Dennis Dreesen
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.
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Using recent developments on locally compact groups, we are able to obtain quantitative results on embeddings into Lebesgue spaces for a large class of HNN extensions.
Stability of amalgamated free products and HNN extensions
本文考察稳定群在有限群上的合合自由积与HNN扩张，分析算子范数、Hilbert-Schmidt及置换三种稳定性，构造了多个稳定或柔性稳定的非顺从群新例。
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.
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.
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…
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.
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…
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.
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.
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…
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…
We show that the type function of a space with finite asymptotic dimension estimates its Hilbert (or any $l^p$) compression. The method allows to obtain the lower bound of the compression of the lamplighter group $Z\wr Z$, which has infinite asymptotic dimension.
Given two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces X,Y,Z and derive a condition, called the (delta-polynomial) path lifting property, such that coarse embeddability of X,Y and Z implies coarse embeddability of X\wr_Z Y. We also give bounds on the compression of X\wr_Z Y in terms of delta and the compressions of X,Y and Z. Next, we investigate the stability of the property of admitting a box space which coarsely embeds i…
We give lower and upper bounds for the separation profile (introduced by Benjamini, Schramm & Timár) for various graphs using the isoperimetric profile, growth and Hilbertian compression. For graphs which have polynomial isoperimetry and growth, we show that the separation profile $\mathrm{Sep}(n)$ is also bounded by powers of $n$. For many amenable groups, we show a lower bound in $n/ \log(n)^a$ and, for any group which has a non-trivial compression exponent in an $L^p$-space, an upper bound in $n/ \log(n)^b$. We show that solvable groups of exponential growth cannot have a separation profile…
From any two median spaces $X,Y$, we construct a new median space $X \circledast Y$, referred to as the diadem product of $X$ and $Y$, and we show that this construction is compatible with wreath products in the following sense: given two finitely generated groups $G,H$ and two (equivariant) coarse embeddings into median spaces $X,Y$, there exist a(n equivariant) coarse embedding $G\wr H \to X \circledast Y$. As an application, we prove that $$α_1(G \wr H) \geq \min(α_1(G),α_1(H))/2 \text{ for all finitely generated groups $G,H$,}$$ where $α_1(\cdot)$ denotes the $\ell^1$-compression. As an ot…
We explicit Ozawa kernels for classical group theoretical constructions, for discrete metric spaces of finite asymptotic dimension, of large enough Hilbert space compression, and for suitable actions of countable groups on metric spaces. We also obtain an alternative proof of stability results concerning Yu's property A.
By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank $\geq 2$ equals 1. This answers a question of J-F. Lafont and S. Prassidis.
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。
For a finitely generated group G and a banach space X let α^*_X(G) (respectively α^#_X(G)) be the supremum over all α\ge 0 such that there exists a Lipschitz mapping (respectively an equivariant mapping) f:G\to X and c>0 such that for all x,y\in G we have \|f(x)-f(y)\|\ge c\cdot d_G(x,y)^α. In particular, the Hilbert compression exponent (respectively the equivariant Hilbert compression exponent) of G is α^*(G)=α^*_{L_2}(G) (respectively α^#(G)= α_{L_2}^#(G)). We show that if X has modulus of smoothness of power type p, then α^#_X(G)\le \frac{1}{pβ^*(G)}. Here β^*(G) is the largest β\ge 0 for …
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