多项式增长Borel图的大尺度几何
Large-scale geometry of Borel graphs of polynomial growth
AI总结：
从渐近几何与描述集合论角度研究多项式增长Borel图，通过构造粗嵌入证明$\mathbb Z^n$自由Borel作用的普适性，解决了超有限性与toast结构等问题。
摘要：
我们从渐近几何和描述集合论的角度研究多项式增长图。研究的起点是Krauthgamer和Lee的一个定理，该定理表明每个连通的多项式增长图都存在一个到$(\mathbb Z^n, \\|\cdot\\|_\infty)$的单射收缩映射，其中$n\in\mathbb N$。我们从多个方面加强并推广了这一结果。特别地，作为对Papasoglu问题的回答，我们构造了从多项式增长图到$\mathbb Z^n$的粗嵌入。此外，我们仅要求$n$与图的渐近多项式增长率呈线性关系；这在“渐近意义”上证实了Levin和Linial、London以及Rabinovich的猜想。（该猜想的精确形式已被Krauthgamer和Lee反驳。）我们所有的结果都是针对Borel图证明的，这使得我们能够解决描述组合学中的一些问题。粗略地说，我们证明了由$\mathbb Z^n$的自由Borel作用生成的图对于多项式增长Borel图类是普适的。这提供了一种将关于$\mathbb Z^n$-作用的结果推广到所有多项式增长Borel图的一般方法。例如，我们主要结果的一个直接推论是，所有多项式增长Borel图都是超有限的，这回答了该领域一个著名的问题。作为另一个例证，我们证明了多项式增长Borel图支持一种称为toast的特定组合结构。我们论证中的一个重要技术工具是计算机科学中的填充分解概念，它与Gromov的渐近维数概念密切相关。在此过程中，我们找到了Papasoglu定理的另一种概率证明，即渐近多项式增长率$ρ<\infty$的图其渐近维数至多为$ρ$，并在Borel设定下建立了相同的界。
英文摘要：
We study graphs of polynomial growth from the perspective of asymptotic geometry and descriptive set theory. The starting point of our investigation is a theorem of Krauthgamer and Lee who showed that every connected graph of polynomial growth admits an injective contraction mapping to $(\mathbb Z^n, \|\cdot\|_\infty)$ for some $n\in\mathbb N$. We strengthen and generalize this result in a number of ways. In particular, answering a question of Papasoglu, we construct coarse embeddings from graphs of polynomial growth to $\mathbb Z^n$. Moreover, we only require $n$ to be linear in the asymptotic polynomial growth rate of the graph; this confirms a conjecture of Levin and Linial, London, and Rabinovich "in the asymptotic sense." (The exact form of the conjecture was refuted by Krauthgamer and Lee.) All our results are proved for Borel graphs, which allows us to settle a number of problems in descriptive combinatorics. Roughly, we prove that graphs generated by free Borel actions of $\mathbb Z^n$ are universal for the class of Borel graphs of polynomial growth. This provides a general method for extending results about $\mathbb Z^n$-actions to all Borel graphs of polynomial growth. For example, an immediate consequence of our main result is that all Borel graphs of polynomial growth are hyperfinite, which answers a well-known question in the area. As another illustration, we show that Borel graphs of polynomial growth support a certain combinatorial structure called toast. An important technical tool in our arguments is the notion of padded decomposition from computer science, which is closely related to the concept of asymptotic dimension due to Gromov. Along the way we find an alternative, probabilistic proof of a theorem of Papasoglu that graphs of asymptotic polynomial growth rate $ρ<\infty$ have asymptotic dimension at most $ρ$ and establish the same bound in the Borel setting.
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