Markov convexity and nonembeddability of the Heisenberg group
Markov convexity and nonembeddability of the Heisenberg group
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海森堡群的马尔可夫凸性和不可嵌入性
DOI：
10.5802/aif.3045
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Sean Li
中科院分区：
文献类型：
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作者：
Sean Li
我们计算了连续无限维Heisenberg群$\mathbb{H}_\infty$的马尔可夫凸不变量，证明了它是马尔可夫4-凸，而对于任意$p < 4$不可能是马尔可夫$p$ -凸。由于马尔可夫凸是一个biLipschitz不变量，而希尔伯特空间是一个Markov 2-凸，这就给出了经典的Pansu和Semmes定理的一个不同的证明，即Heisenberg群不允许在任何欧几里德空间中嵌入biLipschitz。
马尔可夫凸度下界将从展示Laakso图的显式嵌入$G_n$到$\mathbb{H}_\infty$中，该图形最多具有$C n^{1/4} \sqrt{\log n}$失真。我们用它来证明，如果$X$是一个马尔可夫$p$ -凸度量空间，那么半径为$n$的离散Heisenberg群$\mathbb{H}(\mathbb{Z})$的球至少以若干常数倍的畸变嵌入$X$
$$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$
最后，我们表明马尔可夫4-凸性并没有给出二叉树嵌入$B_m$到$\mathbb{H}_\infty$的最佳失真，表明失真在$\sqrt{\log m}$量级。
We compute the Markov convexity invariant of the continuous infinite dimensional Heisenberg group $\mathbb{H}_\infty$ to show that it is Markov 4-convex and cannot be Markov $p$-convex for any $p < 4$. As Markov convexity is a biLipschitz invariant and Hilbert space is Markov 2-convex, this gives a different proof of the classical theorem of Pansu and Semmes that the Heisenberg group does not admit a biLipschitz embedding into any Euclidean space.
The Markov convexity lower bound will follow from exhibiting an explicit embedding of Laakso graphs $G_n$ into $\mathbb{H}_\infty$ that has distortion at most $C n^{1/4} \sqrt{\log n}$. We use this to show that if $X$ is a Markov $p$-convex metric space, then balls of the discrete Heisenberg group $\mathbb{H}(\mathbb{Z})$ of radius $n$ embed into $X$ with distortion at least some constant multiple of
$$\frac{(\log n)^{\frac{1}{p}-\frac{1}{4}}}{\sqrt{\log \log n}}.$$
Finally, we show that Markov 4-convexity does not give the optimal distortion for embeddings of binary trees $B_m$ into $\mathbb{H}_\infty$ by showing that the distortion is on the order of $\sqrt{\log m}$.