We study growth of absolute and homological k-dimensional systoles of arithmetic n-manifolds along congruence coverings. Our main interest is in the growth of systoles of manifolds whose real rank r≥2. We observe, in particular, that in some cases for k=r the growth function tends to oscillate between a power of a logarithm and a power function of the degree of the covering. This is a new phenomenon. We also prove the expected polylogarithmic and constant power bounds for small and large k, respectively.

中文翻译：

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同余覆盖中 k 维收缩的增长

我们研究算术 n 流形的绝对和同调 k 维收缩沿同余覆盖的增长。我们主要感兴趣的是实秩 r≥2 的流形收缩的增长。我们特别观察到，在某些情况下，对于 k=r，增长函数倾向于在对数幂和覆盖度幂函数之间振荡。这是一个新现象。我们还分别证明了小 k 和大 k 的预期多对数和恒定功率界限。