# Systole of congruence coverings of arithmetic hyperbolic manifolds

### Plinio G.P. Murillo

Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea

## 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

$\mathrm{sys}(M_{I})\geq \frac{8}{n(n+1)}\mathrm{log}(\mathrm{vol}(M_{I}))-c,$

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

## Cite this article

Plinio G.P. Murillo, Systole of congruence coverings of arithmetic hyperbolic manifolds. Groups Geom. Dyn. 13 (2019), no. 3, pp. 1083–1102

DOI 10.4171/GGD/515