# Groups, Geometry, and Dynamics

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**Published online first: 2019-05-07**

**DOI: 10.4171/GGD/515**

Systole of congruence coverings of arithmetic hyperbolic manifolds

Plinio G.P. Murillo^{[1]}(1) Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea

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 DÃ³ria we prove that the constant $\frac{8}{n(n+1)}$ is sharp.

*Keywords: *Systole, arithmetic group, congruence subgroup

Murillo Plinio: Systole of congruence coverings of arithmetic hyperbolic manifolds. *Groups Geom. Dyn.* Electronically published on May 7, 2019. doi: 10.4171/GGD/515 (to appear in print)