Growth of k-dimensional systoles in congruence coverings

TL;DR

This paper investigates how the minimal sizes of certain geometric cycles in arithmetic manifolds grow along congruence coverings, revealing new oscillatory behaviors especially when the real rank exceeds one.

Contribution

It introduces the study of growth patterns of k-dimensional systoles in higher rank arithmetic manifolds, highlighting novel oscillatory phenomena for the case when k equals the real rank.

Findings

Growth functions oscillate between logarithmic and polynomial in degree for certain cases.

Established polylogarithmic bounds for small k.

Proved constant power bounds for large k.

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 $r > 1$. 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.