We show that countable metric spaces always have quantum isometry groups, thus extending the class of metric spaces known to possess such universal quantum-group actions. Motivated by this existence problem we define and study the notion of loose embeddability of a metric space $(X,d_X)$ into another, $(Y,d_Y)$: the existence of an injective continuous map that preserves both equalities and inequalities of distances. We show that $0$-dimensional compact metric spaces are "generically" loosely embeddable into the real line, even though not even all countable metric spaces are. We also prove that compact Riemannian manifolds $(M,d)$ equipped with their geodesic distances do not contain a number of distance patterns that rule out loose embeddability into a finite-dimensional Hilbert space, making $(M,d)$ a good candidate for loose embeddability.

中文翻译：

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量子等距和松散嵌入

我们证明可数度量空间总是具有量子等距群，从而扩展了已知具有这种通用量子群作用的度量空间类。受这个存在问题的启发，我们定义并研究了度量空间 $(X,d_X)$ 到另一个 $(Y,d_Y)$ 的松散嵌入性的概念：存在一个内射连续映射，它同时保留了等式和不等式距离。我们证明了 $0$ 维紧凑度量空间“一般地”可以松散地嵌入到实线中，即使并非所有可数度量空间都是如此。我们还证明了配备了测地距离的紧凑黎曼流形 $(M,d)$ 不包含许多距离模式，这些距离模式排除了有限维希尔伯特空间的松散嵌入性，使得 $(M,