arXiv math/06123782006-12math.GRmath.FA

Compression functions of uniform embeddings of groups into Hilbert and Banach spaces

Goulnara Arzhantseva, Cornelia Druţu, Mark Sapir

备注：: 21 pages; version 3: The final version, accepted by Crelle; version 2: corrected misprints, added references, the group has asdim at most 2, not at most 3 as in the first version (thanks to A. Dranishnikov); version 3: took into account referee remarks, added references. the paper is accepted in Crelle

英文摘要：

We construct finitely generated groups with arbitrary prescribed Hilbert space compression αfrom the interval [0,1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence they are exact. In particular, the first examples of groups that are uniformly embeddable into a Hilbert space (respectively, exact, of finite asymptotic dimension) with Hilbert space compression 0 are given. These groups are also the first examples of groups with uniformly convex Banach space compression 0.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Hilbert space compression and exactness for discrete groups

We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those word-hyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.

2004-03-26math.FAmath.GR

Hilbert space compression for free products and HNN-extensions

Given the Hilbert space compression of two groups, we find bounds on the Hilbert space compression of their free product. We also investigate the Hilbert space compression of an HNN-extension of a group relative to a finite normal subgroup or a finite index subgroup.

2010-07-06math.GR

Compression of uniform embeddings into Hilbert space

If one tries to embed a metric space uniformly in Hilbert space, how close to quasi-isometric could the embedding be? We answer this question for finite dimensional CAT(0) cube complexes and for hyperbolic groups. In particular, we show that the Hilbert space compression of any hyperbolic group is 1.

2005-09-05math.FAmath.GRmath.GT

Group Actions on Banach Spaces

We survey the recent developments concerning fixed point properties for group actions on Banach spaces. In the setting of Hilbert spaces such fixed point properties correspond to Kazhdan's property (T). Here we focus on the general, non-Hilbert case, we discuss the methods, examples and several applications.

2014-01-29math.DSmath.FAmath.GR

Hilbert space compression under direct limits and certain group extensions

We find bounds on the Hilbert space compression of the limit of a directed metric system of groups. We also give estimates on the Hilbert space compression of group extensions of a group $H$ by a a word-hyperbolic group or a group of polynomial growth.

2010-09-14math.GR

Wreath products with the integers, proper actions and Hilbert space compression

We prove that the properties of acting metrically properly on some space with walls or some CAT(0) cube complex are closed by taking the wreath product with \Z. We also give a lower bound for the (equivariant) Hilbert space compression of H\wr\Z in terms of the (equivariant) Hilbert space compression of H.

2006-03-29math.GR

更多相关论文收起更多论文

Banach空间上的均匀仿射作用：陪集的生长性

Uniformly affine actions on Banach spaces: growth of cocycles

本文研究了Banach空间中均匀仿射作用的生长性质，针对具有性质(T)的Sp(n,1)群，建立了具有最优生长的均匀Lipschitz仿射作用的存在性。

2026-01-09math.FAmath.GRmath.RT

Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces

We characterize the asymptotic behaviour of the compression associated to a uniform embedding into some Lp-space for a large class of groups including connected Lie groups with exponential growth and word-hyperbolic finitely generated groups. In particular, the Hilbert compression rate of these groups is equal to 1. This also provides new and optimal estimates for the compression of a uniform embedding of the infinite 3-regular tree into some Lp-space. The main part of the paper is devoted to the explicit construction of affine isometric actions of amenable Lie groups on Lp-spaces whose compre…

2007-08-25math.CAmath.GR

Compression bounds for wreath products

We show that if $G$ and $H$ are finitely generated groups whose Hilbert compression exponent is positive, then so is the Hilbert compression exponent of the wreath $G \wr H$. We also prove an analogous result for coarse embeddings of wreath products. In the special case $G=\Z$, $H=\Z \wr \Z$ our result implies that the Hilbert compression exponent of $\Z\wr (\Z\wr\Z)$ is at least 1/4, answering a question posed as to whether it has positive Hilbert compression exponent.

2009-12-06math.GRmath.MG

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…

2005-04-19math.GRmath.GT

The wreath product of Z with Z has Hilbert compression exponent 2/3

Let G be a finitely generated group, equipped with the word metric d associated with some finite set of generators. The Hilbert compression exponent of G is the supremum over all $α\ge 0$ such that there exists a Lipschitz mapping $f:G\to L_2$ and a constant $c>0$ such that for all $x,y\in G$ we have $\|f(x)-f(y)\|_2\ge cd(x,y)^α.$ In \cite{AGS06} it was shown that the Hilbert compression exponent of the wreath product $\Z\bwr \Z $ is at most $\frac34$, and in \cite{NP07} was proved that this exponent is at least $\frac23$. Here we show that $\frac23$ is the correct value. Our proof is based o…

2007-08-13math.FAmath.GRmath.MG