Metrics on diagram groups and uniform embeddings in a Hilbert space

By Goulnara Arzhantseva, Victor S. Guba, Mark V. Sapir. 2004

60 citations recorded by the source corpus.

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 $\mathbb{Z}\wr\mathbb{Z}$ is between $1/2$ and $3/4$, and the Hilbert space compression of $\mathbb{Z}\wr(\mathbb{Z}\wr\mathbb{Z})$ is between 0 and $1/2$. In general, we find a relationship between the growth of $H$ and the Hilbert space compression of $\mathbb{Z}\wr H$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c48910e6431f8e16e9f19d96d9ccd21661c62cef

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Yuval Peres
2. Assaf Naor
3. Yves Cornulier
4. Yves Stalder
5. Alain Valette
6. Romain Tessera

In Geometry and Topology

1. Assaf Naor
2. Yuval Peres
3. Yves Cornulier
4. Alain Valette
5. Yves Stalder
6. Romain Tessera
7. S. Vaes
8. Pablo Shmerkin

In Mathematical Physics

1. Yuval Peres
2. Assaf Naor
3. Yves Cornulier
4. Alain Valette
5. Romain Tessera
6. Yves Stalder
7. S. Vaes
8. Pablo Shmerkin

Citations in the corpus

• Isometric Group Actions on Hilbert Spaces: Growth of Cocycles (2005, background): “or complex hyperbolic space [FH], i.e. SO(n,1), SU(n,1) and their cocompact subgroups (e.g. Fuchsian groups); • a large class of “diagram groups”, including Thompson’s group F of the interval and Z≀Z [AGS]. Also note that by [AGS], 1/2 ≤ B(Z≀Z) ≤ 3/4, and B(Z≀Γ) ≤ 1/2 for every finitely generated gro”
• PROPER ACTIONS OF WREATH PRODUCTS AND GENERALIZATIONS (2009, methodology): “Better constraints on embeddings of Z o Z into a Hilbert space were brought out by [ AGS , ANP].”
• Speed of random walks, isoperimetry and compression of finitely generated groups (2015, unknown): “understand the compression of ∆, which is a diagonal product of groups Γs≀ Z, a first step is to evaluate compression exponent of general wreath product H≀Z. This has been an object of intensive study [AGS06], [NP08], [ANP09], [Tes11], [CSV12]. We develop a novel approach and derive the following exp”
• Lp compression, traveling salesmen, and stable walks (2009, background): “Let G,H be nontrivial finitely generated groups, and assume that H has polynomial growth.”