Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces
Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces
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群上的渐近等周测量和均匀嵌入到 Banach 空间中
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发表时间：
2006
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通讯作者：
R. Tessera
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作者：
R. Tessera
本文对一类群，包括指数增长的连通李群和字双曲生成群，刻画了在Lp-空间（1 < p < ∞）中一致嵌入压缩的可能渐近行为.特别地，这些群的希尔伯特压缩率等于1。这也提供了新的和最佳的估计压缩的一致嵌入的无限3-正则树到某些Lp-空间。本文的主要部分是致力于明确建设的仿射等距作用的顺从连接李群的压缩是渐近最优的Lp-空间。这些结构是基于球内的Lp-等周轮廓的渐近下界。我们计算了所有顺从连通李群和所有1 ≤ p < ∞的这个轮廓的渐近性，提供了这些群的新的几何不变量。我们还涉及到希尔伯特压缩率与其他渐近量，如体积的增长和随机游走的返回概率。
We characterize the possible asymptotic behaviors of the compression associated to a uniform embedding into some Lp-space, with 1 < p < ∞, 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 connected Lie groups on Lp-spaces whose compressions are asymptotically optimal. These constructions are based on an asymptotic lower bound of the Lp-isoperimetric profile inside balls. We compute the asymptotic of this profile for all amenable connected Lie groups and for all 1 ≤ p < ∞, providing new geometric invariants of these groups. We also relate the Hilbert compression rate with other asymptotic quantities such as volume growth and probability of return of random walks.