Growth of k-Dimensional Systoles in Congruence Coverings

Abstract

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.