We introduce the notion of controlled products on metric spaces as a generalization of Gromov products, and construct boundaries by using controlled products, which we call the Gromov boundaries. It is shown that the Gromov boundary with respect to a controlled product on a proper metric space is the ideal boundary of a coarse compactification of the space. It is also shown that there is a bijective correspondence between the set of all coarse equivalence classes of controlled products and the set of all equivalence classes of coarse compactifications.

中文翻译：

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粗压实和受控产品

我们引入度量空间上受控积的概念作为 Gromov 积的推广，并通过使用受控积构造边界，我们称之为 Gromov 边界。结果表明，在适当的度量空间上，关于受控产品的 Gromov 边界是空间粗紧化的理想边界。还表明，受控产品的所有粗等价类集与粗致密化的所有等价类集之间存在双射对应关系。