Title: Products of Finitely-Generated Groups with a Certain Growth Condition Have Fixed Price One Show Chinese title

Title: 有限生成群的乘积具有某种增长条件时具有固定价格一

Abstract: An open problem posed by Gaboriau is whether the product of any two infinite countable groups has fixed price one. We provide an affirmative answer if the two groups are finitely generated and their growths satisfy a specific condition. The proof uses the propagation method to construct a Poisson horoball process as a weak factor of i.i.d., where each horoball is equipped with a marking that depends only on the first coordinate, in an i.i.d. manner. Then, a low-cost graphing of this process is constructed using the markings of the horoballs and adding a percolation with small intensity. Show Chinese abstract

Abstract: 一个由Gaboriau提出的开放问题是，任何两个无限可数群的乘积是否具有固定价格一。 我们提供了一个肯定的答案，如果这两个群是有限生成的，并且它们的增长满足特定条件。 证明使用了传播方法，构造了一个泊松horoball过程作为i.i.d.的弱因子，其中每个horoball都配备了一个标记，该标记仅依赖于第一个坐标，以i.i.d.的方式。 然后，通过使用horoball的标记并添加一个低强度的渗透过程，构建了这个过程的一个低成本图。