Systole of congruence coverings of arithmetic hyperbolic manifolds

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

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 Appendix A together with Cayo Dória we prove that the constant $\frac{8}{n(n+1)}$ is sharp.