphic to any of the known Thompson groups. There are other Thompson groups that are infinite, simple and finitely presented. Here we show that the new group is also finitely presented by calculating an explicit finite presentation.Comment: 35 pages, to appear in J. Algebr

TRAINING OF SPORTING MOVEMENTS OF CHILDREN ON THE BASE OF BIOMECHANICAL ANALYSIS

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