Title: Products of Infinite Countable Groups Have Fixed Price One Show Chinese title

Title: 无限可数群的乘积具有固定价格一

Abstract: We prove that the product of any two infinite countable groups has fixed price one. This resolves a problem posed by Gaboriau. The proof uses the propagation method to construct a Poisson horoball process as a weak factor of i.i.d. Then, a low-cost graphing of this process is constructed by connecting the points of each horoball first, and then adding a percolation with small intensity. The connectedness of this graphing is ensured by proving that the resulting horoballs have the infinite touching property almost surely, if the metric and the other parameters of the construction are chosen carefully. Show Chinese abstract

Abstract: 我们证明了任何两个无限可数群的乘积具有价格一的固定性。 这解决了Gaboriau提出的问题。 证明使用传播方法，将泊松霍罗球过程构造为独立同分布的弱因子。 然后，通过首先连接每个霍罗球的点，再添加一个低强度的渗透过程，构造了该过程的一个低成本图。 通过证明如果仔细选择构造中的度量和其他参数，得到的霍罗球几乎必然具有无限接触性质，从而确保了该图的连通性。