Borel维数增长与超有限性
Borel dimension growth and hyperfiniteness
AI总结：
该研究引入Borel版双参数维数增长函数与球分割算法，证明特定体积增长的Borel图递增并集的超有限性，改进Borel Følner铺砌相关结果，并提出关联Borel顺从性等性质的猜想，推进Weiss问题的研究。
AI中文摘要：
我们证明，所有逐点体积增长至多为$\boldsymbol{\exp(O(r^γ))}$的Borel图的递增并集都是超有限的，其中$γ\approx0.1523$是方程$(1-γ)^3-4γ=0$的唯一实根。这一结果对Weiss关于可数顺从群的Borel作用的超有限性问题的特殊情形给出了肯定回答——该情形针对局部体积增长为$\boldsymbol{\exp(O(r^γ))}$（$γ$取值同上）的可数顺从群。我们还证明，所有次指数体积增长的有界度Borel图都存在Borel Følner铺砌，改进了Downarowicz和Zhang针对由次指数体积增长群的自由Borel作用生成的图的相关结果。我们的主要工具包括：构建Dranishnikov与Sapir提出的度量空间双参数维数增长函数的Borel版本，以及理论计算机科学中的球分割算法。最后我们提出了两个关联Borel顺从性、Borel次指数维数增长与超有限性的猜想，若猜想成立将对Weiss的问题给出肯定回答。
英文摘要：
We show that every increasing union of Borel graphs of pointwise volume growth at most $\exp(O(r^γ))$ is hyperfinite, where $γ\approx 0.1523$ is the unique real root of $(1 - γ)^3 - 4γ= 0$. This implies a positive answer to the special case of Weiss's question on the hyperfiniteness of Borel actions of countable amenable groups, for countable amenable groups locally of volume growth $\exp(O(r^γ))$ for $γ$ as above. We also show that every bounded degree Borel graph of subexponential volume growth has Borel Følner tilings, improving a result of Downarowicz and Zhang for graphs generated by free Borel actions of groups of subexponential volume growth. Our main tools are the development of a Borel analogue of the two-parameter dimension growth function of a metric space as introduced by Dranishnikov and Sapir, and ball carving algorithms from theoretical computer science. We end with a pair of conjectures relating Borel amenability, Borel subexponential dimension growth, and hyperfiniteness, which would imply a positive answer to Weiss's question.
