标题： Aldous--Lyons猜想 I：子群检验 显示英文标题

标题： The Aldous--Lyons Conjecture I: Subgroup Tests

摘要： 本文及其相关论文[BCV24]致力于对Aldous--Lyons猜想[AL07, Ald07]的否定性解决。该猜想起源于概率论，众所周知（参见[Gel18]）等价于自由群的每个不变随机子群都是共紧致的。我们否定了这一最后陈述。在本部分中，我们引入子群测试。这些测试是自由群子群空间到$\{0,1\}$的连续函数上的有限分布。子群测试提供了一个通用框架，可以在其中研究自由群的不变随机子群。经典概念如群的sofic性和群的稳定性自然地出现在这个框架中。通过自由群子群与Schreier图之间的对应关系，可以将子群测试视为某些边标记图的属性测试模型。这种对应关系也提供了与随机网络的联系。子群测试具有值，这些值是在对共紧致不变随机子群进行积分时的渐近最优期望。我们的第一个主要结果是，如果自由群的每个不变随机子群都是共紧致的，那么可以以任何正的加法常数逼近子群测试的值。我们的第二个主要结果是某些非局部博弈与子群测试之间本质上保持值的对应关系。通过将这种对应关系与MIP*=RE [JNV+21]中证明的更强变体的约简相结合，并在相关论文[BCV24]中证明，我们得出结论，逼近子群测试的sofic值与停机问题一样困难，因此是不可判定的。我们两个主要结果的结合证明了自由群存在非共紧致的不变随机子群。 显示英文摘要

摘要： This paper, and its companion [BCV24], are devoted to a negative resolution of the Aldous--Lyons Conjecture [AL07, Ald07]. This conjecture, originated in probability theory, is well known (cf. [Gel18]) to be equivalent to the statement that every invariant random subgroup of the free group is co-sofic. We disprove this last statement. In this part we introduce subgroup tests. These tests are finite distributions over continuous functions from the space of subgroups of the free group to $\{0,1\}$. Subgroup tests provide a general framework in which one can study invariant random subgroups of the free group. Classical notions such as group soficity and group stability arise naturally in this framework. By the correspondence between subgroups of the free group and Schreier graphs, one can view subgroup tests as a property testing model for certain edge-labeled graphs. This correspondence also provides the connection to random networks. Subgroup tests have values, which are their asymptotic optimal expectations when integrated against co-sofic invariant random subgroups. Our first main result is that, if every invariant random subgroup of the free group is co-sofic, then one can approximate the value of a subgroup test up to any positive additive constant. Our second main result is an essentially value preserving correspondence between certain non-local games and subgroup tests. By composing this correspondence with a stronger variant of the reduction in MIP*=RE [JNV+21], proved in the companion paper [BCV24], we deduce that approximating the sofic value of a subgroup test is as hard as the Halting Problem, and in particular, undecidable. The combination of our two main results proves the existence of non co-sofic invariant random subgroups of the free group.