Systole of congruence coverings of arithmetic hyperbolic manifolds

TL;DR

This paper establishes a lower bound on the systole of principal congruence coverings of arithmetic hyperbolic manifolds, generalizing previous results and providing applications to systolic genus and homological codes.

Contribution

It proves a universal lower bound on systoles for congruence coverings of arithmetic hyperbolic manifolds of the first type, extending known results to higher dimensions.

Findings

Systole bounds grow logarithmically with volume

The constant in the systole bound is proven to be sharp

Applications include estimates for systolic genus and homological code distances

Abstract

In this paper we prove that, for any arithmetic hyperbolic $n$-manifold $M$ of the first type, the systole of most of the principal congruence coverings $M_{I}$ satisfy $$sys_{1}(M_{I})\geq \frac{8}{n(n+1)}\log(vol(M_{I}))-c,$$ where $c$ is a constant independent of $I$. This generalizes previous work of Buser and Sarnak, and Katz, Schaps and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In an appendix together with Cayo D\'oria we prove that the constant $\frac{8}{n(n+1)}$ is sharp.