We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in $\unicode[STIX]{x1D6E4}$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.

中文翻译：

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服从一致的循环子群和格嵌入

我们研究可数组类的格嵌入$\unicode[STIX]{x1D6E4}$由最大服从一致递归子群的性质定义${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$是连续的。什么时候${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$来自一个极其近端的动作和包络${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$可以接受$\unicode[STIX]{x1D6E4}$，我们获得了对局部紧群的限制$G$包含一份$\unicode[STIX]{x1D6E4}$作为一个格，特别是关于$G$, 产品分解$G$，以及更普遍的密集映射$G$为局部紧凑群的产物。