We investigate connections between various rigidity and softness properties for discrete quantum groups. After introducing a notion of residual finiteness, we show that it implies the Kirchberg factorization property for the discrete quantum group in question. We also prove the analogue of Kirchberg’s theorem, to the effect that conversely, the factorization property and property (T) jointly imply residual finiteness.We also apply these results to certain classes of discrete quantum groups obtained by means of bicrossed product constructions and study the preservation of the properties (factorization, residual finiteness, property (T)) under extensions of discrete quantum groups.

中文翻译：

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离散量子群的特性（T），特性（F）和剩余有限性

我们研究离散量子组的各种刚度和柔软性之间的联系。在引入剩余有限性的概念之后，我们表明它暗示了所讨论的离散量子群的Kirchberg分解性质。我们还证明了基希贝格定理的类似物，其结果相反，因式分解性质和性质（T）共同暗示了残差有限性。我们还将这些结果应用于通过双叉积构造获得的某些类别的离散量子组，并研究了在离散量子群的扩展下保留特性（分解，剩余有限性，特性（T））。