论文标题
可及组之间的定量测量等效性
Quantitative measure equivalence between amenable groups
论文作者
论文摘要
我们启动了有限生成的组之间的度量等效性（和轨道等效）的定量研究，该研究扩展了$ \ mathrm l^p $测量等效性的经典设置。在本文中，我们的主要重点将放在可正常的群体上，我们证明了刚性和灵活性结果。 在刚度方面，我们证明了Isoperimetric配置文件所满足的一般单调性属性，这在$ \ Mathrm l^1 $测量等效性下尤其意味着其不变性。这使得对两个隔离组之间的度量耦合的可集成方式显式“下限”。该结果也对几何组理论有意外的应用：在可及组之间的粗嵌入下，等级谱是单调的。这有各种应用程序，其中有一个不可数的家族，由$ 3 $可分离的群体组成，它们不会彼此融合在一起。 在灵活性方面，我们在具有规定的集成性条件的可符合的基团之间构建明确的轨道等价。我们的主要工具是Følner平铺序列的新概念。我们在许多情况下显示，从等级曲线得出的边界是锐利的，直到对数因素。我们还从这项研究中推断出，在$ \ mathrm l^1 $轨道等效性下，没有保留两个重要的准偶然体不变性：渐近维度和有限的存在性。
We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus will be on amenable groups, for which we prove both rigidity and flexibility results. On the rigidity side, we prove a general monotonicity property satisfied by the isoperimetric profile, which implies in particular its invariance under $\mathrm L^1$ measure equivalence. This yields explicit "lower bounds" on how integrable a measure coupling between two amenable groups can be. This result also has an unexpected application to geometric group theory: the isoperimetric profile turns out to be monotonous under coarse embedding between amenable groups. This has various applications, among which the existence of an uncountable family of $3$-solvable groups which pairwise do not coarsely embed into one another. On the flexibility side, we construct explicit orbit equivalences between amenable groups with prescribed integrability conditions. Our main tool is a new notion of Følner tiling sequences. We show in a number of instances that the bounds derived from the isoperimetric profile are sharp up to a logarithmic factor. We also deduce from this study that two important quasi-isometry invariants are not preserved under $\mathrm L^1$ orbit equivalence: the asymptotic dimension and finite presentability.
