Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

中文翻译：

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组对的 Furstenberg 边界

Furstenberg 与每个拓扑群相关联$G$普遍的边界$\unicode[STIX]{x2202}(G)$. 如果我们另外考虑一个子群$H<G$, 的相对概念$(G,H)$-boundaries 再次承认一个最大对象$\unicode[STIX]{x2202}(G,H)$. 在离散群的情况下，Bearden 和 Kalantar 引入了一个等价的概念（酉表示的拓扑边界。预印本, 2019,arXiv:1901.10937v1) 作为他们构造的一个非常特殊的例子。然而，类似的普遍性并不总是成立，即使对于离散的群体也是如此。另一方面，它确实在凸紧集方面的仿射重构中成立，它承认一个通用单纯形$\unicode[STIX]{x1D6E5}(G,H)$，即度量的单纯形$\unicode[STIX]{x2202}(G,H)$. 我们确定边界$\unicode[STIX]{x2202}(G,H)$在许多情况下，突出显示可能出现意外的属性。